What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
[+673] [164] Liz
[2013-03-07 06:50:31]
[ soft-question education big-list ]

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," [1] and found that I, too, lament the uninspiring quality of my elementary math education.

I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.

However, I myself am mathematically unsophisticated.

What was the first bit of mathematics that made you realize that math is beautiful?

For the purposes of this children's book, accessible answers would be appreciated.

(32) For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. - Manjil P. Saikia
(20) Wow. Just last night I had a fierce argument with one of the bartenders of my usual watering hole who is a mechanical engineering student. He insisted that he has a better idea than me of what is mathematics. I am so going to print him a copy of Lockhart's text. Thank you for that link! - Asaf Karagila
You might want to take a peek inside "proofs from THE BOOK" by Aigner and Ziegler. A lot of the contents are way to advanced, but not all of it. However, it is all beautiful. (The story behind the book is that there was once a mathematician who wandered the earth looking for proofs. When he found a paricularly beautiful proof he would claim it was "from the Book", which was a book God kept of the perfect, most beautiful proofs. The mathematicians name was Paul Erdős, and when he died some people decided to complie such a Book...) - user1729
(4) Have you ever read the book "The Number Devil"? I think it's a good book to read for that sort of thing. It uses nonstandard terminology the whole way through, focussing on the concepts instead. - Joe Z.
(6) To expand a little on @JoeZeng’s comment, that’s The Number Devil: A Mathematical Adventure, by Hans Magnus Enzensberger, and it’s a true gem. - Brian M. Scott
(32) I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. - Brian M. Scott
(7) I just want to say that this was a refreshing question, especially compared to many of the homework questions that keep popping up on here. Thank you so much for asking this! :) - apnorton
This is the book that really did it for me: amazon.com/Aha-Insight-Martin-Gardner/dp/071671017X - I already liked math, but this book made me aware I could have fun with it too - SeanC
(3) Has anyone noticed that this is question 32 33 34 ? - Jim Green
(19) Although I don't know if it's what you are looking for, try looking up "vihart" on youtube--Even if it's not helpful, I guarantee you will appreciate it. - Bill K
(2) @BrianM.Scott +1 In my opinion, it's sort of like asking when did you first realize that flowers were beautiful? - skullpatrol
(5) In my childhood, there were lots of attempts to get me to like math by discussing patterns of numbers, etc. Those never really did it for me because I always felt there was some trickery I didn't fully understand. What really first inspired me was the first time I picked up a book that first stated the field axioms precisely. It then went on to prove all of the relationships of basic arithmetic that I had kept as disconnected "it-just-works-that-way" facts in my head for years. It was that beautiful reasoning that got me hooked. - yamad
(4) I'm a programmer. For me, after learning low-level UI, and eventually Flash programming. You started out with a canvas with a bunch of inanimate objects. However, some simple Algebra in the Actionscript started to make things move. The more math I put in, the more the program felt alive. It was a great feeling. - Joe Plante
(1) I can't remember a time when I thought mathematics was beautiful. I have been forced to do it since I was like 5 and then later by teachers in school and college. I hate it. - l--''''''---------''''''''''''
(1) @АртёмЦарионов, hopefully Liz will make this book so that doesn't happen to more people! And hopefully hanging around this site will change your mind :) - Will
@yamad what book was that? Or at least an example of one such book? - Osteoboon
When I was young I thought it was pretty clever that I could evaluate 111111111^2 after seeing the pattern in long-multiplication of 11*11 and 111*111 - wim
I remember in 2nd/3rd grade when I first learned to count by 10's. Groups of 10 items. I was amazed, that I could count that high that fast, by doing "10, 20, 30, 40, 50" - chovy
(2) Even though I have plenty of reputation on stackoverflow, I am denied the ability to answer this question because I am new to the math forum. I think this is a poor decision made by the stack exchange team. Reputation should transfer when it comes to making decisions about preventing poor answer quality. Anyway, couple ideas: unit circle, binary math (how your iphone works at the lowest level), simplest algebra (solving for x). All of these should be consumable by 2nd/3rd grade, some sooner. - Robert Christian
(2) I'm in the same boat as Rob. My own personal magic moment was discovering the glider gun in Conway's game of life. The fact that you could build a mathematical object which spawned "life", based on a few simple rules, was awesome. Up until that point I had used cellular automata just to make pretty patterns which soon fizzled out. On seeing the glider gun in action, I got a glimpse of a much bigger picture. en.wikipedia.org/wiki/Gun_(cellular_automaton) - Puzbie
(2) And this is even more awesome: en.wikipedia.org/wiki/… - Puzbie
(22) I think it's a shame that this question was voted closed... - Will
(2) @KevinF In my case, I'd say the two first "real math" books happened to be Linear Algebra by Shilov and Calculus by Spivak. Shilov, it turns out, was wildly inappropriate for such an introduction, but it impressed me nonetheless. Spivak (such wonderful writing) and Calculus I by Apostol (also wonderfully clear, but not so playful as Spivak) have great introductory chapters that build the basic properties of real numbers, but assume the reader (like me) isn't so familiar with proof-based math. - yamad
(5) Meta: There is a meta thread discussing this question. - user1729
(8) The moment I realized there's no magic in mathematics, and I could actually prove (and come up with) all this stuff myself. Now that was a magic moment! - yatima2975
@rob: Certainly you can answer some other question first to get one measly upvote in order to post here. I fully support the idea of no privilege transfer. Protection of many questions does not come from "lack of software experience", but rather "lack of familiarity with the community norms". One upvote somewhere else is not a huge chore. - Asaf Karagila
This has nothing to do with mathematics. I was truly delighted when I figured that every thing you see or hear happened in the past. Later I came to know everything has a finite speed. - dineshdileep
Regardless of the problem itself, for many it is the feeling of "I didn't really expect that to work" and then it does and then it becomes part of your reality. That might be hard to convey with a book, since the problems are all going to be contrived and the reader knows it. - DanielV
Learning binary, personally. - Akiva Weinberger
I support you in this, i didn't realize the beauty until i was 15, but if i had when i was younger i would have had a much better childhood - Eric Lawson
The non-existence theorems or facts that certain equations cannot be solved, like FLT... - DVD
This is my new favorite question, it is just beautiful! - Simply Beautiful Art
I wish I asked my elementary teachers all the math questions in my head, I would've learned so much! - Simply Beautiful Art
For me, it was when I studied denumerable and uncountable sets, going to infinity and beyond. Then Cantor's set was even a further intellectual orgasm. - Benedict Voltaire
I must say Partitions.... - tatan
[+320] [2013-03-07 07:02:51] Will

This wasn't the first, but it's definitely awesome:

This is a proof of the Pythagorean theorem [1], and it uses no words!

[1] https://en.wikipedia.org/wiki/Pythagorean_theorem

I'm not a math expert at all (quite the contrary), but to me it seems completely logical. Why should this be awesome? - 11684
(17) @11684: This picture is a proof of the Pythagorean Theorem, which says that $a^2 + b^2 = c^2$ in right triangles. - Jesse Madnick
(5) Wow... You're right this is amazing. Perhaps it's an idea to add that to your answer? (Accessibility was a requirement.) - 11684
This video also nicely demonstrates the Pythagorean theorem. - Kibbee
this video demonstrates the (a+b)^2. - knowledge_seeker
(3) This picture would be even greater with some shading - Moshe
(7) @double_squeeze, the idea is that the area of both big squares is the same, and the area of the two sets of four triangles is the same. In the first picture, we see that the remainder of the area after taking away the four triangles is $a^2 + b^2$. In the second picture, the very same remainder is just $c^2$. So, we have $a^2 + b^2 = c^2$. It should be clear that one can draw a similar picture for any right triangle. QED. - Will
@double_squeeze, Also, who claimed the videos above are proofs of the Pythagorean Theorem? The commenters refer to them as demonstrations, and state what they demonstrate. Seems fine to me. - Will
(20) This is not a proof, this is THE proof of the Pythagorean Theorem. If you traveled back in time, and told Pythagoras himself that $a^2 + b^2 = c^2$, he either wouldn't understand you, or (after you defined what the square root means) wouldn't believe you. - vsz
(1) Allowing a little algbera, one wouldn't need the first picture, only the second: $(a+b)^2 = a^2 + b^2 + 2ab$ gives the area of the big square in terms of the two lengths. The triangles are right triangles and all four are congruent (SAS). The area for triangles is $\frac{1}{2}ab$ and there are 4 of them, for a total area of $2ab$. Since we are removing them from the big square to get the inner square, we are left with the identity: $(a+b)^2 - 4\frac{1}{2}ab = a^2 + b^2 = c^2$. - Assad Ebrahim
(1) This one immediately sprang to my mind when I read the question. - flup
(7) @AKE, the point of this picture is that it doesn't require 'a little algebra'. - jwg
@jwg, exactly :) - Will
(2) @jwg: I get that. My comment wasn't suggesting that the first picture is unnecessary. It was rather to observe that depending on what 'first principles' one wishes to allow, one can build up a proof with different tools. So a pure geometric proof -- yes, both pictures, brilliant. Allow a little algebra with your geometry, and only one picture is required. With algebra only --- well, then you don't have a fact at all! -- Pythagorean theorem is the result of an axiom of a certain kind of metric space.... - Assad Ebrahim
en.wikipedia.org/wiki/File:Pythag_anim.gif is an animation showing the same thing. - Lee Sleek
Wow it's great. Why didn't anybody show this to me before? - lcv
(1) We should add an image of this to every interstellar spacecraft we ever launch. - gerrit
it's really awesome +1 - jeanne clement
I believe that the provider of that proof was an Indian mathematician of long ago named Bhaskara. I remember seeing the diagram on the right, which really is the heart of the proof -- even without the left-hand diagram. In the center of the RH diagram appeared the word "Behold!" in one of the languages of India. - Senex Ægypti Parvi
1
[+288] [2013-03-07 07:42:39] Devdatta Tengshe

For me it was the Times Table of $9$.

We are usually forced to memorize the multiplication tables in school. I remember looking at the table for $9$, and seeing that the digit in ten's place increased by one, while the digit in the one's place decreased by one.

$$\begin{array}{r|r} \times & 9 \\ \hline 1 & 9 \\ 2 & 18 \\ 3 & 27 \\ 4 & 36 \\ 5 & 45 \\ 6 & 54 \\ 7 & 63 \\ 8 & 72 \\ 9 & 81 \\ 10 & 90 \end{array}$$

After this, I realized that I could always add $10$ and subtract $1$ to get the next result. For a $7$ year old, this was the greatest discovery ever made.

And that your hands could give you the answer immediately: $7 \times 9$ = hold down your $7$th finger, leaves $6$ fingers on left of held down finger, and $3$ on right: $63$.. works all the way up to $9\times10$, beautiful.

(62) Fo me, it was the "flipping" of the digits along 9's times table. (e.g. $9 \times 9 = 81$ and $9 \times 2 = 18$ or $9 \times 3 = 27$ and $9 \times 8 = 72$. - Evan Teitelman
(5) Reminds me of when I realized that "skip counting" (like $3, 6, 9, 12, \ldots$) is the distributive law. - Jesse Madnick
(6) Add 10 and subtract 1 to get the result. I like that :) - archy
(23) Also, both digits of the result, sum 9. (18, 27, 36...) - Francisco R
I never learnt the times table of 9. I was so proud I found this trick that I preferred to do the mental arithmetic. +1 - Laurent Couvidou
This same rule applies to any base with (base-1) as the cyclic number. - zzzzBov
(1) Don't we all remember figuring out the shortcut for figuring out 99x99? 99x100-99=9801? - Stu
(11) I love this example, because it shows how the abstract side — finding patterns, understanding the reasons for them — arises directly out of the concrete side. Too often (as in the Lockhart) the two sides are presented as in opposition to each other — really, they’re intimately connected the whole way. - Peter LeFanu Lumsdaine
(1) There was a song about this on Square One TV years ago: "Nine Nine Nine" with the refrain "Nine nine nine, that crazy number nine. Times any number you can find, it all comes back to nine!" - Ari Brodsky
(1) I was going to upvote, but there were 72 upvotes, and I didn't want to change that. - Scott Rippey
@Scott There are 80 now, so your golden opportunity has arrived! - MJD
The way I still do x9 is like 7x9 is 70-7, and 8x9 is 80-8. - AaronLS
I'm not really an aficiando, however i^e pi = -1 always struck a fantastic cord with me. Perhaps it was a different equation. But for some reason I found it particularly amazing that something that did not exist to the constant of the circumfrence of a cirlce equating to minus 1 felt whimsical. - Brandon Bertelsen
(1) Adding to a possible chapter on the 9 times table, there's also the fact that the digital sum of a multiple of 9 is always a smaller multiple of 9. A corollary is that if you repeatedly take the digital sum of a multiple of 9, you will always eventually reach 9. In my experience, laymen really appreciate this theorem. - Jack M
@Devdatta I remember exactly where I was on the walk home from elementary school when I realized this fact! - Brian Gordon
This was my first discover as well! - Abramo
(1) Blew my mind when I realised the same laws apply for the last number in any numerical system! In octal for example, the number 7 follows the same patterns: 7*3 = 25, 7*4 = 34, 7*5 = 43, 7*6 = 52, 7*7 = 61. Beautiful. - Connell
@Ari Brodsky: Thx, that's great - Nikolaj-K
This too takes me way back. One of the first things I remember. - Al Jebr
Diagonally down the times table is the squares, and the difference in the difference in squares is two, which makes a whole lot of sense because the second derivative of the square function is 2 and MVT! - bjb568
(2) My teacher wrote that table on a blackboard and told us: look, when you add those digits you always get 9. Everybody was like: meh, do we need to learn that too? And I was like, wow that's awesome because I just need to memorize only the first digit and I can calculate the second digit by subtracting the first one from 9. Then I noticed I don't have to memorize the first digit either, because it's always one less than the one that's multiplying 9. That was very a important point in time, when I realized I don't have to memorize stuff, instead just look for patterns. Not just in math. - Cano64
There are 243 (9*9*3) upvotes: please, no one break this! :) - Numeri
2
[+235] [2013-03-07 10:12:17] Matt Lacey

Whether this is 'simple' enough is debatable... the method to generate the Mandelbrot set is likely to be far too complicated for the book in question, but the mathematical expression that's at its heart couldn't be much simpler.

$z_{n+1} = {z_n}^2 + c$

After implementing the Mandelbrot set I learned about the Buddhabrot, which is basically a way of rendering the stages of the Mandelbrot algorithm, and after some considerable processing time I had a render:

I then tweaked my input parameters to 'zoom in' on a particular area, and when I saw the result my jaw hit the floor. This is when I saw the true beauty in mathematics beyond 'nice' results. Again, it's probably too advanced for your book because of the steps involved in creating the visual, but maybe it'd make for a nice final hurrah to inspire further exploration? It still boggles my mind to see such amazing results from something so simple.

(2) What software was used to render these fractals? - Asaf Karagila
(23) It was just a C program I wrote myself after reading about the Mandelbrot sait in one of A. K. Dewney's books. When I was a kid I was fascinated with the Mandelbrot set but never understood it, after reading a brief description of the algorithm in the book as an experienced developer it surprised me how simple it was. Took about half an hour to do the Mandelbrot set code, but getting nice renders of the Buddhabrot was harder - not least because it was taking several hours to perform a high-resolution render. - Matt Lacey
I see. Very nice! - Asaf Karagila
(1) @AsafKaragila, there is some more information in a blog post I wrote about it here: laceysnr.com/2010/10/attack-of-buddahbrot.html - Matt Lacey
(99) Would you make the code open source? - Wouter Zeldenthuis
(11) Hoooooooooly. That Buddhabrot looks like a nebula. - Joe Z.
(7) I cant imagine a child reading about a recursive formula with complex numbers and be amazed. This is surely beautiful and in a sense is simple, but not for a child. - Integral
(8) @Integral You would do it the other way around. Show the pretty pictures, maybe show an animation, or allow them to play with some fractal software that allows you to zoom in. Then mention that it’s just a short formula behind it. I’d consider most answers here too complicated for children but with fractals they have something they can immediately see and be amazed of. - poke
(2) @poke: I don't think that's right. If you ask the average fractal "artist" what is a fractal they will simply tell you something about modifying the parameters in a pregiven formula and playing with the palette. I had these conversations many a time before. Showing pretty and breathtaking photos is great, and it is beautiful when you have the means to understand them already, but to have these means you need at least some prior experience (in mathematics, or programming for "hands on" generating experience like LaceySnr had). - Asaf Karagila
(1) you mean A.K. Dewdney? (there's a d.) He was one of my profs at the University of Western Ontario, in London Ontario Canada. Here's his webpage: csd.uwo.ca/faculty/akd/akd.html - Warren P
(3) There are simpler fractals that are simple enough for a child to understand conceptually that still produce pretty pictures. ex en.wikipedia.org/wiki/Barnsley_fern - Dan Neely
Wow very nice renderings! After reading this post I found myself looking at lots of pretty pictures on google. Made my day! - Dave
@WarrenP Yes! Typo on my part :( I've got four of his books and love them. - Matt Lacey
(3) @WouterZeldenthuis I'll dig out the source if I can find it and see if it's still working. It's not pretty, I never optimised it for performance or anything just got it working as quickly as I could :) - Matt Lacey
@LaceySnr I'd definitely like to get a copy of the source and hack away at it! It'd be an awesome way to finally get around to learning C :) Do you have a GitHub.com account? That's probably the easiest way of publishing it. - Will
(3) Yeah I do have an account on git hub though nothing of real use on there right now. If I can a) find the code, and b) get it working I'll be sure to put it up. I might fix it to render on screen first, it actually just dumps RAW RGB data to a file which I was importing with Gimp :) - Matt Lacey
@WouterZeldenthuis look into fractint. - ErikE
@LaceySnr: While you're at it, maybe add CUDA and OCL support for faster calculations! :-) - Asaf Karagila
(2) @integral, don't underestimate a child's will to learn. I was exposed to fractals when I was 12-13 and was compelled to understand how they worked. That meant learning complex numbers and differential equations many years before my peers. - Joni
(3) Your images look like pictures seen in astronomy books...I wonder if there is a mathematical formula that would plot the universe... - zerosofthezeta
(2) @Integral You would be surprised at what children are capable of. There is a reason so many child prodigies are gifted at math. It's because math is one of the few disciplines out there that has no context, and requires no life experience to master. - dinstruction
@LaceySnr Any luck on that source code :) - Anubian Noob
(2) Good call! Forgot to find it... on a new machine now... but I'll try and dig it out and get it onto github. - Matt Lacey
The first image looks like Cartman from South Park... - Jp McCarthy
@LaceySnr: so what's about the source code? :) - dbanet
(1) @dbanet I'll get this up in the next day or two... after your prompt I dug it out and got it building on Windows, just need to do a bit more clean up (it's very hacky, I wrote it just for kicks while I was off sick). - Matt Lacey
(1) @dbanet Apologies for the delay! First daughter was born and that got me a tad sidetracked ;) The basic code is now up on GitHub. It's terrible, I wrote it originally while sick and just for kicks so wasn't bothered about magic numbers and the like. Hopefully you can get it working though! github.com/mattlacey/BuddhaBrot.git - Matt Lacey
(4) @WouterZeldenthuis A couple of years later, and the source is available (and still bad) github.com/mattlacey/BuddhaBrot.git - Matt Lacey
@LaceySnr Fantastic! That's really great! I will take a look ASAP. Then we could communicate through Github issues (I'm dbanet there too), pull requests and so on. Thank you very-very much! - dbanet
@dbanet Oh there'll be more than a few issues :) It was never intended to be public so it's a complete hack job, could do with some optisation and command line parameter support etc. :) - Matt Lacey
Beautiful! Sadly, I don't think a children's book could use this. :( - Simply Beautiful Art
3
[+192] [2013-03-07 12:48:36] zkanoca

I used to love naughty $37$.

$37 \times 3 = 111;$

$37 \times 6 = 222;$

$37 \times 9 = 333;$

$37 \times 12 = 444;$

$37 \times 15 = 555;$

$37 \times 18 = 666;$

$37 \times 21 = 777;$

$37 \times 24 = 888;$

$37 \times 27 = 999;$

(13) That is naughty! - Loki Clock
(86) But, there isn't a naught in any of that... - Gary S. Weaver
If you want a slightly more contrived one, try 271 × 41 = 11111. - Joe Z.
Or 239 × 4649 = 1111111. - Joe Z.
(9) Wow - 37 is my new favourite number! - Steve Chambers
(54) If you like 37, you'll love double-37: 37037*3=111111, 37037*6=222222, 37037*9=33333... - MiniQuark
(4) And triple-37 (37037037), quadrule-37 (37037037037)... - MiniQuark
(27) And in general any repetition of 037. For the infinite case, take 1/27. - Joe Z.
(1) Is there some sort of repository of these mental math tricks? I'd love to see more of this stuff - AlanH
Maybe I was wrong about naming 37 as naughty. English is not my primary language. So if you have a suggestion for this heading, please tell me. - zkanoca
(2) Here's one more. $$15873 \times 7 = 111111, 15873 \times 14 = 222222 \cdots$$ - Satish Thulva
(8) It's fun to look at these as curious surprises, but the secret is simply that 37 * 3 = 111. Then most of these products seem more obvious, such as 111 * 2 = 222, 111 * 3 = 333, etc. Is that spoiling the fun? I personally enjoy both the mystery and the resolution together. The fact that there is a nice resolution makes it cooler. - Tyler
(3) And it's not too hard to prove that every integer that is not a multiple of 2 or 5 can be multiplied by some integer to make a string of consecutive ones. - user21820
4
[+167] [2013-03-07 07:37:42] Jair Taylor

I found it completely amazing that the angles in a triangle always added up to 180 degrees. No matter how you drew a triangle, you could measure the angles with a protractor and they always add up to about 180 degrees, like magic. Even more amazing when I realized it wasn't some rule of thumb or approximation, but true in some deeper sense for the ideal, platonic triangle.

(435) When I came home and told my father, he drew a triangle on the skin of an orange. All angles were 90°. I was deeply disturbed. - gerrit
(19) The sum of the angles of a triangle is 180 degrees only in an Euclidean geometry (where the shortest distance between any two points is a straight line). On an orange, the shortest distance between two points is a curve. - utnapistim
(195) your father is awesome =) - Glougloubarbaki
(2) @utnapistim technically the shortest distance between two points on an orange is still a straight line... through the orange. Incidentally, the 180° triangle thing was first point at which I realised mathematics was beautiful. - LordScree
(1) Could you find any example of the triangle on the curved surface, i can't imagine it - Buksy
(25) @Buksy: Imagine the earth (as a sphere). Draw a line from the north pole to the equator, then a quarter of the way around the equator, then back north to the pole. All angles are 90°. - Christian Mann
(17) This is indeed a very pretty result, but lost on children. The reason it's lost is because they've been told this rule in school already, it's been "spoiled". I was taught this in middle school, and my attitude was "what's this? just another rule for doing calculations? okay.". It was only years later I understood how pretty it is. - Jack M
@gerrit: a million times yes - Greg
(1) @gerrit: Troll dad it seems? I remember my teacher in the last year of elementary school telling us it's impossible to draw a triangle whose angles don't add up to 180°. I wish I had known about non-Euclidean geometries back then and taken a soccerball to illustrate. Then again, I'm sure they would not have accepted my example since they are not really triangles since they have curved sides. Sigh... - Raskolnikov
(1) When I was mathematically naive, around age seven or so, I felt that as a triangle was increased in size, its angles should decrease. - Lubin
@Lubin Considering many people, if asked to draw a number line, will start out with large spaces and as they get to the edge of the page make them smaller until it forms a logarithmic curve, could you have perhaps been modelling the triangles getting bigger in a hyperbolic plane? - Loki Clock
(2) This works best for children if you ask them to make a triangle out of paper, then cut it up, and place all the angles to make a flat line - Andrey
@JackM : I completely agree. It was for that reason, that I had just learnt off in school that the angles sum to 180, that when I found out that a sphere can have angles that sum to whatever you want (between 0 and $2\pi$ I suppose) that I thought the curved geometry beautiful. As soon as I started reading the question I thought of a trinagle on a cuvred surface as being beautiful. It's interesting to see that others did too. - Flint72
Isn't this result based on the postulate of parallels which, along with it's negations, has been shown to be consistent in geometry? - inkievoyd
5
[+147] [2013-03-07 07:10:15] Kaster

The first "math thing" that just blew my mind was the identity $$e^{i\pi} = -1$$ Namely the fact that the two independently discovered transcendent numbers and imaginary one so simply and elegantly bound.

In the marginally rearranged form $$e^{iπ}+1=0$$ it uses absolutely nothing but nine essential concepts in mathematics:

• five of the most essential numbers, $\{0,1,i,e,π\}$,
• three essential operations, { addition, multiplication and exponentiation }, and
• the essential relation of equality.

(79) Didn't the title say childrens book? If you knew the concept behind $e^{i\pi} = -1$ as a child then you must be a brillant prodigy. - Q.matin
(9) I was about 15 :), but my bad overlooked that part. - Kaster
(8) You are still a prodigy then because I didn't learn that till now and I am 20. :) - Q.matin
Some schools simply teach it early, although it does take a great curiosity and a great mind to truly understand it so young. I was taught that identity at 17 and still dont quite understand why it turned out so elegantly (well, I remember it's a series...) - markovchain
(68) I've always thought that it's kinda cheating to teach this to someone who doesn't fully understand complex numbers. I remember I first heard about this when I was about 16, and I thought it was some miraculous numerical coincidence, when in reality this exponentiation doesn't work like the one you know, so this identity does not mean what you think it means at that age. - Javier
(1) Though we don't make this explicit, THIS is OUR AXIOM! we set eiπ=−1 - Dheeraj Bhaskar
@DheeB not true, we never take it as a starting point. We take Euler's formula for the complex exponent and identity above is just a consequence. - Kaster
(3) @Kaster e^{ix} = \cos x + i\sin x \ . This is just a notation. This is not a mathematical result. When you choose this notation, isn't eiπ=−1 also a notational consequence we chose? kindly correct me if i'm wrong. - Dheeraj Bhaskar
@Kaster what my intention was that that these constants aren't in the same equation by a mathematical fact,it's by notation. - Dheeraj Bhaskar
(11) @DheeB oh really? Complex exponential:Proofs - Kaster
(7) @Kaster Woah! that opened my Eyes! This is MY moment. Math IS beautiful. Thanks for that Kaster. - Dheeraj Bhaskar
@Kaster I kinda feel silly now. - Dheeraj Bhaskar
(1) I usually don't think anything that xkcd has mentioned is suitable for children. - Alvin Wong
(6) As a friend at math camp once said to me, "$\pi$ is essentially the number defined to make this statement true." When I thought about that, it all seemed less magical and more circular. However when you first learn about $e$, $i$, and $\pi$ it isn't clear how they're related yet. So I guess it's still cool? - Dylan Yott
I found this result pretty cool when I was about 17, I read a book called Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills which was cool, it had a cool poem but I can't remember it at the moment :( - hmmmm
(2) All the comments talking about how this isn't really amazing seem to be missing part of the point: why should adding $i^2=-1$ give you a number system that can coherently and intimately represent 2D geometry? Exponentials taking the role of rotation is really remarkable. You can't just take all the complex machinery behind the formula as obvious, and then say that the formula looks simple. It may not be witchcraft, but it's still fantastic. - Robert Mastragostino
(14) @Dylan: many arguments involving $\pi$ are circular :) [couldn't resist bad pun, sorry..] - Marek
(2) @Dylan $\pi$ was around a lot longer than $\sin(x)$. Define $\pi$ as the ratio of the circumference of the circle to its diameter, $e$ as, e.g., the positive real number for which $de^x/dx = e^x$, and $i$ by $i^2=-1$. Define $(\sin(\theta),\cos(\theta))$ as the $x$ and $y$ coordinates of the point at angle $\theta$ with respect to the positive $x$-axis on the unit circle, with the origin at $(0,0)$. The shocking bit is when you realize the Taylor expansion of $e^x$ is equal to the Taylor expansion of $\cos(\theta) + i\sin(\theta)$. $e^x = \cos(\theta) + i\sin(\theta)$ isn't just convention! - Douglas B. Staple
I never learnt complex analysis, but I think that what happens is that one tries to extend the function $x\rightarrow e^x,x\in\mathbb{R}$ to an analytic function defined on all of the complex plane. I think the new function (call it $f$) satisfies $f(i\pi)=-1$ - Amr
I prefer $e^{iπ}+1=0$, using five essential numbers $\{0,1,e,i,π\}$, three essential operations: { addition, multiplication and exponentiation }, the essential relation of equality and nothing else. - PJTraill
6
[+137] [2013-03-07 07:08:41] MJD

I remember being very pleased at an early age, perhaps five or six, by the following bits of calculator tinkering, among others:

• 12345679 × $n$ × 9 = nnnnnnnnn.
• The cyclic behavior of the decimal expansions of $\frac n7$. For example, $4\times 0.142857\ldots = 0.571428\ldots$.
• The reciprocity of digit patterns in numbers and their reciprocals. For example, $\frac12 = 0.5$ and $\frac15 = 0.2$; $\frac14 = 0.25$ and $\frac 1{2.5} = 0.4$. This is the earliest pattern I can remember observing completely on my own. Similarly, I enjoyed that the decimal expansions of $\frac1{2^n}$ (0.5, 0.25, 0.125…) look like powers of 5.
• The attraction of the map $x\mapsto \sqrt x$ to 1, regardless of the (positive) starting point. I liked that numbers greater than 1 were attracted downwards, and numbers less than 1 were attracted upwards. Later on I noticed, from looking at the calculator, that $\sqrt{1+x} \approx 1+\frac x2$ when $x$ is small; for example $\sqrt{1.0005} \approx 1.0002499$, and similarly when $x$ is negative. When this useful fact recurred later in calculus and real analysis classes, I was already familiar with it.

When I got a little older, I loved that I could find an $n$th-degree polynomial to pass through $n+1$ arbitrarily chosen points, and that if I made up the points knowing the polynomial ahead of time, the method would magically produce the polynomial I had used in the first place. I spent hours doing this.

I also spent hours graphing functions, and observing the way the shapes changed as I varied the parameters. I accumulated a looseleaf binder full of these graphs, which I still have.

As a teenager, I was thrilled to observe that although the number "2 in a pentagon" in the Steinhaus–Moser notation [1] is far too enormous to calculate, it is a trivial matter to observe that its decimal expansion ends with a 6.

I realize that your book wants to discredit the notion that math is merely a series of calculations, but I have always been fascinated by calculation, and I sometimes think, as the authors of Concrete Mathematics say in the introduction, that we do not always give enough attention to matters of technique. Calculation is interesting, for theoretical and practical reasons, and a lot of very deep mathematics arises from the desire to calculate.

[1] http://en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation

(18) Related: $\displaystyle \frac{1}{7}=0.(142857)(142857)(142857)(142857)(142857)\ldots$ and $$7\times 142857=999999$$ $\displaystyle \frac{1}{13}=0.(076923)(076923)(076923)(076923)(076923)\ldots$ and $$13\times 076923=999999$$ $$\vdots$$ - Git Gud
And if you keep going, you can also find things like 5291 * 189. - Joe Z.
(1) You sound like one of those savant kids.. - Dheeraj Bhaskar
(11) @DheeB Perhaps it sounds like that, but I was not. - MJD
I didn't see the reciprocity of the digit functions thing, but I found the rest of those with a calculator and some boredom as well. I once ended up trying to play a game of continuously multiplying 1 by 2 and seeing how close I could get the calculator to 99999999 without overflowing. - Joe Z.
Of course, this would get pretty boring after 67108864 if I just kept starting over again, but I just cleared the error sign on the calculator when it read "1.3421772" and kept going. - Joe Z.
(8) Wow, you sound like me. Also hitting the cosine button on the calculator converges 0.739..., like $\sqrt{x}$. Rational approximations of $\pi$. But 1/7 was THE thing for me when I was nine. I looked at all the $x/n$ up to $n = 25$. All the questions I had about $1/7$ weren't answered until I took Abstract Algebra in college, a decade of curiosity! About that time, I switched to a math major. Even now I don't have a good answer to why the multiples of 3 are missing in the decimal expansion, other than the expansion has to be 6 digits long, which means some digits are missing. - Michael E2
(1) @MichaelE2: Here's a simple answer to your question: the multiples of 3 are missing because the corresponding fractions of 7 just happen to miss those intervals on the real line. Notice that when you do long division, the repeating digits correspond to repeating remainders after bringing down a zero. - Joe Z.
(5) 12345679 x 8 = 98765432 - jwg
(2) @JoeZeng Thanks. I have a similar one: The fractions occurring at equal intervals, you have to cast out four digits equally spaced, starting with 0. It happens that every third one works perfectly, which seems accidental. Is there a reason that connects divisibility by 3 to the prime 7? That's the question I had as a child. I think the question is a bit childish, in that I dreamt a certain sort of answer might exist, but our explanations are probably better and closer to the "real reason." Not all dreams come true, but they propel us into searching for answers. - Michael E2
(1) I would say it's mostly just a coincidence that 10 minus 6 is 4. - Joe Z.
@MJD This is actually the first beautiful math result I remember. I remember multiplying the numbers out by hand with great effort, and the final result seemed like such magic. It may have been what triggered me into becoming a mathematician. The second result I remember was when my dad showed me how to add all the numbers from 1 to n without actually doing the additions explicitly. WoW! - Peder
"I spent hours doing this." Neeeeeeeerdddddd. Just kidding :) - Bennett Gardiner
7
[+135] [2013-03-07 11:04:48] poke

Adding to LaceySnr’s answer, I’d like to mention fractals in general. While fractals will probably count as a higher application of maths, they are very often very visually beautiful. So you could easily show a picture of a fractal and explain that there is just a simple formula behind it all.

Some more examples:

[1] http://www.bit-101.com/chaos/fern.html
[2] http://www.bit-101.com/chaos/tree2.html
[3] http://vimeo.com/20687741
[4] http://youtu.be/xVnk_UV8hII

(25) holy crap the last 3.... - im so confused
(9) All the fractals shown in this and other answers are too complex for little kids, but the basic concept is easy for them to understand. Start with a tree with each branch being another tree. It doubles as a introduction to recursion for those more inclined towards programming than pure math. - Izkata
(9) @Izkata But they are visually appealing and that will spark the interest in it. As a kid I wouldn’t care how to prove that 2 is irrational, probably because I wouldn’t know what irrational is to begin with, or that there is some equality with some weird constants I don’t even know ($e^{i\pi} = -1$). It’s things like fractals, or other natural things like Fibonacci flowers that makes math interesting beyond just numbers. - poke
(2) This is fantastic. - josh
(6) @Izkata Perhaps fractal theory is too complicated for kids, but using fractal generation software might be within their reach. And if they start playing with that, they'll be learning a whole ton of math in a way that is incredibly fun. - Kevin
I suggest including the formula, to really stoke those curiosity, W.T.F. fires. Of course the Mandelbrot Set takes the cake with: $z_{n+1} = z_{n}^2 + c$ - BobStein-VisiBone
(1) The last picture is breath-taking. It looks like a kingdom :) - Prism
@poke : Awesome !!! I didn't even know these fractals ! Very nice ! - jibounet
8
[+127] [2013-03-07 07:31:19] Git Gud

This isn't what did it for me, but it's fairly simple and quite nice:

$$0.9999999999\ldots =1$$

(19) +1: This one, or actually the binary version $0.\overline{1} = 1$, was a source of early fascination for me. - copper.hat
(1) Unfortunately, this tends to confuse more people than it inspires. - MikeFHay
(27) I remember thinking "ahhhh I broke maths!!!" when I first stumbled over this. - Ben
(4) I remember heated discussions in school why this makes no sense… - poke
(1) Can someone explain? - user11393
@infact Let $x=0.99999...$ then $10x=9.99999....$ so subtracting $9x=9 \rightarrow x=1$. - Ben
@Ben but $9.99999... - 9 \,=\, 0.999999... \,!=\, 1$ ... - user11393
(9) @infact We are performing $10x-x=9.99999..... -0.999999......=9$ - Ben
(2) @Ben, Oh, the importance of commas... ;) - Will
(45) An "argument" that I find to convince a surprisingly high number of people goes like this: do you agree that $1/3 = 0.3333...$? And that $2/3 = 0.6666....$? Well, how about $3/3 = 0.9999....$? - Jesse Madnick
you can "prove" this by typing 1 / 3 * 3 into a simple calculator. - azimut
I just spent far more time then I should have trying to wrap my head around this. Thanks for that one! - John
(12) A YouTube video titled "9.999... reasons that .999... = 1": youtube.com/watch?v=TINfzxSnnIE - Kevin
@Ben That was the only argument that I was willing to accept prior to learning calculus and having my assumptions about infinitesimals corrected. - Dan Neely
(1) @poke I teach maths to 16-19 year olds. I look forward to this argument every year! - AndrewC
@AndrewC That’s good to hear because you seem to be prepared for it! Unfortunately my teacher wasn’t really able to explain this well, which is a shame because it’s a nice thing to show… - poke
(1) @poke I normally use three arguments, one the 10x-x argument above, one the 1/9,2/9,3/9...8/9,9/9 argument, but first I give them a geometric sequence with first term a=0.9 and common ratio r=0.1, and get them to add up the first few terms, then find the sum to infinity using the formula they've been using all week a/(1-r). We have a talk about the difference between adding up a very large finite number of terms (calculator thinks it's 1, be we know it's not), and a genuinely infinite number of terms. - AndrewC
(3) @kutschkem Why that's a silly question! Of course it must be $0.0000...1$ right? - Cruncher
Just found a fairly long wikipedia article on this en.wikipedia.org/wiki/0.999... - kutschkem
(1) @Cruncher which is zero. $\lim_{n \to \infty} 10^{-n} = 0$, as does $\lim_{n \to \infty} 10^{-n} b$, which corresponds to any $0.000...b_0...b_n$, where $b_k$ is the kth digit of the decimal expansion of b. - Cameron Martin
(2) @CameronMartin Precisely. My comment was a coy take on standard arguments for $0.999.. \neq 1$ - Cruncher
@Cruncher The interesting thing was, in around 3rd grade when I played around with how foolish people were for not understanding that $0.999... = 1$, I realized that $1 - 0.999... = 0.00...01$ and I got thinking that $0.999...$ is not actually $1$, but it is arbitrarily close to $1$ such that it is $1$, because $0.0...01$ clearly isn't $0$; it's just closer to $0$ than any number, and it was equal to $0$ because it has no unique representation. Or so I thought at that age. Interestingly, I still use that way of thinking to explain infinitesimals and limits to myself. - Justin
9
[+125] [2013-03-07 17:33:19] user17762

Here is my favorite classic illustrating the power and beauty of mathematical argument. Consider the question:

Question: Can an irrational number raised to an irrational number be rational?

Answer: One of the classic answer goes as follows. Consider the number $x=\sqrt{2}^\sqrt{2}$. If $x$ is rational, we are done. If $x$ is irrational, then consider $x^{\sqrt2}$, which is $2$ and now we are done.

(15) I love those "anyway" proofs. - Joe Z.
(79) The really amusing thing is that after the proof is given, we still don't know whether $\sqrt{2}^{\sqrt{2}}$ is rational or not! - Jesse Madnick
Turns out it's irrational. - Joe Z.
my knowledge of powers is a bit rusty, how can i show that it is $\sqrt{2}^2$? - ted
(8) @ted $(a^{b})^c = a^{bc}$. Hence, $(\sqrt2^{\sqrt2})^{\sqrt2} = \sqrt{2}^{\sqrt2 \cdot \sqrt2} = \sqrt2^2 = 2$. - user17762
BeautifulHandsome^^ - GarouDan
This is a fascinating argument, without doubt. But also without doubt, it is way too advanced for a children's book. - azimut
(6) How about $e^{\ln 2} = 2$? - Jas Ter
(7) @SimenK. It is significantly more difficult to prove that either $e$ or $\log 2$ is irrational than it is to prove that $\sqrt 2$ is. - guest196883
@JesseMadnick, we do now! :-) - Zain Patel
(1) How about $\sqrt 2 ^{\log_2(9)} = 3$ ? Here it is easy to show that $\log_2(9)$ is irrational since $\log_2(9) = \frac{a}{b}$ would mean $9^b=2^a$ which is not possible if $a,b>0$ are positive integers. - Watson
10
[+104] [2013-03-07 15:18:09] BobStein-VisiBone

The number of pennies stacked in a triangle $(1,3,6,10,\cdots)$ is along one diagonal line of Pascal's Triangle. The number of spheres stacked in a tetrahedron $(1,4,10,20,\cdots)$ is the line next to it. The next line is the number of hyperspheres in a pentachoron.

I was about $10$ and living in a hotel and home sick from school, stacking up pennies and "red hots" in pyramids, etc. I made a table of these numbers. Noticing the simple addition rule in the table, I extrapolated to the $4$th, $5$th, dimensions. Later when I learned of Pascal's triangle that moment was probably my biggest joy of mathematics, realizing I'd run into that years before.

(14) Back in School i noticed that each row of pascals triangle is a multiple of 11: 11^0, 11^1, 11^2 etc. - Nils Werner
@NilsWerner I'm not sure that's entirely accurate. How then do you handle numbers beyond single-digits, or are you perhaps working in some 'arbitrary base'? - Sean Allred
(8) @vermiculus: I think what Nils Werner is saying is that the $n$th row consists of the numbers $\binom{n}{k}$, and that $\sum \binom{n}{k} 10^k = (10 + 1)^n = 11^n$. Of course some of the $\binom{n}{k}$ will have more than one digit. - Jesse Madnick
(1) Mark odds and evens in Pascal's triangle and you get something nice :) - Manishearth
(3) @vermiculus Shift multiples of 10 one position to the left, so 1 5 10 10 5 1 becomes 1 6 1 0 5 1 - Nils Werner
@NilsWerner That... that is awesome. Right there. That's awesome. - Sean Allred
@JesseMadnick I tried to do the proof for that equation in Uni... I couldn't do it :-D - Nils Werner
@NilsWerner's observation holds in any base, e.g. hexadecimal $11_{hex}^5 = 15AA51_{hex}$ - BobStein-VisiBone
Hint: color in multiples of $n$. Hint 2: consider the difference in the patterns when n is prime versus composite. - Jack M
Great treatment of what he calls triangular, tetrahedral, Pentatope and other Hypertetrahedral Polytope Numbers at Ron Hubbard's Blob. - BobStein-VisiBone
11
[+101] [2013-03-07 11:01:14] azimut

As a child, the Fibonacci numbers [1] $$1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\;\ldots$$ were very fascinating to me. They are named after the the Italian mathematician Fibonacci [2], who described these numbers in his 1202 book Liber abaci [3] modeling a growing rabbit population:

Formally, the Fibonacci numbers $F_n$ are defined recursively by $$F_1 = 1, \quad F_2 = 1, \quad F_{n+2} = F_{n+1} + F_n$$ It was a lot of fun to compute them, one after the other, and to collect the results in ever-growing tables: $$F_3 = F_2 + F_1 = 1 + 1 = \mathbf{2}\\F_4 = F_3 + F_2 = 2 + 1 = \mathbf{3}\\F_5 = F_4 + F_3 = 3 + 2 = \mathbf{5}\\F_6 = F_5 + F_4 = 5 + 3 = \mathbf{8}\\F_7 = F_6 + F_5 = 8 + 5 = \mathbf{13}\\\vdots$$

At some point, I asked myself the question: To compute $F_{10}$, do I really have to compute all the Fibonacci numbers up to $F_9$ beforehand? So I tried to figure out some formula where you can plug in $n$, do some basic arithmetics, and get $F_n$ as a result. I've spent a lot of time on this. However no matter how hard I tried, I didn't succeed.

After a while I found the closed form $$F_n = \frac{1}{\sqrt{5}} \left(\left(\frac{1 + \sqrt{5}}2\right)^{\!n} - \left(\frac{1 - \sqrt{5}}{2}\right)^{\!n}\right)$$ in some book. I was paralyzed.

How can it happen that such an easy recurrence formula needs to be described by such a complex expression? Where do the square roots come from, and why does the expression still always evaluate to an integer in the end? And, most importantly: How on earth can one find such a formula??

[1] http://en.wikipedia.org/wiki/Fibonacci_number
[2] http://en.wikipedia.org/wiki/Fibonacci
[3] http://en.wikipedia.org/wiki/Liber_Abaci

(10) I first discovered the Fibonacci sequence by playing with an 8-digit calculator. There's a bug in most standard 8-digit calculators (the kind you get as promotional gifts, for example) so that if you press "1 + = + = + = ...", it will give you the Fibonacci sequence. - Joe Z.
(14) In order to find this formula, you usually will use linear algebra, and the property that linear recurrences can always be represented in matrix form. In particular, the two exponential terms correspond to the eigenvalues of the matrix $\left [ \begin{matrix}1 & 1 \\ 1 & 0\end{matrix} \right ]$, which, as you could probably guess, has a characteristic polynomial of $x^2 - x - 1$. - Joe Z.
(3) JoeZeng: Wow, that's a nice bug-abuse :) Nowadays I know, of course, how to derive the formula. Besides the linear algebra method you indicated (my favorite), there is also a standard way to do it by generating functions. - azimut
(3) There's a way to derive this using only plain algebra, which would be nicer for a children's book. Let a = (1+sqrt(5))/2 and b = (1-sqrt(5))/2. Show that a^2 = a + 1 (and similarly for b). Multiply by a and simplify to get a^3 = 2a + 1, and show that you can repeat this to get a^n = F(n)*a + F(n-1) (and similarly for b). Take a^n - b^n, simplify and rearrange and you're done. - Michael Shaw
(3) A bug? I thought it was a feature? - Raskolnikov
(1) One of the first beautiful maths i saw (at college) was indeed the connection between analysis and linear algebra. I first saw the proof of this formula in analysis and it was not very elegant. Then i saw it in linear algebra , using the matrix @JoeZ. illustrated. I loved the approach and was surprised by it's elegance. Still i was even more surprised about the connection between these two subjects. - sigmatau
By the way, here is a fun puzzle related to the above comment: What decimal digit occurs at 100 places after the decimal point in the decimal representation of $(1+\sqrt{2})^{500}$? - Gowtham
Assume that there exists a number $\phi$ such that $\phi^2=\phi+1$. It's easy to show that $\phi^3=\phi^2+\phi=2\phi+1$. Try making a table of $\phi^n$ for a few $n$. What do you see? - Akiva Weinberger
Here's a nice explanation of the solution. en.wikibooks.org/wiki/Linear_Algebra/Topic:_Linear_Recurrenc‌​es - silgon
12
[+84] [2013-03-07 10:30:37] cormullion

My son loved this when he was little - patterns everywhere:

(1) Oh, I would've loved this, too. Very cool! - Jesse Madnick
(6) Except the white boxes, they doesn't have pattern! XD - ᴊ ᴀ s ᴏ ɴ
(62) Why isn't 2 red, 3 yellow, 5 green etc? - AidanO
(4) crayons can be provided... :) - cormullion
(1) @AidanO none of the prime numbers are coloured - Omar Kooheji
(29) @OmarKooheji: yes, but the pattern would be more natural if the prime numbers themselves were coloured — as solid squares of whatever colour they’re given in subsequent composites. // The difficult question is: what colour, if any, should 1 be? - Peter LeFanu Lumsdaine
(1) @aidanO Yes, that's certainly an obvious thing to do. Instinctively I avoided doing it that way at the time, and would do the same now. The thing is, primes are different: not just bigger coloured squares, but different from other numbers, and related to each other as strongly if not more so than to their multiples. And then, what colour is, say, 97? (Well, it later gets a colour automatically, since the chart originally went up to 1000.) But would that colour indicate the number's primality well? That's why the white works, for me. (I didn't get into the nature of 1... :) - cormullion
@PeterLeFanuLumsdaine 1 must be colored with the transparent color! - Pål GD
(6) Pretty cool! People fascinated by this might also like prime factorization diagrams: datapointed.net/visualizations/math/factorization/… - Matt W-D
(1) @cormullion I think the fact that there is a full box on one specific colour would indicate that it's prime. As to the question what colour is 97, you could just as easily as what colour is 3, it has a colour once you assign it. Either way, after staring at it for 30seconds trying to figure out what it represented I thought it was pretty cool :) would give it a few +1s! - AidanO
(3) @AidanO, I had an idea once of coloring the primes (including 97) with a hue $[0...1)$ derived from $\{ \log_2 n \}$ that is, the fractional part of the log base 2. This would provide a unique color for all primes because any two numbers with identical fractional parts would have to be related by a power of 2. - BobStein-VisiBone
@Aidian: Because the prime powers are already fully colored and this could potentially add confusion. - user02138
Make the primes fully colored circles! - BeniBela
Regardless of the fact that this is an old post I owuld just like to add that this is really really cool - KBusc
Prime Climb multiplication table. It's a really fun game that targets both children and adults. - brimborium
13
[+83] [2013-03-07 15:51:45] Tomas Greif

There is a well known story about Karl Friedrich Gauss when he was in elementary school. His teacher got mad at the class and told them to add the numbers 1 to 100 and give him the answer by the end of the class. About 30 seconds later Gauss gave him the answer.

The other kids were adding the numbers like this:

$$1 + 2 + 3 + ... + 99 + 100 = ?$$

But Gauss rearranged the numbers to add them like this:

$$(1 + 100) + (2 + 99) + (3 + 98) + ... + (50 + 51) = ?$$

If you notice every pair of numbers adds up to 101. There are 50 pairs of numbers, so the answer is $$50 * 101 = 5050$$ Of course Gauss came up with the answer about 20 times faster than the other kids.

In general to find the sum of all the numbers from 1 to n:

$$1 + 2 + 3 + 4 + ... + n = (1 + n) * \bigg(\frac{n}2\bigg)$$ That is "1 plus n quantity times n divided by 2".

(5) This was also one of my favorites, perhaps because of the story. I'll suggest a slight improvement in the last statement. It's easier to imagine it as number of items times average value, i.e. $n \frac{n+1}{2}.$ Then it becomes common sense and no need to remember any formula. - karakfa
14
[+80] [2013-03-07 10:06:30] Dan Petersen

When I was a kid my parents explained basic arithmetic to me. After thinking for a while I told them that multiplying is difficult because you need to remember if $a \cdot b$ means $a+a+\ldots + a$ ($b$ times) or $b + b + \ldots + b$ ($a$ times). I was truly amazed by their answer.

(24) What was their answer? - User 17670
(1) Yeah, what was their answer? - goblin
(4) At a guess, "No you don't" :P - Ben Millwood
(11) @Dan Petersen It's bad manners to tell this story and not tell us what the answer was. - Git Gud
(14) I assumed their answer was the trivial one: "They're the same, you don't have to remember it". - user5501
(10) Maybe it was "Uuuuh, bedtime!"? : - ) - k.stm
(52) The answer should be obvious to anyone using this site, and it would spoil the humour of the story to spell it out explicitly. - John Gowers
Let's take an example: 5*5=25. The reason you find it so easy is because you have used it so many times in your life. You don't actually have to 'actively remember' things. Another example is 4+5=9, one 'real way' of finding out that 4+5=9 is by drawing 4 circles and 5 circles, then counting how many total circles there are. We're just so used to addition and multiplication that they have become second nature. An analogy is that when we come across new math in our life, it requires effort to learn it, but after lots of using that math it becomes second nature. - raindrop
I decide to sign up just to downvote because you didn't tell us the "their answer" ! but I have not enough rep. Will be back until I have the privilage. - Baby
(7) @Donkey_2009: I have been using this site for three years up to a decent amount of reputation, and I have no idea what the answer should be. I don't think I'm the only one. - Martin Argerami
@Donkey_2009 you are of course saying wrong. the answers has a major role in this criteria. I think you have not read the question rigorously. - Sufyan Naeem
(1) @MartinArgerami the answer would be explaining to the son that it doesn't matter which one you do because multiplication is commutative. - MCT
15
[+65] [2013-03-07 09:12:40] Kobi

I always thought cycles in decimal fractions were magic, until I realized I can easily create whichever cycle I wanted:

• ${1\over9} = 0.111...$
• ${12\over99} = 0.12\ 12\ 12...$
• ${1234\over9999} = 0.1234\ 1234...$

I failed a number theory exam because the professor did not know this trick and said I needed to prove it.

It's just as much fun in binary... Take 1/10. Then, 1 / 5 / 2 = 3 / 15 / 2 = (binary) 0.0 0011 0011 0011 .... where 15 in binary is 1111 so any binary value repeated every 4 digits is evenly divisible by 15. - ErikE
(40) You sound like you don't agree with the professor, but of course you need to prove it! - Kundor
You can use the result from this question to prove that you can create any cycle in any base. - Joe Z.
(2) Isn't it obvious by long division? I knew it since I was a kid.. - user21820
does somebody know a proof why this works? - user159517
(2) One proof is very similar to the proof $0.999... = 1$. If we know ${1\over9999} = 0.00010001...$, then $1234{1\over9999} = 1234 * 0.00010001...$, ${1234\over9999} = 0.1234 1234...$ - Kobi
16
[+63] [2013-03-07 06:58:16] Trevor Wilson

I don't remember what the first beautiful piece of math I encountered was, but here are a couple of candidates:

• Proof that the square root of 2 is irrational

• Euclid's proof that there are infinitely many prime numbers

(13) +1: I must confess that I have showed those two proofs to many people, hoping that I will be able to recreate in them the awe and fascination that those proofs ignited in me. Unfortunately, although some did find it a neat curiosity, I have failed to kindle in them a permanent interest for mathematics. - user5501
(3) When I first saw the proof for infinitely many primes it blew my mind as well! - sdm350
If I want to know if seeing that proof will blow my mind, where would I go read it, well explained? - Warren P
@WarrenP This page has a modernized version of the original argument with some explanation and also several other proofs on the linked page: primes.utm.edu/notes/proofs/infinite/euclids.html. Here is a new proof that is perhaps the simplest: jstor.org/stable/10.2307/27642094 - Trevor Wilson
Which proof of $\sqrt{2}$ being irrational? - Caleb Jares
@Caleb I was thinking of the usual proof by infinite descent - Trevor Wilson
I find it quite beautiful that Euclid knew there were infinitely many prime numbers back in 1782 but even with the supercomputers around today we don't know (for certain) whether there are an infinite number of twin primes. - Steve Chambers
(5) @SteveChambers I hope you find it even more beautiful to learn that Euclid lived over 2000 years before 1782! - Erick Wong
(7) @ErickWong If he lived over 2000 years he probably wasn't very beautiful toward the end :-J - Trevor Wilson
@ErickWong Oops, looks like 1782 was James Williamson's translation of Euclid's elements! Should check my references more carefully... - Steve Chambers
(1) Exactly these two examples are provided in 'Mathematician's apology' - Alex
17
[+63] [2013-03-07 07:29:12] Asaf Karagila

The fact that you can always divide something by two. That is an amazing discovery my dad tells me I made as a toddler.

I think that ever since I remember abstract mathematics was a fascination of mine, even before I knew what it was (because it was obvious school mathematics wasn't that).

Another fact I stumbled upon as a teenager and fascinated me was that if you hold a magnifying glass over a ruled paper the parallel lines bend, and eventually meet at the edge of the glass. That, in a nutshell, is a non-Euclidean geometry.

(6) That is a really amazing discovery. It reminds me of my own happy discovery that iterating $x\mapsto x/2$ produces numerals that look like powers of 5. Did you do your division with paper or with some sort of calculator? - MJD
@MJD: Dividing by two? I have no idea. That's a story my dad keeps telling (the age varies from 3 to 5, and I am inclined to believe 5 is the correct age). He says I knew that if you have a floor tile, you can break it in half, and again and again and again. I don't think I came across many calculators at that age. - Asaf Karagila
(3) @Asaf It reminds me of my discovery that $a+b\geq 2\sqrt{ab}$ by trial and error when I was in 8th grade. - Ishan Banerjee
@Ishan: That's pretty nice! - Asaf Karagila
18
[+62] [2013-03-07 08:17:33] Kaz Dragon

For me, it was the discovery that the sum of the digits in all multiples of three are themselves multiples of three, and you can recursively sum them to get to 3, 6, or 9 (i.e. an 'easy' multiple of three)

E.g.

The sum of the digits in $13845$ is $21$,

The sum of the digits in $21$ is $3$

Edit: Should probably add that what made this useful to me was that numbers that are not multiples of three do not have this pattern.

I always found it fascinating! - Integral
(9) Or, more generally, that the ultimate result of this digital summing is the same as the remainder when dividing by 9. - cobaltduck
(2) This generalises via arithmetic modulo 10 to produce tests for divisibility by other numbers based on similar properties of their decimal digits. - user3490
this is the one I was going to submit, and it's easy for a 7 year old to grasp, even if the proof is difficult at that stage. - Alnitak
I discovered a special case of this in third (?) grade: 3 times 37 is 111. It caught my attention -- how could THAT happen? - SixWingedSeraph
This is the first bit of maths (rather than learning stuff) that I can remembre - Mark Bennet
Just a bit of a shame that it is so tied up with the human genome. - PJTraill
19
[+54] [2013-03-07 09:55:06] quoniam

When I was a child, I spent the whole summer at a camping at the coast of Catalonia [1]. There I was always around my grandfather. He himself had no proper education and never finished school. Nevertheless he liked to read books on his own, about many things, grammar, the French language, mechanics, mathematics...

I remember he taught me many things. He was the first to explain me, as I fell asleep in his arms, under the starry night, that the Earth was a ball, and that there were people underneath the ground where we stood, on the other side of the planet, who were standing upside down without falling, because we were all attracted to the center of the ball. I did not understand, at that moment, how was that possible. But I trusted him and knew that there were many things I did not understand about the world.

One particular thing related to mathematics that he told me and that got me thinking, making myself questions and reaching the boundaries of my mind, was that one frog could try to jump her way across a puddle (we also went together to catch frogs), jump first to the half of it, then to the half of the remaining half and so on, and that after an infinite number of jumps she would arrive at the other shore.

This was, I think, one of the first things that made me feel that the world or that reality itself was infinitely bigger, more complex and more beautiful that anything we could understand or even begin to grasp. I guess this sense of real magic is what makes me have a special love for mathematics.

[1] https://en.wikipedia.org/wiki/Catalonia

20
[+53] [2013-03-07 12:50:58] krikara

Who remembers magic squares? Those sparked my interest in mathematics.

(16) Althought this isn't suitable for children. My interest in magic squares was only sparked much later in life, during my undergrad, where I realised that the set of $n\times n$ magic squares (over a field $k$) form a subring of the matrix ring $M_n(k)$. I also spent many hours trying to work out the dimension of the ring as a $k$-vector space. - Dan Rust
(10) How isn't this suitable for children? Finishing one out that has already been started (but has some missing elements) can be a fun task for someone just learning addition... - apnorton
I was first introduced this in 4th grade. We were given a blank square (with the exception of 5 in the middle) and the goal was to make the sum of each row/column equivalent. It was a great way to challenge ourselves in math. - krikara
What I loved about the magic square was that the numbers form a symmetric pattern. To remember the order of the numbers, you just need to remember the pattern. - Paddy Landau
(11) @anorton: Daniel's comment's opening clause "Although this isn't suitable for children..." referred to the rest of his comment, not to the answer. ("Although the following isn't suitable for children,...") - ShreevatsaR
I think if introduced the right way, part of that could be appropriate for children as well. Of course not the multiplication part, but the fact that you can add two magic squares (component-wise addition is easy to understand) and still get an magic square (except for the condition "numbers from 1 to 9") should be well accessible. Also the fact that you can multiply with a number and again get a magic square. Which are already the two operations of a vector space (and when, much later, they learn about vector spaces, maybe they remember those properties and recognize it as vector space). - celtschk
I love that you can use this magic square to provide a game isomorphic to tic tac toe: The "sum to 15" game. I will let you figure out the rules. Show them the sum to 15 game, and then see how much easier it is to play once you know the isomorphism. - Steven Gubkin
21
[+51] [2013-03-07 08:46:30] user1551

Here are some things that I found interesting back when I was in junior high school. I hope they are not too advanced for young children:

• Archimedes' method for computing areas and volumes [1] (which is really cool).
• The "limit" card magic. Take 27 cards from an ordinary deck of playing cards. Invite your audience to pick one of them, without telling the choice. Deal the 27 cards into three stacks, say $A, B$ and $C$, each containing 9 cards. The deal order is $A\to B\to C\to A\to B\to\cdots\to C$. Ask the audience which stack contains the chosen card. Collect the three stacks into one deck, where the stack containing the chosen card is placed in the middle. Repeat this deal-and-ask procedure twice more (so, thrice in total). Now the chosen card is the middle one in the stack as told by the audience.
• The remainder of a whole number, when divided by $3$, is the remainder of the sum of its digits when divided by $3$.
• The cyclic decimal expansion you get when a whole number is divided by $7$.
• $1+2+\ldots+n=\frac{n(n+1)}2$. $$n\left\{ \begin{array}{ccccc} \bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\bullet&\bullet&\color{red}\bullet\\ \end{array}\right.$$ (Actually $1^2+2^2+\ldots+n^2=\frac{n(n+1)(2n+1)}6$ is even more interesting, but its proof is certainly too advanced for most young children.)
• The (slanted) cross section of a cone has a symmetric shape (an ellipse). (Provided that the cross section does not cut into the base of the cone, of course.) This is rather inobvious to me because I thought the slant will break symmetry.
[1] http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2243&pf=1

(12) +1 for visual 1 + 2 + ... + n = n*(n+1)/2. - Rauni
+1 for $n(n+1)/2$. I "proved" this in the 5th grade using the same method :). - Jacob
Actually, the one that's interesting is the sum of the first $n$ odd integers. - asmeurer
@asmeurer Yes, that's more interesting! - user1551
You can prove the formula for the sum of squares using the same wort of picture: stack a 1x1x1 on a 2x2x1 on a 3x3x1, on a ... on a nxnx1 block. Six such things form a nx(n+1)x(2n+1) block. If you actually made the shapes, I think this would be just as accessible to young children as your proof of the sum of the first n integers. - Steven Gubkin
I've just learnt that the card trick I mentioned in the above is called Gergonne’s Card Trick, named after Joseph Diaz Gergonne, who first published an analysis of the trick circa 1813 in Annales de Mathématiques. - user1551
22
[+46] [2013-03-09 20:48:20] Matt W-D

A few things come to mind:

Here's a beautiful JavaScript demo of these graphs being generated: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

• Even as an adult, I think continued fractions [6] and generalized continued fractions [7] are amazing. One of the simplest is the golden ratio [8]: $$\varphi = 1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ And this identity is downright incredible:

$$\frac{\pi}{2} = 1+\cfrac{1}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}$$

I should stop myself now... But math is really filled with astounding phenomena like I've mentioned above...

[1] http://en.wikipedia.org/wiki/Bean_machine
[2] http://en.wikipedia.org/wiki/Central_limit_theorem
[3] http://en.wikipedia.org/wiki/Bean_machine
[4] http://mathlesstraveled.com/2012/11/05/more-factorization-diagrams/
[5] http://mathlesstraveled.com/2012/11/05/more-factorization-diagrams/
[6] http://en.wikipedia.org/wiki/Continued_fraction
[7] http://en.wikipedia.org/wiki/Continued_fraction
[8] http://en.wikipedia.org/wiki/Golden_ratio

(2) I've never seen that one for $\pi$ before! It's much better than the one with integers, which has no pattern at all. - Ryan Reich
(1) I wonder if there exist an intuitive argument explaining the pattern that appears in the continued fraction of $\tfrac\pi2.$ - Hakim
(1) The Galton device makes a really important (philosophical?) point that kids can understand: through math, you can be completely certain that a pattern will emerge from the aggregate behavior of a large number of individuals, even if those individuals don't communicate with each other or organize, and even if they act completely randomly! There was (is?) one at the Museum of Science and Industry in LA when I was a kid, and I remember trying it over and over again -- how on earth did all those balls know where to line up? - Tad
23
[+42] [2013-03-07 15:40:44] Jordan Gray

I always had a peripheral understanding that there was something more to maths than working out how your change or divvying up sweets with your siblings. But the day I really, truly understood was when I learned about $\pi$.

$\pi$ was magical to me. For one thing, it's a funny-looking Greek letter with a funny-sounding name. But, more captivatingly, it introduced me to an epiphany: that somewhere, on some level, the fundamental structure of reality itself could be understood through mathematics.

Let's assume your childen understand what a circle is, and how to measure things with a measuring tape. Introduce them to circumference and diameter. Give them a table with three columns—circumference, diameter and "the secret of circles"—and a big tape measure. Tell them to go out and measure as many circles as they can find: plates, car tires, stop signs, plant pots, lines on a basketball court… anything so long as it's circular. Let them loose.

Later, once they're done measuring everything in the neighbourhood, hand them a calculator and tell them to go through each of their circles and divide the circumference by the diameter, and write the number they get in the mystery third column. Tell them that a pattern will start to appear, and they need to see if they can spot it.

Once they're done, you can explain to them that the reason the first couple of numbers is the same is because there's a number, a magical number, that tells us a secret about every circle in the universe—from rings we wear on our fingers to the sun and moon in the sky and the whole planet Earth. No matter how big or how small, how grand or how humble, every single circle is a bit more than three times bigger around than it is from one side to the other. This number is so special that it has its own name, pi, and its own special letter, $\pi$. It's not three and it's not four—it's somewhere just after three, and we can't write down exactly where because it goes on forever. Luckily, we only really need to know the first few numbers most of the time, so we can use this magical number whenever we need it.

The sense of revelation that came from knowing that every circle in the universe is connected by this strange, special number stayed with me for a long time, and is at least partly responsible for my love of mathematics in later life.

(8) That's an awesome idea for something to do with your kids. +1 - Will
(3) Actually the number is tau and it's less magical (but no less beautiful) when you think of every circle as a radius spinning around a point. - AlexChaffee
(1) Even better still would be the more beautiful number $\tau$. $\tau$ uses the radius instead of the diameter, which is what formula for circumference, area, volume and even hyperspheres use. See tauday.com - John Nicholson
24
[+39] [2013-03-07 07:39:15] Carl

The game of Nim and its solution are pretty cool. The proof might be a bit difficult, but I think kids would love to learn a game like that and how to beat their parents at it.

There's a lot of other fun mathematical games like that too. But I think the first thing I learned that turned me towards mathematics was the existence of multiple infinities, and things like Hilbert's infinite hotel [1].

(11) +1, Hilbert's hotel could be great for a children's book! See en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel. - Will
If one knows base two, the optimal strategy for Nim is pretty simple. I didn't see this until much later, but this would have been a favorite of mine if I'd seen it earlier. - robjohn
25

Everybody loves fractals. I think this one - The Dragon Curve - is particularly easy to explain, and it is very surprising and aesthetically pleasing:

Here's a video I've seen which explains how it comes about: The Dragon Curve from Numberphile [1]

(5) I have a friend who is disgusted by pictures of fractals. Might relate to trypophobia. - Nikolaj-K
(1) I seem to recall this fractal making an appearance in Jurassic Park (the book). - icurays1
Check out the video I attached at the bottom of the picture, it states that appearance in Jurassic Park! - Adar Hefer
(1) I found by myself this fractal by folding and reopening a long strip of paper. The strip was actually the edge (with holes) of the continuous paper used in old pin printers. - Emanuele Paolini
26
[+36] [2013-03-07 09:50:41] chris

It was presented to me like this: There exists a number that you can square, subtract itself, and you'll get 1. Or, you can inverse the number, add 1, and you'll get the number back. What a beautiful number, I thought. Of course, I later realized the number was just a solution to:

$$x^2 - x - 1 = 0.$$

However, I was really impressed when later I learned this number also shows up in nature in the patterns of plant growth. Wow! Who would have thought?

[1] https://en.wikipedia.org/wiki/Golden_ratio

(3) The way I first saw it was like this: there is a number that, if you add one to it, becomes its square. Then they showed that you could easily calculate it from just that information with only the quadratic formula (which I had thought was boring and had never suspected of hiding anything beautiful), and that if you kept multiplying it by itself, you'd get a simple expression with fibonacci numbers as coefficients. That was a happy day. - Michael Shaw
(3) Showing that number on a calculator really drives its magic home. Square it, and all the digits after the decimal point stay the same. Or hit the 1/x button, same thing. - Joubarc
(4) I think it's more impressive that if you subtract one from that number, you get its reciprocial. - Zsbán Ambrus
(4) youtube.com/watch?v=ahXIMUkSXX0 Vi Hart explains why the "magic" is actually completely inevitable, and the connection with the Fibonacci numbers. - Ben Millwood
27
[+36] [2013-03-07 12:11:57] user2128456
Oh yes. Ooohhhhh yes. - Sean Allred
Ooohhhhh yes indeed! I can still remember the theme music now, and picture the pentagrams arms expanding into golden rectangles. - Lucas
You can watch it again online: youtube.com/watch?v=YRD4gb0p5RM - Lucas
@Lucas removed for copyright infringement. hurray for copyright law, making the world a better place for everyone. - Alexander Gruber
28
[+35] [2013-03-07 22:32:48] Carl Smith

### Realising why zero is not nothing, and understanding numbers

I first understood the difference between zero and none when thinking about thermometer readings. If you had a ton of thermometers scattered around the world, and you collected their readings periodically and put them in a database, what would you do if any thermometer was broken? If you just put a zero reading, you'll screw up your averages, but if you put a null value, you can handle broken thermometers easily.

That made me realise what a number is.

29
[+34] [2013-03-17 04:21:50] Hypercube

I was in elementary school, drawing 3D shapes in class while bored. I drew cubes by drawing two overlapping squares and connecting the vertices, like the top row in this image:

Then I thought, what if I did the same procedure, but to a cube? So I drew four squares and connected the vertices, like this:

I was struck by the beauty of the resulting image, with its intricate structure of star-like patterns. Here's a static version:

It was years later that I discovered, to much fascination, that this was in fact the four-dimensional analogue of the cube: the hypercube. Hence my username.

Edit: Another thing I remember thinking about when I was younger was that I could not always draw a straight line through three points, but was surprised to find that it would always work for two points.

IIRC Emmy Noether also used to draw such pictures to visualise higher dimensions as a child. - Marc van Dongen
30
[+33] [2013-03-07 12:09:49] rsp

It was probably not the first thing that made me realize that math is beautiful, but it was something that amazed me the most and still does to this day: The fact that the Mandelbrot set is not only infinite - in a way that eg. the Koch snowflake is infinite - but that it is infinitely complex, the complexity never ends, you can zoom it forever and you will never find exactly the same patterns, the information that is contained in it is infinite and yet it is described by such a simple formula.

It made me wonder whether math was discovered or created, whether things like the Mandelbrot set existed independently from their discovery or not, whether the infinitely complex pictures existed if they were never seen etc.

I remember the sleepless nights in elementary school when I was writing programs to explore the Mandelbrot set, to find nice looking colors, to animate it - impossible to do live at that time so I had to learn how to script some animation program that I had, wait an hour to realize that I had the colors wrong, change one number, wait another hour, rinse and repeat.

I didn't know about complex numbers at that time. I only knew that I was looking at something most amazing in the world and just couldn't stop exploring. Fractals became my obsession and were probably one of the reasons why I started programming more seriously.

I was fascinated by the fact that I could zoom it so much that it was like finding some proton on the face of Earth and zooming it to the size of a planet, and then looking at that planet-sized proton with an electron microscope. I could print what I found and I knew that no one in the Universe has ever seen it before me and no one will ever be able to find it even after looking on my printout - the scale was so amazing.

I remember how I got scared when I eventually saw large pixels in my Mandelbrot set! Finally I realized that I hit the limits of the floating point number precision on my 386 but I knew that the complexity of the Mandelbrot set was there, somewhere, even if I couldn't see it with my computer at that time.

Those are some of my favorite pictures that I posted to Wikipedia:

Cool Mandelbrot:

Calm Mandelbrot:

Hot Mandelbrot:

One of those pictures was magnified 248,034,982,258 times - probably the Cool Mandelbrot but I'm not sure because strangely all of them have the same description on Wikimedia Commons (something had to go wrong when they were copied from Wikipedia to Wikimedia Commons).

I would be honored if you'd like to use those pictures in your book. If you need higher resolution pictures or more information about them then I might be able to find something in some very very old backups.

Good luck with the book!

[1] http://en.wikipedia.org/wiki/User:Rfl/Fractals

(2) I feel the earth shift a little every time I try to comprehend that the formula for this is $z_{n+1} \leftarrow z_{n}^2 + c$ - BobStein-VisiBone
31
[+30] [2013-03-08 18:37:15] icurays1

For me, the result that really captured my imagination was the divergence of the harmonic series:

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots=\infty$$

It combines some wonderful ideas about the infinite and the infinitesimal, and it seemed (at the time) completely absurd to me that adding infinitely small numbers could result in an infinitely large one.

As an illustration of this idea, say we have a big pile of 1-foot square boards. We stack the first board on the second, hanging half-way (6 inches) over the edge. Then we stack the third on top the second, hanging 1/3 of the way (4 inches) over the edge. The forth is stacked on, hanging 1/4 of the way (3 inches) over the edge. The fifth...you get the idea. At first glance, one might think that our pile can only extend horizontally a finite distance - we might take bets that it gets at most 2 feet, or maybe 5 or 10 feet horizontally. But it turns out that if we have enough boards (negligibly thin, say), we could build a bridge across any river, any ocean, in fact we could build a bridge across the entire universe this way.

Here is a Wolfram demonstration of this, although their stack is upside-down from how I have described it: http://demonstrations.wolfram.com/OverhangingCards/

(2) The stack the way you've described it will collapse, I believe. You need to do it the way the Wolfram demonstration does. - Joe Z.
For years I remembered (from physics) that there was a balancing construction based on the harmonic series, but I could only think of the wrong way to do it. It was very frustrating until I saw the picture. Also, it is much less impressive now: rather than the theorem being "there exists a stack such that for all distances it extends past that distance", it is merely "for all distances, there exists a stack...". Cute, but verging on "so what". - Ryan Reich
The harmonic series arises in improbable places: math.stackexchange.com/questions/1113984/… - kjetil b halvorsen
32
[+27] [2013-03-07 11:19:12] Roberto Tyley

That the roots of $z^n-1 = 0$ start to form a circle as $n$ increases. It starts out with the simple solution, the quadratic which you've already seen, then the complex plane comes in for $z^3$ and all of a sudden it's like "Hey! Those form circle!"

(20) Well, I was amazed, and 15 years of age - not yet a man, despite my best efforts. - Roberto Tyley
:D This is how I got interested in complex analysis! - Simply Beautiful Art
33
[+25] [2013-03-07 17:48:23] Danko Durbić

Many years ago, before I knew multiplication, I wrote numbers 1 to 10 in a row:

1  2  3  4  5  6  7  8  9 10


Then I wrote a second row, just for the fun of it, starting with 2, increasing each number by 2:

2  4  6  8 10 12 14 16 18 20


And then a next row, starting with 3, with an increment of 3, and so on, until I got:

 1  2  3  4  5  6  7  8  9  10
2  4  6  8 10 12 14 16 18  20
3  6  9 12 15 18 21 24 27  30
...
9 18 27 36 45 54 63 72 81  90
10 20 30 40 50 60 70 80 90 100


I showed this to my parents, and they told me it was this thing called the "multiplication table" and explained how it worked. I was amazed.

Still today I'm very proud that I reinvented the multiplication table :)

:D What's more fun was when I made my exponential table! It didn't go out 10 places... Maybe I'll remake it today. - Simply Beautiful Art
34
[+23] [2013-03-07 11:32:26] Aky

Arithmetic series might be interesting: straightforward to explain and amenable to pictorial representation ...and the child might love the fact that they've learnt how to do huge sums that might stump many (non-mathematical) adults.

You could show how $1 + 2 + 3 + \cdots + 100$ could be worked out by pairing numbers from opposite ends of the sum together $(1 + 100) + (2 + 99) + \cdots + (50 + 51) = \underbrace{(101 + 101 + \cdots )}_{\text{50 terms}} = 5050$.

or by adding the series to itself with terms running in ascending and descending order $1 + 2 + \cdots + 99 + 100$

$100 + 99 + \cdots + 2 + 1$

to get $101 + 101 + ... = 101 \times 100$ which is twice the sum.

(6) The story could even be about a little boy named Carl Freidrich... - Will
+1 was going to post this story about Gauss myself - BlueRaja - Danny Pflughoeft
(4) I also remember a teacher (must have been around 10yo) asking us to calculate the sum of all numbers from 1 to 100, letting us sweat uselessly, then showing us how to pair the numbers. He liked to play tricks on us, but the lessons were always useful. Well, sometimes he played us for fools by being wrong on purpose - once he "calculated" that there were about half of the days in the years that were holidays to show we shouldn't complain about school. Took me years to realize he had counted a lot of days twice (like, 2 months holiday & 52 week-ends.) - Joubarc
(7) @Joubarc: That reminds me of a joke. A clerk asks his boss for a raise, and the boss calculates how many days the clerk works there. There were 366 days in that year, and he worked for 8 hours a day, so it became 122. Then, he had to subtract 52 Saturdays, two weeks' vacation, four bank holidays, and 52 Sundays, for a grand total of no days at all. "And you have the nerve to ask me for a raise!" - Joe Z.
35
[+22] [2013-03-23 20:32:16] BobStein-VisiBone

Thanks to @FacebookAnswers for suggesting Conway's Game of Life [1], a cellular automaton devised by John Conway in 1970.

## A Gosper Glider Gun

With its patterns, oscillators, spaceships, glider guns (the minimalist Gosper Gun is shown above), breeders [2], Turing Machines [3], and the many derivatives, this "game" has spawnd much thinking and imagining.

## A generation $\approx 10^{28}$ Turing Machine in Golly

Of course it's a challenge to replicate the wonders in a static book, but there's great potential for the CD, ebook, or website.

[1] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
[2] http://en.wikipedia.org/wiki/Breeder_%28cellular_automaton%29
[3] http://en.wikipedia.org/wiki/File:Turing_Machine_in_Golly.png

36
[+20] [2013-03-07 07:46:45] Robert Israel

Euclidean geometry was the first thing that got me (about grade 9 or 10). That's where I first found out that

1) There is such a thing as mathematical proof (rather than just calculation).

2) Mathematics is not a closed subject: new and interesting results can still be found.

37
[+20] [2013-03-07 13:48:14] Qwerty

This was my favourite equation. I was 16 or so, when my father showed it to me. I was amazed, and I programmed an application which drew this:

The interval should be <-6;6> maybe. I made it a looong time ago after all ;)

[1] http://imageshack.us/f/221/heartwl.png/
[2] http://imageshack.us/f/23/heart01q.jpg/
[3] http://imageshack.us/photo/my-images/268/heart21.bmp/

38
[+20] [2013-03-07 14:26:27] Kcvin

My favorite was when I was asked:

"If you were to save 1 penny on day one, and double your money for a month every day after that, how much money would you have?"

One I realized the answer was $10,737,418.24 I was flabbergasted. That was when I was able to understand that there is a mathematical model/equation for just about everything in this world; now that's beautiful. (4) My answer was going to be about that fable of the grains of rice on a chess board.. mathforum.org/sanders/geometry/GP11Fable.html Its very similar to your answer though. - gordatron 39 [+20] [2013-03-07 17:49:20] trex005 The following riddle blew my mind when I was a kid. Three men went into a hotel. At the front desk they were told that the room would be \$30, so they each gave \$10. After the men went to their room the manager realized they booked a room that was only \$25, so he gave the bell boy \$5 in ones to take back to the men. On his way, he thought, "5 can not be evenly divided by 3 men", so he pocketed two and gave the other three to the men, one to each. So, effectively each man paid \$9 for the room, a total of \$27. Remember, the bell had \$2 in his pocket. \$27 the men paid + \$2 the bell kept = \$29. Where did the extra dollar they paid go? (4) I love this one. Thank you. - Stu (4) What the hell?! Ah, I got it. Crafty.. - Thomas (4) Where did the extra dollar they paid go? Taxes! - BeniBela My father told me this one, and I was momentarily puzzled by it too. I wouldn't say that its goal is to show that math is beatiful, though; rather, it's intended to make one doubt about it. :-) - leonbloy (1) Great illustration of the need to check your sign. - BobStein-VisiBone 40 [+20] [2013-03-08 09:58:50] Deanna For me, it was topology [1], and beautiful Klein bottle [2] and Möbius strip [3]. Related to this was the realisation that a coffee cup is topologically identical to a doughnut: This still fascinates me to this day despite not being involved in advanced maths at all. Coincidentally, I learnt about this from a maths book for children written ~30 years ago :) [1] http://en.wikipedia.org/wiki/Topology [2] http://en.wikipedia.org/wiki/Klein_bottle [3] http://en.wikipedia.org/wiki/M%C3%B6bius_strip (7) Wait, is that a Klein bottle in a Klein bottle in a Klein bottle? - Joe Z. (6) @JoeZ. What does "in" mean, when you're talking about a Klein bottle? - Andreas Blass (1) You tell me. :P - Joe Z. I learnt about topology for the first time from a book called "Experiments in topology" by Stephen Barr. I was hooked for life on topology at that point! - Grumpy Parsnip I was talking to a friend once about how the No Retraction Theorem implies you can't get through a wall without putting a hole in it, and she replied "But a wall is a rectangle, and this is about a disk." At that point, I knew I was in too deep to get out. - Robert I've just found the book, and apparently it wasn't for children after all! Time Life, Mathematics (Found via the topology picture on this page. - Deanna 41 [+16] [2013-03-07 15:26:48] apnorton Two instances where I thought math was amazing: 1. In like 4th grade or whenever you learn areas of rectangles, one of the exercises in my book was to estimate the area of some squiggly shape overlaid on a rectangular grid. I thought this was pretty cool, and reasoned that if you could make the grid "smaller" (higher resolution), you could be more accurate. I mentioned this to my mom, who proceeded to tell me that was basically how Calculus 2 worked. :) That was very fun for me. 2. Deriving the quadratic formula in Algebra 1. That was fun--it showed that some totally un-intuitive formula could be easily found using other previously found results. (1) Your first sentence is incorrect. :) I had the "grid" epiphany, too! - Akiva Weinberger 42 [+16] [2013-03-07 20:37:18] user65621 The fact that Gabriel's horn [1] has infinite surface area, but finite volume, hence you can "fill it with paint, but you can never cover the whole surface". Gabriel's horn (also called Torricelli's trumpet) is the graph of$y =1/x$for$x\geq 1$rotated around the$x$axis. [1] http://en.wikipedia.org/wiki/Gabriel's_Horn 43 [+15] [2013-03-15 00:35:00] badjr In elementary school, my math teacher taught us this trick for the 9 multiplication values: (2) That's neat!${}{}$- Elmar Zander 44 [+13] [2013-03-07 07:14:51] amWhy It is really difficult to remember my days as an elementary student. I just remember how beautiful I found math to be: the connections I saw between everything I was learning, the beauty of the patterns, the sense-making, the sheer marvel of it all. I cannot pinpoint one "fact" I learned, or one particular "ah-ha!" moment (there were so many), but I attribute my love for math as much to the freedom I was given to actively inquire and explore mathematics, as much as to the many marvels I discovered in this way. I recall being encouraged (by remarkable teachers) to explore, ask questions, and try to find answers to those questions. I was given a lot a lee-way, apart from classroom lessons, to pursue the connections and patterns I saw, to conjecture, and confirm conjectures, or find counterexamples. Given this encouragement and flexibility, I found mathematics to be akin to solving mysteries. I wondered about what I was learning, and was able to anticipate what this would lead to, before receiving formal instruction. And this was terribly satisfying: the wonder, the pursuit, the discovery, and even "invention" (for myself) of things I would later find to be true. So in a sense, I discovered as much about math as I learned through formal instruction, and didn't get trapped into the mechanistic learn-a-rule/apply-the-rule/produce-an-answer mode which so many students come to define as "doing math." So it wasn't so much a matter of the facts I learned that drew me to, and keeps me enamored by, math: it was/is the activities of mathematics: the process of questioning why certain relationships hold, conjecturing, exploring, testing, discovering and chasing down implications, constructing an understanding, and defending or rejecting my hypotheses, and on and on... Nice Amy! Thanks for sharing us. - Resident Dementor 45 [+13] [2013-03-07 07:31:47] Michael0x2a For me, I suppose it was Pascal's triangle. I was first formally introduced to it in one of my high school math classes, where my teacher explained Pascal's triangle, and challenged us to find as many patterns as we could in it. We spent a decent chunk of time doing so, and I was amazed by how a simple rule to generate a simple pattern of numbers could yield so many interesting patterns and properties [1]. I also found it cool how Pascal's triangle could be used to solve a variety of patterns from binomial distribution to the problem where you try and find the total number of paths on a grid [2] assuming you can only travel in two directions, and demonstrated to me how mathematics is a lot more interconnected then I thought. [1] http://ptri1.tripod.com/ [2] http://betterexplained.com/articles/navigate-a-grid-using-combinations-and-permutations/ 46 [+13] [2013-03-07 16:33:48] Siméon When I was maybe 8 or 9, the following trick was showed to me as a sanity check for calculation by hand. 1. Take two numbers, let's say$358$and$77$. 2. Sum up the digits until you get a single digit number. $$\begin{array}{r} 358 \to 16 \to 7\\ 77 \to 14 \to 5\\ \end{array}$$ 3. Do the same with the two sums $$\begin{gather} 7 + 5 = 12 \to 3\quad\text{and}\\ 358+77 = 435 \to 12 \to 3 \end{gather}$$ 4. You get the same result? Try with other numbers. Be Amazed! 5. Best of all? It also works with products: $$\begin{gather} 7 \times 5 = 35 \to 8\quad\text{and}\\ 358 \times 77 = 27566 \to 26 \to 8 \end{gather}$$ I could not believe this always worked, it looked at the same time so beautiful and magic! A few years later, I was finally able to prove it by myself. I was so happy! (6) This method is called "casting out nines", and works due to modular arithmetic. - Joe Z. (1) Indeed, but I did not know modular arithmetic when I was 8. The name of the method where I am from would translate "proof by nine". - Siméon 47 [+12] [2013-03-07 16:22:08] Oliver E. Anderson First of all I must say that I really appreciate the idea of such a book. I wish I was exposed to such a book when I was younger as it was relatively late in my life (high school)I started appreciating mathematics. Anyway here is something I consider to be beautiful and simple, that you might find of interest: The Pigeon hole principle and it's applications. The pigeon hole principle goes something like this: Assume that you have some pigeons and some holes, and you want to put your pigeons into the holes, then if you have more pigeons than holes at least one of the holes must contain more than one pigeon. For example if I have 3 pigeons, but only two holes then one of the holes must contain at least two pigeons. The more mathematical way to state this is that if you have a set$X$consisting of$n$elements and another set$Y$consisting of$m$elements and$n > m$then there cannot exist an injective function from$X$to$Y$. Now this statement is fairly obvious and I am sure most people can understand this. But this statement shows up a lot in various disguises. Here is an example I think is pretty cool: Suppose a group of people are at a party. Each person may introduce himself/herself and shake hands with someone else at the party. I claim that there will always be at least two persons who have shaken the same amount of hands. Here is a proof of that statement: Suppose there are$n$people at the party. Then a given person can either shake$0, \;1 \ldots n-1$different peoples hands. That is$n$different possibilities, however if there is a person who shakes$0$hands (that is he doesn't shake hands with anyone) then there can't be a person who shakes hands with$n-1$persons (that is he shakes hands with everyone except himself), and conversely if a person shakes hands with everyone, then it is not possible that someone else doesn't shake hands with anyone. So there are only$n-1$possibilities but there are$n$people, so thinking of the people as pigeons and the possibilities as holes we see that we have$n$pigeons and$n-1$holes so at least two pigeons must go into the same hole, that is at least two people must shake the same amount of hands at the party. You can read more about the pigeonhole principle here: http://en.wikipedia.org/wiki/Pigeonhole_principle You can present this as a suitcase problem. If all your suitcases can hold five shirts, but you have six shirts, no matter how you pack them you will always need at least two suitcases. - Joe Z. While it's not exactly the same concept as the pigeonhole principle, it is definitely similar. - Joe Z. 48 [+12] [2013-03-09 12:15:51] Alessandro Jacopson I remember that when I was five, I made this reasoning "if I can write the digits$0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \space$then I will be able to write all the numbers". … in theory, that is. - k.stm (5) For reasonable values of all. - Chris Cudmore (1) If you can write 1+ and 0, then you can write all numbers! (Might take a while for a few of them.) - Akiva Weinberger 49 [+11] [2013-03-07 07:15:40] user64788 1) Modular arithmetic fascinated me. I could not believe that with just a few tools, I could find the remainder left when$3^{100}$is divided by 8. ($3^2\equiv 1$mod$8$and hence the result.) 2) Euclid's proof of the infinitude of primes. (Let the number of primes be finite. Let them be$P=\{p_1,p_2,\dots,p_r\}$. Take$k=p_1p_2\dots p_r+1$. None of the primes in$P$divides$k$, hence$k$is a prime or divisible by a prime not in$P$, and so we have a contradiction.) Modular arithmetic is definitely amazing. - Thomas The example 1) amazed me long time in a very similar way! - Gottfried Helms 50 [+11] [2013-03-07 18:00:44] OmnipresentAbsence I guess this is not as special as the other ones, but this is how mathematics amazed me for the very first time: I just turned 4 years old (I still vividly remember this), and my mother bought 4 cartons of eggs, a dozen per carton. My mother, trying to challenge me, asked how many eggs we bought in total, and after a short while I said 48 (I've always had a knack for arithmetic and I guess intuitively I knew it was$12 \times 4$). My mother was amazed, and she asked me how I did it. At this point I wasn't formally introduced to any mathematics (no multiplication and division, just basic addition and subtraction using our hands). So when I tried to show using my fingers how I got to 48 by taking 12 four times, it took me a lot longer, and my mother decided to teach me multiplication right then and there. This was the beginning of my interest. The more I think about this story, the more beautiful it gets. I implement the lesson I learn everytime someone says mathematics is useless!! Ask them to do$12+12+12+12$with their fingers. While a lovely story indeed, the last part will only reinforce the idea that mathematics is about calculating things, or making calculations more efficient. It's not. - Asaf Karagila (1) @AsafKaragila I agree, but it's the part of math everyone will get in school, and the part that everyone, no matter how 'uneducated', uses and finds useful. Of course, if one makes such a statement, you're probably not going to convince them showing beautiful mandelbrot sets, because those are fairly useless to a layman. - OmnipresentAbsence Do you try to convince people that poetry is useful? How about music? Mathematics is useful because it is interesting, not because it has convenient byproducts like calculations. - Asaf Karagila (1) @AsafKaragila No I don't, because music and poetry are barely ever mandatory, while mathematics is (everybody has to learn some mathematics, while music and poetry are hobbies). So convincing people poetry and music are useful is not needed. Of course, you can let people think what they think, but I think it's more beneficial to us all if people know how 'mathy' or world is. Comparing mathematics to music/poetry is like comparing biology to dancing, or physics to acting. - OmnipresentAbsence (1) If you've read the lament linked to in the question, you'll know that this sort of argument is exactly what kills mathematics education in schools. - Joe Z. @AsafKaragila I disagree. I don't think this story reinforces the idea that mathematics is only about calculations. He found a clever way to solve a problem and at that age what else could it be other than something with numbers and basic operations? From his point of view it wasn't just a calculation since he didn't even learn multiplication yet. At a young age numbers are the only things in math you can be fascinated with and there is nothing wrong with that. - Pratyush Sarkar @JoeZeng Sorry for the late reply, but I just read this paper and remembered your comment. What an absolute piece of mediocrity. The only lament I have is wasting my time on that paper. Full of melodramatic semi-truths and utopic mathematician's dreams. I'm a high school student myself and I can honestly say it's bullshit. Not even sorry for my language here. - OmnipresentAbsence @OmnipresentAbsence Lockhart would declare you a casualty, that you've internalized the attitudes that the teachers imposed on you about math. - Joe Z. 51 [+11] [2013-03-07 18:24:54] April Drake Solving for an unknown. 2x = 4 so x = 2. Beautiful. (4) It becomes hideous when forced to solve endless amount of them. - Phonics The Hedgehog 52 [+10] [2013-03-07 09:19:47] goblin Hilbert's infinite hotel [1], the realization that$\mathbb{Z}$is equinumerous with$\mathbb{N}$, and the uncountability of the set of all functions$\mathbb{N} \rightarrow \{0,1\}$. Basically: if it involves infinity, it's interesting. [1] https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel I think the uncountability of$2^\mathbb{N}$might be too complex conceptually for a children's book, even though the proof is simple. - Trevor Wilson 53 [+10] [2013-03-07 11:57:26] Szop Pracz When I was young I found a riddle: • Think about a number • Multiply it by 3 • Add 1 • Multiply it by 3 • Add the number you thought at the beginning • Tell me the result:) The number you thought about is your result without the digit 3 at the end, so i.e. if your result is 53, then you thought about 5. (1) This kind of joke was very commom im my school ^^ - Integral (1) @drjimbob: You're missing the "add the number you thought at the beginning" line, which makes the result be$(9x+3) + x = 10x + 3$. - ShreevatsaR 54 [+10] [2013-03-08 20:40:41] grep I remember my own observation about Pythagorean triples. I already knew that$3^2+4^2=5^2$and$5^2+12^2=13^2$, and realized that the same trick can be done starting with any odd number$n$, and the other two will be serial numbers that add up to$n^2$. For example, starting with$n=7$, we get$24+25=7^2$, and finally$7^2+24^2=25^2$. 55 [+9] [2013-03-07 19:29:57] Stu When I saw my first list of mathematical axioms (algebraic in this case). This was when I was 11. 1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal. 2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal. 3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal. 4. If equal quantities be divided by the same or equal quantities, the quotients will be equal. 5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered. 6. If a quantity be both multiplied and divided by another, the value of the former will not be altered. 7. If to unequal quantities, equals be added, the greater will give the greater sum. 8. If from unequal quantities, equals be subtracted, the greater will give the greater remainder. 9. If unequal quantities be multiplied by equals, the greater will give the greater product. 10. If unequal quantities be divided by equals, the greater will give the greater quotient. 11. Quantities which are respectively equal to any other quantity are equal to each other. 12. The whole of a quantity is greater than a part. It was an almost religious experience, as in "if you take these on faith, the rest can be proven". I compared these to axioms of religious faith. There was, is, nor will there ever be any comparison. In short, seeing this list sold me on rationality forever. Damned ! I'm not the only one ! I could have wrote every single word of this post... - Julien (2) "In short, seeing this list sold me on rationality forever." But what about those pesky Cauchy sequences that never converge? - Joe Z. (1) @Joe: SHUSH!!!!! - Stu Tell the Pythagorean 1-1 diagonal to shush. - Joe Z. I did, but it's almost like it slides right off. - Stu 56 [+9] [2013-03-07 19:58:06] Guilherme Gregores One of the things I really like in math is the probability. One of the best examples that I like is on the film 21 [1]. You are in a program show and you have three doors: One: With the prize, and the other two with monsters; The presenter tells you to pick up a door. When you finally choose a door, he asks "Are sure about it? Then for some reason he decides to open one of the wrong doors and asks you: "Are you going to stay with your door or change it?" and he says "But remember I know where the prize is". So what should have you do? [1] https://en.wikipedia.org/wiki/21_%282008_film%29 (10) Goats aren't monsters! D: - Joe Z. (1) Of course, switch. The first time I have 1/3 chance of being right. The other two doors 2/3. Now, one wrong door is opened, meaning that the remaining door has the unified 2/3 chance of being right. I am flappergasped how many people do not see this. This is where I demand them to envision this example with 1 million doors, you chose one at random. Then the moderator opens 999.998 other doors leaving you with your first choice, and the one that was not openened. Do you really think not switching would be smart? - k0pernikus (10) Isn't this called the Monty Hall Problem? - funkymushroom @kopernikus: I read that exact argument in a science magazine when I was 10, and I thought, of course even with 9999999 other doors switching does not matter......I guess intuition differs between people! - user54609 (1) I remember my teacher telling us this problem, but she had trouble explaining why it was that way. Most of my classmates didn't believe it was better to switch. But I wrote a Basic program on my Commodore 128 running 1000 iterations and found out that it really resulted in 50% chance of winning when switching doors. Several years later I stumbled upon a good explanation of why it is better to switch. - Anlo @Anlo: If you got a 50% chance of winning, you may have done something wrong. It should have been 67%. - user2357112 Ahh, right, I meant 67% :-) - Anlo 57 [+9] [2013-03-15 21:23:22] Raskolnikov The possibilities of abstraction. This liberated me. Until I was about 13, I always had trouble with solving problems involving proportionality and inverse proportionality. Until I learned about variables. When I realized that you could just put a symbol instead of the number you don't know and just perform computations with it until everything simplifies in a way you can find back the number, I had a feeling of unstoppability. The power of abstraction is so great that I'm very saddened by our current educational system in which it has nearly disappeared. All the students I get are struggling with symbols there were I have always seen them as my friends. "our current educational system" = "the current Belgium educational system"? Or a more general statement about the state of affairs? - Peter Mortensen @Peter Mortensen: to be honest, I only have first hand experience with the Belgian educational system, and a little with the Dutch and the French systems. But I have a feeling this generalises for Western education in general. - Raskolnikov 58 [+9] [2013-10-19 21:02:38] user67133 When I was a kid I realized that $$0^2 + 1\ (\text{the first odd number}) = 1^2$$ $$1^2 + 3\ (\text{the second odd number}) = 2^2$$ $$2^2 + 5\ (\text{the third odd number}) = 3^2$$ and so on... I checked it for A LOT of numbers :D Years passed before someone taught me the basics of multiplication of polynomial and hence that $$(x + 1)^2 = x^2 + 2x + 1.$$ I know that this may sound stupid, but I was very young, and I had a great time filling pages with numbers to check my conjecture!!! As a hint, you could always say$\sum(odds)=x^2$. - Simply Beautiful Art 59 [+8] [2013-03-07 08:10:42] uday1889 I believe it was when I was in 5th grade. I used to enjoy adding the digits of the plate numbers of vehicles until it resulted in a single digit result. I was excited to realize that all I had to do was eliminate nines from the number. Example 9468 is 9 (removing 9,6+3, 8+1), 3454 is 7 (what remains after removing 5+4). It's simple but it sure made travelling fun for me. (3) That (including the removing of nines) was almost a compulsion for me... well,. it still is (i'm 46) - leonbloy I still do it too...everytime I'm driving and I have things weighing on my mind...I guess some things never stop entertaining you... - uday1889 In Ontario, license plates only have three digits on them :\ - Joe Z. Then you have a better task ahead of you. Represent the alphabets as numbers and concatenate then add. I envy the fun you're gonna have :) - uday1889 I do it all the time too! :D - Y Y 60 [+8] [2014-02-27 20:30:05] debap I must have been very small, around three of four, when I suddenly dashed out of my room, full of excitement, wanting to show my dad something that had made a great impression on me. I held a book, it's front cover facing me. In a flash, I gave it two half-turns. One upside-down, the other left to right. This is what came out: I held my breath, as the trick wasn't over yet. Sure enough, two same quick moves and -lo and behold- the front cover was facing me properly again, just as in the beginning. "Look! Dad!" :) That surely must have been my first conscious encounter with symmetry. I held the memory dearly close for a number of years but then forgot about it completely. It came back to me, only very recently, after going through the first pages of Nathan Carter's Visual Group Theory [1] and seeing this image: [1] http://rads.stackoverflow.com/amzn/click/088385757X What was Dad's reaction? - Did I remember that he was sick in bed, and smiled. - debap 61 [+7] [2013-03-07 09:13:29] markovchain It's not the first one that made me love math (what made me love math isn't math itself at all, but rather someone pointing out to me that I was pretty good at math -- and then I proceeded to like math haha), but this is the most amazing discovery I made when I was 15. $$Pr(X = r) = \frac{(nCr)(x-1)^{n-r}}{x^{n}}$$ Which is really just a restatement of the binomial distribution: $$Pr(X = r) = (nCr)(p^{r})(p^{n-r})$$ where$p = 1/x$, so it works only makes sense for integer values. For example, the chances of choosing one blue jar out of 10 differently colored ones would be$x=10$, but also$p=0.1$. I discovered it after one week of exhaustively listing down all the permutations of the letters n, t, g, and b and figuring out what patterns they looked like when you took only 1, 2, 3, and 4 elements. Then I went ahead and added more and more letters until I arrived at that formula by inspection. In my opinion, it isn't the math itself that makes kids dislike math. It's all the people around them who dislike math who make kids dislike math. I agree with that opinion! You should check out the OP's link to Paul Lockhart's essay. - Will 62 [+7] [2013-03-07 10:45:42] Angelika Complex numbers was awesome to me  i² = - 1  (1) $$\lim_{n\to\infty}\left(1+\frac in\right)^n=\cos1+i\sin1$$which can be proven geometrically. Expanding it out with the binomial theorem gives us series representations of the cosine and some of 1. I thought I'd share it with you, because it's fairly amazing. - Akiva Weinberger If you haven't seen limits yet, I'm basically saying that$(1+\frac i{1000})^{1000}\approx\cos1+i\sin1$, and that it gets more and more accurate as you replace 1000 by bigger numbers. (EDIT: radians, of course.) - Akiva Weinberger 63 [+7] [2013-03-07 15:20:32] user65551 Pi has always fascinated me. The notion that perimeter of every possible circle imaginable divided by its diameter always results Pi is astonishing. 64 [+7] [2013-03-07 20:20:34] Mr.Mindor Not sure if it was the first, but one very early one for me was realizing if you knew the square of an integer, you could easily step to the next one by adding the known square, the original integer and the next integer together. Know$2^{2}$Want$3^23^2 = 2^2 +2 +3 3^2 = 4+2 +33^2 = 9$or$(n+1)^2 = n^2 + n + (n +1)$Another although much later point for me was when Calculus just clicked (4th time taking Calc 1). It was like cracking The Matrix. I could see derivatives and integrals in everything around me and the relationship between the trig functions suddenly made sense. Also the magic of Fourier and Laplace transformations. 65 [+7] [2013-03-07 22:59:29] dtldarek I discover that math is beautiful again and again, I suspect there is no end to this. Some of things that blew my mind: • The very first thing that hooked me to science in general was the ability to model reality. When I was a child my father showed me how differential equations (by then I had no idea what it was) could simulate reality (it was masses connected by springs and dampers with some simple graphical representation). • Prime numbers. The natural numbers are intrinsic to the world and so a prime numbers. If there is a sentient alien race, they are aware of prime numbers (or they will be someday). • The numbers$e$and$\pi$, their interconnection, and importance in both continuous and discrete worlds. • Fourier transform (both continuous and discrete). There is no denying: Fourier transform is just awesome, the sheer number of applications, implications, similar transforms, etc. speaks for itself. • The 3D proof of Desargues' theorem [1]. It's one of those illuminative cases where considering a harder case simplifies the problem. • The probabilistic method [2]. The first time I saw it, it was very inspiring, also the person of Paul Erdős [3] is very inspiring too (and funny, e.g. the title of "Permanent Visiting Professor"). • Randomness, randomized algorithms, and the P vs. NP problem [4]. There is something incredible in the fact that for some problems the only fast solutions we know are randomized, any known deterministic approach is way slower (the canonical example being checking if the symbolic determinant of a matrix containing variables is zero). • ... this ever-growing list continues... Cheers! [1] http://en.wikipedia.org/wiki/Desargues%27_theorem [2] http://en.wikipedia.org/wiki/Probabilistic_method [3] http://en.wikipedia.org/wiki/Paul_Erd%C5%91s [4] http://en.wikipedia.org/wiki/P_versus_NP_problem 66 [+7] [2013-03-08 11:25:01] Ben As a child, I liked drawing. When I realized that there was an easy way of telling whether it is possible to draw a given figure in a single stroke, I was intrigued. I read this in a popular mathematics book and it can be easily explained to a child. (if there is 0 or 2 intersection with odd degree, the figure can be drawn in a single stroke) I discovered that by myself, at about eleven, but at the moment didn't imagine it could be a mathematically relevant fact. - leonbloy 67 [+7] [2013-03-08 21:14:29] rundavidrun $$\sqrt{\sqrt{\dotsb\sqrt{x}}} = 1$$ (or its more precise version:$lim_{n \rightarrow \infty} \sqrt[n]{x}$, for x positive) As a kid I would always type in a number in my calculator and then keep hitting the square root key until the display went to 1. I would also do this with other keys on the calculator to see what would happen (some would blow up past the capacity of the floating point storage and some would go to 0, some to 1). (5) Actually$\sqrt[n]{x}\to 1$, as$n\to\infty$. - Asaf Karagila 68 [+7] [2014-08-21 10:28:23] RE60K Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae. The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the GrafEq formula-graphing program he developed. The formula is an inequality defined by: $${1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$ where$\lfloor \cdot \rfloor$denotes the floor function and$\mathrm{mod}$is the modulo operation. Let k equal the following 543-digit integer: 960 939 379 918 958 884 971 672 962 127 852 754 715 004 339 660 129 306 651 505 519 271 702 802 395 266 424 689 642 842 174 350 718 121 267 153 782 770 623 355 993 237 280 874 144 307 891 325 963 941 337 723 487 857 735 749 823 926 629 715 517 173 716 995 165 232 890 538 221 612 403 238 855 866 184 013 235 585 136 048 828 693 337 902 491 454 229 288 667 081 096 184 496 091 705 183 454 067 827 731 551 705 405 381 627 380 967 602 565 625 016 981 482 083 418 783 163 849 115 590 225 610 003 652 351 370 343 874 461 848 378 737 238 198 224 849 863 465 033 159 410 054 974 700 593 138 339 226 497 249 461 751 545 728 366 702 369 745 461 014 655 997 933 798 537 483 143 786 841 806 593 422 227 898 388 722 980 000 748 404 719 If one graphs the set of points$(x, y)$in$0 \le x < 106$and$k \le y < k + 17$satisfying the inequality given above, the resulting graph looks like this (note that the axes in this plot have been reversed, otherwise the picture comes out upside-down): 69 [+7] [2014-11-20 02:20:52] alexy13 Parallel lines. I was amazed to find out that they would never, ever meet. 70 [+6] [2013-03-07 07:09:36] Aang These amazed me quite a lot when I first saw them:$1.$Prove that$|(a,b)| =|\Bbb R|$,$\forall a,b\in\Bbb R$and$a<b$.$2.$Both$\Bbb Q$and$\Bbb R\setminus \Bbb Q$are dense in$\Bbb R$, but$\Bbb Q$is countable set while$\Bbb R\setminus \Bbb Q$is uncountable. (8) At what age?${}$- mrf In high school( then, i was around$12$) - Aang I didn't even know the cocept of negative number at that age, not to mention real numbers. - user1551 (2) My brother used to tell me these kind of things from an early age. - Aang (2) You learned about dense, countable and uncountable sets at that age?? Its hard to believe...and its very strange that you have to get so far to see something you considered beautiful. - Integral I didn't know what density was till I was about 19, when I was a child there was no way I would have had any idea! I think as someone already said I was probably still trying to get my head around negative numbers! In fact thinking about it it would be quite a while till I knew what the reals and rationals were! - hmmmm What is your first point supposed to mean? Am I the only one who can’t parse this? - k.stm @K.Stm. The first point is that the size of any open interval is the same as the size of all the real numbers. This can be seen by stretching a function like tan to create a bijection between the two. You can read more here: en.wikipedia.org/wiki/Cardinality_of_the_continuum - tttppp (3) @tttppp … I read$(a,b)$as an tuple and completely forgot it could also denote an open interval (which I write$(a..b)$now) – even after searching for different meanings! It’s times like this that I wonder whether I have some serious brain condition. - k.stm @K.Stm. Ah ok! So |(a,b)| = 2 :-) - tttppp 71 [+6] [2013-03-07 08:48:45] nuthan Pythagorean theorem If$(a, b, c)$is a Pythagorean triple, then so is$(ka, kb, kc)$for any positive integer$k$.$3^2 + 4^2 = 5^2$72 [+6] [2013-03-07 15:56:19] Integral In my school when I learned about Cartesian coordinate system was shocking! Because that was the time I learned that was possible to make drawings with numbers. Unlike most people here, I didn't have so much fun playing with numbers, but everything changed when I realized that I could convert numbers (more precisely, ordered pairs) in drawings over the coordinate plane. And YES, that was a lot of fun. 73 [+6] [2013-03-07 18:04:35] Mike Graf My two favorite mathy things (not mentioned in other answers) were the powers of 11 11 ^ 0 = 1 (1) 11 ^ 1 = 11 (1, 1) 11 ^ 2 = 121 (1, (1+1), 1) 11 ^ 3 = 1331 (1, (1+2), (2+1), 1) 11 ^ 4 = 14641 (1, 1+3, 3+3, 3+1, 1) 11 ^ 5 = 161051 (1, 1+4, (bump 1 to left) 4+6, (bump 1 to left) 6+4, 4+1, 1) 11 ^ 6 = 1771561 (1,1+6, 6+1, 1+0, 0+5, 5+1, 1)  and the estimated relationship btwn powers of 2 and powers of 10 and how they diverge (hard drive manufacturers think this is beautiful :P ) 2^10 ~= 10^3 (1024, 1000) "kilo" 2^20 ~= 10^6 (1048576, 1000000) "mega" 2^30 ~= 10^9 (1073741824, 100000000) "giga" 2^40 ~= 10^12 (1099511627776 , 1000000000000) "tera" 2^n ~= 10^floor(n/3) (where n is a multiple of 10)  (3) If you look closely, the first few powers of 11 are rows of Pascal's Triangle. This is not a coincidence. - Joe Z. Hell, I just noticed that you can consider this as kind of a discrete convolution or polynomial multiplication! - flawr 74 [+6] [2013-03-07 20:52:29] Michael T I found a book by Isaac Asimov called "Asimov On Numbers" which is a compilation of his essays related to math and numbers. It was all very fascinating - things like why Roman numerals [1] are inefficient, why zero was such a groundbreaking number to invent, and things like that. You might want to see if you can find that book to get some inspiration. [1] http://en.wikipedia.org/wiki/Roman_numerals 75 [+6] [2013-03-08 18:15:20] AlexHeuman I remember reading a magazine when I was a kid that asked this question. If you have 6 pieces of spaghetti that extend as long as you please, and you cross them so that they create as many overlaps as 6 sticks allow. How many cross points do you have. Then it asked how many would it be for 17 spaghetti sticks, could you figure it out for any number of spaghetti? And I remember concluding that$\frac {n(n-1)}2$is the formula for finding the answer. I was excited at the time. Looking back now I see how elementary that was. Here is a visual way to see it: So, I started by drawing out a strand of spaghetti assuming that they could be as long as you please and as thin as you please, then I start with one and work my way up to 5 counting how many times they cross. So then for 5 sticks of spaghetti I labeled all of the crossings. I did this for 6 as well just to see what was happening. I noticed that the number of crossings on each strand of spaghetti was the number of total spaghetti - 1 because it didn't cross itself. So from now on I will refer to the number of spaghetti as n. So to count the number of crossing I knew it was$n-1$crossings for every stick and$n$sticks so the total number of crossings was$(n-1)n$and I noticed that each crossing occurs on two separate sticks, because one crossing is the crossing of two sticks to the total number of crossings is half of the number in the diagram so it was$\frac 12 (n-1)n = \frac {n^2-n}2$P.S. sorry for using the words crossing and sticks instead of points and lines. It was something that stuck because of the spaghetti analogy in my head. I didn't realize I was doing it until it was too late. (1) Another different but related question is how many pieces of a round pie can you cut using$n$straight cuts. The answer turns out to be$\displaystyle \frac{n(n+1)}{2} + 1$. - Joe Z. 76 [+6] [2014-01-09 22:00:01] Michael The fact that$\Bbb C$is algebraically closed. About 12 years old, after I just learned about quadratic equation such as$x^2=a$may or may not have solutions, my mother told me about complex numbers: you attach the number$i=\sqrt{-1}$to real numbers and after that$x^2=-1$have solutions. "Nah", I said, "that doesn't help much: although you now have solutions for$x^2=a$for$a$in the old number system, which are reals, you still don't have solutions of$x^2=a$in the new number system, which are complex. You still don't have a solution of$x^2=i$, for example. And having complex solutions for some of the complex numbers is no better than having real solutions for some of the real numbers." Then she showed me the roots of$x^2=i$and explained that$x^2=a$has complex solutions for any complex$a$. The ingenuity of the complex numbers impressed me a lot. Then she told me about polynomials of degree higher than 2 and that they all have roots in the same field of complex numbers, that you don't need to "attach"$n^{th}$root of$-1$or any other number in order to have any polynomial of degree$n$have roots, that$\sqrt{-1}$is sufficient for them all. And I was impressed even further. Slightly relevant: consider the "Dual numbers," which are the real numbers with the addition of$\epsilon$($\epsilon$), such that$\epsilon^2=0$. In this new number system, you can't always divide ($\frac1\epsilon={}?$), but it's still an interesting thing to work with. (Less useful than the complexes—I think—bit still cool. Note that$p(x+\epsilon)=p(x)+p'(x)\epsilon$. Try deriving the product rule with this.) - Akiva Weinberger 77 [+6] [2015-02-12 23:13:57] hasnohat I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers. I was granted immunity from this idea at a very young age. When I was in 3rd grade (possibly 4th, my memory is a little fuzzy) my father showed my sister and I knotplot (http://knotplot.com/), a piece of software that one of his friends was working with. He explained that his friend was a knot theorist, a kind of mathematician who studied knots. I had no idea what that really meant, of course. All I knew was that studying knots was apparently a thing that mathematicians did. It wasn't a lot, but it was enough. Whenever the North American math curriculum tried to trick me into thinking that math was about numbers, my brain would reply "You say that... but what about knots?" It was my vaccine. It also helped that my physicist father insisted on teaching us the interesting bits of math whenever it seemed appropriate. He told us about negative numbers the moment we learned about subtraction (I then obnoxiously quizzed my classmates on the playground. "What's$5-6$?", "Zero?", "Nope! Minus one." It's a wonder I didn't have more friends...). He showed us imaginary numbers when we learned about squares (the mysterious$i$became another mental vaccine.) When I mentioned$\pi$, he countered by telling me about$e$. The thing about math is, once you have a general curiosity, you'll start discovering interesting things all over the place. Once that starts happening, you're doomed. Math will never be boring again. 78 [+5] [2013-03-07 07:05:52] Sniper Clown • Like Trevor Wilson, I was awed by human ingenuity where just by looking at "few" given numbers, one can deduce that there are infinitely many primes from reason alone and doing basic operations. ( Here is more on Euclid's proof [1].) • As freshman undergrad in college, always loved Theoni Pappas' Joy of Mathematics before being introduced to Raymond Smullyan, Charles Seife (Biography of Zero), Rudy Rucker and Hoftstadter's books. • As far as Math.SE's question is concerned, this was an interesting brain teaser and simplicity at best [2]. ===================EDITED THE FOLLOWING BELOW==================== I just realized although the above have been influential, but earliest memory of the workings of mathematics came in the manner of following magic trick aged six or seven: Effect: Performer asks someone to write down a random long number 4567829872367783456753745673456347567346534756 and he writes the another line of the matching digit, so let's say she writes 1263347567346534756378567563434543534543534545 and after that performer asks another audience member to approach the blackboard on dimly lit stage. Let's say the next random line 8636652432653465243621432436565456465456465454 and random number and as he writes 5555555555555555555555555555555555555999999990 the performer quickly writes below 4444444444444444444444444444444444444000000009 and then pauses. Then he continues his patter: Now I could not have possibly known what digits you would have chosen, right ladies and gentlemen? Well, let me gather my thoughts for a while and clear my mind....as I attempt to add this rather cumbersome mess in just the time of writing it down. Then he approaches the blackboard and without hesitation calculates the answer: 24567829872367783456753745673456347567346534754 which of course proves to be correct. Method: There is of course no telepathy and the trick is entirely mathematical in nature. If the reader wants the audience to choose the first line make sure the last digit does not end in 0 or 1. So what the practitioner would do is matching the digits of audience write the complement of the number adding to 9. Say the line is of a 10-digit sequence of 5s then the performer should write a 10-digit sequence of 4s. To add the whole block one simply copies down the first line with 2 in front of it and subtracting 2 from the last digit. Hence the need for no 0 or 1 in the first line. Tips: To make it realistic, make sure it is a cumbersome mess that is not too big of a block. Because say one smart aleck chooses all digits of 0000... and then when performer writes 999999... it may be a give away. Strike a balance between how big the mammoth block should be to appeal awe from students and the reality of randomness in the numbers. The rest is, of course, all up to the showmanship of mentalist. [1] http://primes.utm.edu/notes/proofs/infinite/euclids.html [2] https://math.stackexchange.com/questions/87870/are-the-solutions-of-xxxx-cdot-cdot-cdot-2-correct (1) +1 for the first bullet point! I had the same enthralling reaction to the Euclid's proof that there are infinitely many primes. It was the first time in my life that I realized there was more to mathematics than mindless calculation, and it fundamentally changed the course of my life. - user5501 79 [+5] [2013-03-07 07:27:36] user63181 I remember in geometry using direct reasoning once and another by the absurd, and I manage to show that lines are parallel, intersecting at a point, a triangle is isosceles, it is inscribed in a circle ..... I was fascinated by geometry. 80 [+5] [2013-03-07 11:23:59] Loki Clock I didn't so much discover mathematics was beautiful as I discovered everything I found beautiful was mathematics. I would have said the dodecahedron was my favorite elementary shape when I was little, but as a teenager I was exposed to the fourth dimension. I became obsessed with symmetry and analogy-based objects, and with the Johnson solids, which gave me a then-ineffable feeling of filling out the quality of symmetry that it always creates good shapes, that there are rules you can make that name exactly the set of good shapes. That "looks right" and "looks wrong" can be made precise, and hence the feeling can be explained, you can learn what it is you're noticing about those shapes that you wouldn't feel when you look at a 3D mesh of a face or a blanket. 81 [+5] [2013-03-07 13:26:18] Soul Reaver To me, it was probably an old animated book "Och ta geometria" [1] (eng. "Oh that geometry") written and illustrated by Zlatko Šporer, Nedejko Dragić. In the form of funny comix (check the link for samples), this book introduces basics of geometry from points, segments (not sure if this is the correct name), lines, flat figures and their area to cartesian coordinate system. This was probably the catalyst for my interest in math. [1] http://biblioteka-hades.pl/aukcje/9211/zdjecia/20893 What a wonderful book! Is there a way to see more pages? - Georges Elencwajg It's pre-web- and pre-pdf-era book ;) So it can be hard to find it now in digital form. I've got lucky to find link I posted. You can try looking for it by authors' names, here is link to google books info about original book (mine was a polish translation). btw, I've found this book in polish auction site archive, and it's price was below 1 EUR ... very good book for a very nice price - Soul Reaver Thanks for the information, @Soul Reaver. - Georges Elencwajg 82 [+5] [2013-03-07 15:11:09] Joe Z. If you've ever heard of$3,529,411,764,705,882$being multiplied by$3/2$to give$5,294,117,647,058,823$(which is the same as the 3 being shifted to the back), you might consider including that in the book. There are lots of other examples, like$285,714$turning into$428,571$(moving the 4 from back to front) when multiplied by$3/2$, or the front digit of$842,105,263,157,894,736$moving to the back four times in a row when you divide it by$2$. (There's a leading zero in front of the last term, though.) Difficult to observe without e.g. the unlimited precision of Python. - BobStein-VisiBone (1) Or without some good ol' pencil and paper. - Joe Z. (2) Here's the trick: Take the decimal expansion of 1/(any prime number). That will give you lots of ratios to work with. - Joe Z. 83 [+5] [2013-03-07 15:55:24] Peter For me it was when I realized that with sine and cosine I could draw a circle! (2) And then experimenting with those functions to draw cool parametric shapes on the calculator ^^ - Thomas 84 [+5] [2013-03-07 23:17:51] Godot In the elementary school, when I was learning about maps on the geography lessons, I was amazed by the concept of the scale of a map. It might sound silly now but I was very happy when I understood the relation between ratios of distances and ratios of areas (and ratios of volumes:). It somehow provoked me to thinking about what length, area and volumes really are, how to define them. And how to define what a map is. Of course I got familiar with precise definitions much, much later :) 85 [+5] [2013-03-08 11:14:48] mk.. I felt like an Einstein [1] and was really interested in mathematics when I myself discovered the truth behind a^0 =1. That is, a^0 = (a)^(1-1) = a^1/a^1 = 1 Yeah, I know this is simple.. But generally it is taught as a formula. Instead this one can be used to change the way of thinking... Also, multiplication is repeated addition... This used to fascinate me a lot... 2 * 3 = 6 that is, 2 + 2 +2 4 * 3 = 4 + 4 + 4 5 * 8 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 And then in the end you can say that, for very big numbers, you cant sit adding all of them and hence, multiplication is the shortcut to add all of them :) I am not a writer... But probably you can take some god examples to explain what I am trying to say here... I think this will be really interesting approach to teach multiplication! All the best for your book. Do let us know the name of the book. We will also cherish it... :) [1] http://en.wikipedia.org/wiki/Albert_Einstein Continuous? Do you mean repeated? - Asaf Karagila Yaa, repeated..!! - mk.. y down vote? You dint like it, thats y?? Funny.. - mk.. @Sandy, I liked it +1 - user58512 86 [+5] [2013-03-08 17:39:13] Todd Wilcox When I was in second grade we memorized the times tables through$9$. At the very end of the year, our teacher taught us simple single-digit division. I was floored: "We can reverse multiply?!?!" I think that got me to pay more attention in math. The first thing that really cemented my love of math was learning set theory in seventh grade (widely reviled as "the new math" by parents and politicians in the U.S.). I wasn't hooked for life until 11th grade when we were given the definition of a relation as a subset of the cartesian cross-product between two sets. I still remember getting chills when I understood that. I personally know that the way my seventh-grade class taught set theory was woefully inadequate, especially the way they introduced us to number sets. You don't define the real numbers as the union of the rationals and irrationals! - Joe Z. 87 [+5] [2013-03-09 09:03:59] pwned If I wrote a book, a few pages would be dedicated to visualizing square root through blocks like this: A kid can put cards on a table and count the edge rectangles to figure out the approximate square root of any number. With the help of some legos you can even teach cube root! 88 [+5] [2013-07-20 18:06:50] Gil Kalai I found the formula$(a+b)^2=a^2 + 2ab + b^2$that my father told me at a young age fascinating. (And also that$(a+b)(a-b)=a^2-b^2$.) Overall, it seems that a parents duty is to teach his children two of the following: (a) to ride bicycles, (b) To play chess, (c) The formula for$(a+b)^2$, and my father took (b) and (c). My mother let me read her high-school calculus book (incidentally one of the authors had the same last name as mine) and there what I found really fascinating (but I could not understand) is that you can add a variable to a triangle. (This was a misunderstanding of what$f(x+\Delta)$means.) 89 [+5] [2013-08-09 13:37:52] Gerard This might seem very elementary: but amazed me when I was a child. The fact that$a\times b = b \times a$. I would keep drawing boxes on the number line of different lengths and then discovering that they fit snugly into one another. Still seems amazing. 90 [+5] [2014-01-10 10:15:11] Alexander Frei The fact that you can add natural numbers successively in the order you prefer and that you can split subtraction: ...I remember that day when got taught this in class which made me really excited so I had to tell my Mum =D I love diagram, haha :) - breeden 91 [+5] [2014-01-30 20:12:05] Unwisdom I have to admit that although I'd frequently been told that mathematics was "beautiful", I didn't really get that while I was in school - even high school. I enjoyed mathematics, and saw plenty of things that were fun, and even cute, but I never really understood any ideas with sufficient depth to think of them as beautiful. When I did encounter ideas that I found beautiful, it was in my first year at college. In fact, there were two closely related ideas in quick succession. We were just being introduced to vector spaces. This was the first time I'd seen an abstract space, but it didn't really seem to mean much except as a fancy way to talk about high dimensional Euclidean spaces. But then I saw my first example of a vector space that didn't just look like the vectors I'd seen in high school. It was the space of infinitely differentiable functions:$C^{\infty}$. We were shown the linear operators associated with two common differential equations (exponential growth and simple harmonic motion): \begin{eqnarray} &\frac{\textrm{d}\phantom{y}}{\textrm{d}t} - kI \\ &\frac{\textrm{d}^{2}\phantom{y}}{\textrm{dt}^2} + kI. \end{eqnarray} We saw the fairly routine proofs that these were linear operators on$C^{\infty}$, but then came the magical part: The solution sets to these differential equations were subspaces of$C^{\infty}$, the canonical solutions I was familiar with were basis sets for these solution spaces, and the solution spaces were actually the nullspaces of these operators! Later (maybe even in that same lecture) we saw how linear regression - the hitherto tedious process of finding the "line of best fit" - could be understood as a linear projection$P$operator onto a two dimensional subspace of the data space. Given a data vector$\mathbf{x}$, the projected vector$P\mathbf{x}$represented the line that was closest to the data - the line of best fit - and the difference term$\mathbf{x}-P\mathbf{x}$represented the error term. I was astonished at how much more elegant this was than the clunky formulas I'd had to memorize in high school. 92 [+4] [2013-03-07 10:48:54] wibix I think the first thing that amazed me in this way was$\pi$. An irrationnal number, which means it has an infinite number of digits, which involves humans can't manage it, we can't know it on the whole, but already the Greeks discovered it. They knew it has something to do in the circumference or the area of a circle, that is, they could manipulate it, and I find this unbelievable. 93 [+4] [2013-03-07 13:46:53] HerrSerker My mother repeatedly tells this story about me. In German television there is a series called Telekolleg [1] (not Kellog you silly, more like in college) which is broadcated for remote learning. One series deals with Math. I was about 5 or 6 years old, when I sat in front of the TV watching this Telekolleg Mathematik series, turning to my mother and insisting: 'This is a good programme, you have to watch this'. I don't remember what the exact topic was, perhaps quadratic function graphs. [1] https://de.wikipedia.org/wiki/Telekolleg#Fernsehlehrg.C3.A4nge 94 [+4] [2013-03-07 13:50:13] kanchirk The first interesting mathematics problem I remember in my limited memory is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... It never totals to TWO :-) (1) Sounds like Zeno's Dichotomy paradox. - BobStein-VisiBone 95 [+4] [2013-03-07 13:59:54] Qwerty This was probably the very first mathematic riddle which absolutely got me. It is called Algebrogram in my language, but I couldn't find a reference in English. I was attending mathematic group after normal school (at age 11-14) and then I made few of my own for my classmates. I loved it ^^ You use characters instead of numbers and you construct some words. You then let others solve it. F O R T Y T E N T E N --------- S I X T Y  Solution: 2 9 7 8 6 8 5 0 8 5 0 --------- 3 1 4 8 6  It was common to construct sentences as well, but it is kind of hard. This is only an example, which is unsolvable ;) You could specify if there were some other operations or you could let your solvers find it out by themselves.  O U R H O U S E H A S - T E N ------------- W I N D O W S  (2) In English these are sometimes known as cryptarithmetic puzzles. - Will @Will Thank you very much!, sir. ;) - Qwerty +1! As a kid I loved the Sideways Arithmetic From Wayside School books because they were full of these fun puzzles. amazon.com/Sideways-Arithmetic-Wayside-School-Sachar/dp/… - Mattias Andersson In computer science, they are known as constraint problems for which there are constraint solvers. This one has "all variables are different", and "1100 O + 110 U + R + 10100 H + 11 S + 10 A + 100 T + 11 E + N = 1000000 W + 100000 I + 10000 N + 1000 D + 100 O + 10 W + S" constraints. - pwned 96 [+4] [2013-03-07 14:53:35] Joe Z. I'm not sure there was a first bit; realizing the beauty of mathematics was a gradual process for me, turning it from a fun little thing I was doing into a full-fledged appreciation. One of the more recent things, I suppose, is some of the patterns that appear in modular arithmetic. The concepts of continued fractions and aliasing in signal processing are closely related. When continuously adding 9 to a number, the ones digit appears to decrease by 1 constantly. If you mark all the multiples of 3 on a 10-by-10 grid, they form diagonal stripes down the page. Things like that, which actually have quite significant uses in real life, are things that make math beautiful (and tricky!) to me. 97 [+4] [2013-03-07 16:44:27] Nick I like cars and automotive racing and such. What got me real interested in it were two things: The first, to a great extent, in Calculus: •$\displaystyle \frac{d}{dt}\ \text{Displacement} = \text{Speed}$•$\displaystyle \frac{d}{dt}\ \text{Speed} = \text{Acceleration}$•$\displaystyle \frac{d}{dt}\ \text{Acceleration} = \text{Jerk}$It all made sense to me after that! Then there was a problem in my Cal. book about calculating the force of a piston in an engine. I can't quite remember it, but it was basically:$\text{Force} = \text{RPM}^3$or something similarly extreme. It relates to the automotive aphorism: Power doesn't kill motors, RPM does. (3) So whenever I commented that someone is a jerk, I was deriving this from their acceleration? :-) - Asaf Karagila (1) @AsafKaragila Sir, I'd like to present to you your well deserved award, 'worst joke ever'! - OmnipresentAbsence (2) @OmnipresentAbsence: For this? Nah, I have had infinitely worse jokes before, and in probability$1$I will have infinitely many more. - Asaf Karagila (1) @AsafKaragila I know, I'm just kidding. I actually chuckled because of how cheesy the joke was - OmnipresentAbsence (1) @AsafKaragila, Hey, "Jerk" is what the book mentioned, not my word! You d/dx Acceleration. - Nick (2) d/dt jerk is called "jounce", I believe. - Joe Z. @JoeZ. Yes, that is correct. It is also known as snap I believe and it is followed by crackle, pop, lock, and drop. It just sounds cooler with snap instead of jounce. Also,$\int\text{Displacement}dt=\text{Absement}$- Simply Beautiful Art 98 [+4] [2013-03-07 17:29:19] Udo Klein I was completely baffled when I learned the approach of C.F. Gauß for summing 1+2+3+...+100. Of course I would have gone for the hard way as well and I was deeply impressed when I learned that this equates to 1+100 + 2+99 + 3+98 + ... = 50*101 = 5050. The next big thing for me was when I discovered that you can reduce multiplication to looking up squares by the identity a*b = ((a+b)(a+b)-(a-b)(a-b))/4 However by that time I was already hooked. 99 [+4] [2013-03-07 18:03:40] Rich Homolka Though a lot have been said (I too worked out Pascals triangle as a kid) no one has (yet) mentioned Gauss' method for adding sequential numbers. It may be apocryphal [1] but the story I heard was that a teacher wanted busy work, so she told the class to add the numbers 1-100, thinking that would take forever. Gauss was smart, he knew that the pair 100+1 was the same as the pair 99+2, the same as the pair 98+3... and now that he paired these numbers off, he now had 100/2 or 50 pairs of them. 50 pairs of 101 was 5050. He told the teacher the answer way before it was expected, and shocked them. The coolness of the story is that it's probably at the level of your audience, something they can do and experiment with. and the guy's a legend too. [1] http://www.jimloy.com/algebra/gauss.htm (1) There had been a mentioning of Gauss, more than one person actually. Look closer. - Asaf Karagila See americanscientist.org/issues/pub/gausss-day-of-reckoning which researches the earliest versions of this story. They have rather little in common with your version. - Robert Israel 100 [+4] [2013-03-07 19:35:40] Ark The simple and commonly used sum, and divide of apples. I was really bad at math, and using objects instead of numbers really teached me how to love (math, LOL). It's amazing how math can be used on anything. Haha, when I had to 'teach' my fellow 8th graders Algebra, I would commonly answer their question by saying "One zebra plus two zebras equals 3 zebras" and I would mess around with the numbers as needed. Also, when asked what$x+y$equaled, I might reply "[that] a zebra plus an elephant does not equal a horse", implying that perhaps you simply can't add$x$and$y$together and must keep them separate. My friends quickly caught on. :D (accomplishments) - Simply Beautiful Art 101 [+4] [2013-03-07 21:10:58] Thomas I would have to say that it was the square root. There was (ans still is) something very fascinating about being able to recover the number that was multiplied by itself. If I know that$x^2 = 9$then I knew that$x$could be$3$(just thinking about positive numbers here). And I thought that it was crazy how one could also take square roots of numbers that aren't actually squares themselves. 102 [+4] [2013-03-08 01:35:17] fluffy A Fibonacci spiral and the way that at large enough scales it converges on the golden ratio. Also the golden ratio. 103 [+4] [2013-03-08 09:11:08] Joubarc Not an example of my own youth I've followed a small seminaryseminar on how to teach math a few years ago, and one of the things the teacher mentioned was that counter-intuitive results were more likely to mark the kids in a way they would start to try to understand why the results wasn't what they expected. The example he gave us was fairly simple: Imagine you ran a rope around Earth's diameter, lying on the ground. Then, add 1 meter to the length of the rope, keeping its shape as a circle (let's forget mountains and pretend Earth is just a ball for a while) - at what distance of the ground will the rope be? For most people, adding one meter to such a long rope is negligible so that there's simply no way it would be far from the ground. Convincing them that it's actually nearly 16cm above the ground is fun to do. As far as I remember, that example was extracted from a book, full of such examples and historical references which are also useful to show math isn't just a boring school obligation; but I can't find the name of the book right now. (1) Seminary? There's a religion of mathematical pedagogy? - Joe Z. (1) Not that I know, but now that you mention it, maybe it isn't such a bad idea. - Joubarc 104 [+4] [2013-03-08 09:46:38] miku From an interview with Vladimir Arnold [1] (NOTICES OF THE AMS, Vol. 44, No. 4): Please tell us a little bit about your early education. Were you already interested in mathematics as a child? ... The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day? I spent a whole day thinking on this oldie, and the solution (based on what is now called scaling arguments, dimensional analysis, or toricvariety theory, depending on your taste) came as a revelation. The feeling of discovery that I had then (1949) was exactly the same as in all the subsequent much more serious problems—be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970) or between singularities of caustics and of wave fronts and simple Lie algebra and Coxeter groups (1972). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main mo- tivation in mathematics. [1] http://hans.math.upenn.edu/Arnold/Arnold-interview1997.pdf I don't get the question at all, are we assuming they left at different times due to the earth being curved.. while also assuming they walk in a straight line as if the earth was flat? - user58512 @caveman: No, they just walk at different speeds. - ShreevatsaR 105 [+4] [2013-03-08 17:22:55] jcolebrand Mine was the discovery of sets in higher order math classes, and how all the lower math classes including physics theories were strictly derived from higher order calculus, and all of the formulas I had ever learned became such simple child's toys. I don't think those belong in a children's text, however. (2) When you realize that taking derivatives is so simple, you look back and realize, "I can't believe people use this as an example of difficult mathematics!" - Joe Z. (1) I had a similar moment of realization for reducing polynomials, back in middle school when my Sunday School teacher used a really long rational polynomial expression as an example of a "problem that's too hard for you to solve" (it was part of a teaching package). She had to resort to using trigonometry and asking me how I would calculate$\tan 35^\circ$, which I didn't know at the time. The polynomial ended up being something contrived, but it did actually reduce quite a bit. - Joe Z. (2) Of course, now when I look back at it, I think, that wasn't actually hard! - Joe Z. 106 [+4] [2013-03-15 10:01:31] Elmar Zander When I was still pretty young (I don't remember my exact age) I was very proud that I could already compute with decimal fractions which nobody I knew in my age could at the time. Around that time my aunt had a student for a visit in her home, and he talked to me about math, and asked me to compute$1/3+2/3. I asked to how many digits and he said as many as you like. So, I sat down and computed it to 10 digits or something: \begin{align} 0.3333333333\\ \underline{+0.6666666666}\\ 0.9999999999 \end{align} Proudly, I presented my result. He said well done, but it's way easier \begin{align} \frac13+ \frac23= \frac{1+2}3= \frac{3}3=1. \end{align} The beauty in this impressed me a lot and kind of got me started in math. 107 [+4] [2013-03-24 10:18:10] lakshman The commutative law doesn't hold for some series. I think this is an amazing fact to teach. http://www.math.tamu.edu/~tvogel/gallery/node10.html The example in the link amazed me. Similarly: finite sums of rationals are rational, but infinite sums need not be—for a similar reason to why the commutative law doesn't work for infinite series! - Akiva Weinberger 108 [+4] [2013-04-23 11:03:04] Plop When I was 10, I read a math booklet, that talked about Euler characteristic. There were drawings of all Plato's polyhedrons, and I counted, and realize that their Euler characteristic was always 2. I was amazed, asked my mom, math teacher, if she knew anything about it, and she told me she didn't. Now I'm 21, and I am just starting to be math-mature enough to understand this theorem. Maths are beautiful :D ! Oh, no, you'll be crushed when I show you a "donut-ish" shaped polyhedron! Euler characteristic is 0! (Unless you've seen this already.) :) - Akiva Weinberger 109 [+4] [2013-05-05 04:57:00] Maria Ines Parnisari The fact that you can't divide by zero always amazed me. I once read the following analogy: Imagine you go to a shop with 100 dollars in your pocket, and imagine that everything in the shop costs 1 dollar. How many things can you buy? 100. What if instead of 1 dollar, each thing costed0.5? How many things can you buy? 200. Now imagine that everything is free. How many things can you buy? Obviously, this question doesn't make sense anymore, because things are free, so you can take 0, or 1, or 2, or...

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[+4] [2013-05-05 07:51:56] Daniel McLaury

If you're writing a children's book on mathematics, please start by reading some excellent children's books dealing with mathematics. Here are some books I have fond memories of:

• The Man Who Counted
• The Phantom Tollbooth
• Flatland
• Alice in Wonderland / Through the Looking Glass
• Everything by Martin Gardner
• Godel, Escher, Bach: An Eternal Golden Thread

111

I'm not sure if this is suitable, but for me, the power of Mathematics lies in the absoluteness of its proofs. This is the only discipline where you can prove something to be true and it will stand up to the test of time, where no textbooks need replacing and facts are always right. (I'm assuming we don't make fundamental changes in axioms and what not!) This cannot be found in any other human endeavour and I find this to be very reassuring!

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[+4] [2013-12-09 07:10:44] waj cheema

For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1 I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of 3" and was also featured in a Harold and Kumar Movie, it renewed my love for math and is and always has been one of my favorite poems! :

I’m sure that I will always be A lonely number like root three

The three is all that’s good and right, Why must my three keep out of sight Beneath the vicious square root sign, I wish instead I were a nine

For nine could thwart this evil trick, with just some quick arithmetic

I know I’ll never see the sun, as 1.7321 Such is my reality, a sad irrationality

When hark! What is this I see, Another square root of a three

As quietly co-waltzing by, Together now we multiply To form a number we prefer, Rejoicing as an integer

We break free from our mortal bonds With the wave of magic wands

Our square root signs become unglued Your love for me has been renewed

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[+4] [2014-08-21 10:06:15] Phira

The realization that you can go on counting forever.

How simple! How beautiful! Kids may love this. - TobiMcNamobi
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[+4] [2014-08-23 00:19:13] Asier Calbet

One of my most memorable moments in mathematics was when I was attempting to prove the formula for the volume of a sphere on my own. I hadn't been taught calculus yet and had no idea about it, but I was convinced I could solve the problem. I used an infinite amount of small disks and added their volume ( essentially the limit of a riemann sum, an integral, but I didn' know that at the time) I made the disks a certain height, worked out the sum using sums of consecutive squares and then made the height equal zero. And voila, I got the right volume! Later I found out I had re-discovered a part of calculus. The realisation that different people can independently discover mathematical truths and techniques was beautiful to me.

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[+4] [2014-11-12 20:58:05] user1483

At about 10 or 11 I discovered that the area of a circle was half the circumference multiplied by the radius.

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[+3] [2013-03-07 07:50:58] bryansis2010

The one that I was particularly intrigued in my late years was the execution of the proof of Gambler's Ruin. However, it might be too deep for small children.

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[+3] [2013-03-07 10:32:31] Manoj R
1. Take your age, and reverse it.
2. Subtract smaller number from bigger.
3. Add the digits of subtraction.
4. You get 9.

(8) What if I'm 22? - Kobi
(2) What if I am 4? - Jack Aidley
(2) What if I am 100? - Trevor Wilson
Let the kid figure out the answers. - Manoj R
Doesn't the answerer mean...Subtract the smaller number from the bigger one? As originally stated, this would result in a negative. More correctly, he could have said...Subtract the smaller value of the two (original age, and reversed age) from the larger value of the two. - funkymushroom
(3) What if I'm 27.9? - Asaf Karagila
(2) ^ Then your answer will be expressed in terms of the p-adic numbers. - Joe Z.
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[+3] [2013-03-08 14:38:03] vegemite4me

Compared to most answers this is certainly not going to blow anyone away, but at the time it did amaze me. Our maths teacher asked us how long it would take us to get home if, we only walked half the way home, and then half the way of what was left, and then half the way of what was left, etc, etc. The realisation that if you kept dividing something by two (no matter how many times), you would never get to zero.

(1) Well, you'd take the time it takes you to go halfway home, plus half of that, plus half of this next amount, and so on. - Joe Z.
119

I recall being told about binary numbers when I was about 7 or 8 years old, and the idea that numbers could be represented otherwise than in base 10 must have fascinated me. Later in school I was mildly disappointed to learn that $\pi$ cannot be expressed in any simple way, as a ratio or using any of the mathematics I knew at that time.

Modular arithmetic is something that I more or less found out about on my own, surely prompted by its usefulness in handling operations on the twelve pitch classes.

It is a very entertaining practical experiment to fold a Möbius strip with paper and tape, then cut it once, and why not twice. It's not very intuitive what is going to happen!

At some point I remember trying to figure out how to generalize the factorial to real numbers. Of course I failed, and it took a few years before I saw the Gamma function in some book.

Huge numbers may provoke curiosity. After addition and multiplication there is exponentiation, and then towers. Just showing that you can construct numbers such as $x^{a^{b^{\ldots}}}$ can be interesting, and even more that some towers with infinite numbers of terms converge (but that is certainly fairly advanced).

For more reading I recommend Lakoff and Núñez, Where Mathematics Comes From.

As for the "Huge numbers" you may wish to check the "big numbers" tag or Googology. - Simply Beautiful Art
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[+3] [2013-07-01 02:48:33] Brian Rushton

The first time I heard that 3 times 5 is the same as 5 times 3, I was really intrigued, and I've been hooked ever since. It is pretty weird when you think that five groups of three people is as many as three groups of five.

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[+2] [2013-03-07 10:14:36] user827918
1. 17 + 20 = 8
2. 17 − 20 = 26
3. 17 · 20 = 21
4. 17^(−1) = 12 (inverse of 17)

I got really upset when I saw this. The professor explained, to do network communication you will need to understand this.

I found maths awesome after dealing with these. What we are normally learning can not always help (it's real numbers mathematics). But the best things deal with fields [1]. Therefore, the below is the explanation of the above meaningless things.

(i) Addition: 17 + 20 = 8 since 37 mod 29 = 8

(ii) Subtraction: 17 − 20 = 26 since −3 mod 29 = 26

(iii) Multiplication: 17 · 20 = 21 since 340 mod 29 = 21

(iv) Inversion: 17^(−1) = 12 since 17 · 12 mod 29 = 1

The elements of F29 [2] are {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 28}

[1] http://en.wikipedia.org/wiki/Field_%28mathematics%29
[2] http://beyondtechs.blogspot.com/2012/09/prime-fields-example.html

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[+2] [2013-03-07 15:50:09] Theo

I was pretty good at math from an early age, but what was the clincher for me was the existence of non-Euclidean geometry [1]. In grade 6 my math professor gave me a book on axiomatic Euclidean geometry, and I was totally blown away that the parallel postulate was just that, a postulate, and not an undisputable true fact. If upmto that point I just considered (school) math easy, from that moment I realized is incredibly beautiful. I did not look back since.

[1] https://en.wikipedia.org/wiki/Non-Euclidean_geometry

See my answer to this question for a simple experiment about non-Euclidean geometry (and the story of how I learned about it). - Asaf Karagila
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[+2] [2014-02-02 17:31:03] leonbloy

Not an experience of mine, but I'm currently reading The Greeks by H. D. F. Kitto [1] and I think this page deserves to be here:

But let us not be too superior to those Greeks who "shut their eyes." They kept something else wide open, namely their minds, and although the eye-shutting retarded the growth of science, the mind-opening led to things perhaps equally important, metaphysics and mathematics.

Mathematics are perhaps the most characteristic of all the Greek discoveries, and the one that excited them most. We shall be more understanding of those who shut their eyes to facts if first of all we keep in mind the Greek conviction that the Universe is a logical whole, and therefore simple (despite appearances) and probably symmetrical, and then try to imagine the impact of their minds on elementary mathematics.

It happens that I myself—if I may be personal for a moment—was enabled to do this by an insomnia-beguiling piece of mathematical research that I once did myself. (Mathematical readers are permitted to smile.) It occurred to me to wonder what was the difference between the square of a number and the product of its next-door neighbors. $10 \times 10$ proved to be $100$, and $11 \times 9 = 99$—one less. It was interesting to find that the difference between $6 \times 6$ and $7 \times 5$ was just the same, and with growing excitement I discovered, and algebraically proved, the law that this product must always be one less than the square. The next step was to consider the behavior of next-door neighbors but one, and it was with great delight that I disclosed to myself a whole system of numerical behavior of which my mathematical teachers had left me (I am glad to say) in complete ignorance. With increasing wonder I worked out the series to $10 \times 10 = 100$; $9 \times 11 = 99$; $8 \times 12 = 96$; $7 \times 13 = 91$… and found that the differences were, successively, $1, 3, 5, 7, \ldots$, the odd-number series. Even more marvelous was the discovery that if each successive product is subtracted from the original $100$, there is produced the series $1, 4, 9, 16, \ldots$. They had never told me, and I had never suspected, that Numbers play these grave and beautiful games with each other, from everlasting to everlasting, independently (apparently) of time, space, and the human mind. It was an impressive peep into a new and perfect universe.

( original source image [2])

[2] https://i.stack.imgur.com/8wLai.jpg

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[+2] [2014-03-03 07:00:04] JChau

I first discovered that math was beautiful upon learning the divisibility rules. At that point I was just like "IT WORKS IT WORKS! HOW DID PEOPLE KNOW THAT?!" I remember I once stayed up to test the divisibility rule of dividing by $8$ (if the last three numbers in the dividend are divisible by $8$ then the whole number is divisible by $8$).

I've been very proud when I had found out that for divisibility by 7. After that, of course I tried many more and got a feeling for the procedure. Maybe I was already too grown up - it was too easy, after all, and had in my feelings no real "beauty" because every modulus had its own rule and so I didn't try to make a full-fleshed toolbox. - Gottfried Helms
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[+2] [2014-06-21 13:24:31] Edoardo Lanari

Maybe the fact that the homotopy category of a model category is equivalent to the full subcategory of fibrant-cofibrant objects with homotopy classes of morphisms.

(13) Congratulations for finding that out as a child. - azimut
(2) Dry humor? (Hope so.) - Did
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[+2] [2014-10-15 01:53:51] J Marcos

Beremiz, an Arab mathematician, arrives on foot to a bedouin camp, accompanied by a friend that rides a camel, only to find three boys having a dispute about the testament of their recently deceased father. In the will, the father had left his herd of camels to be divided among his sons, in the following proportion: the older son should receive half of the camels, the middle son should receive a third of the camels, the younger son should receive one in each nine camels. The problem is that his herd was composed of 35 camels... So, the boys reckoned that, in order not to fail their father's instructions, they would have to reserve at most 17 camels to the older one, at most 11 camels to the middle one, and at most 3 camels to the younger one. This looked bad, not only because the divisions all left remainders, but because in fact there would still be a few camels attributed to no one after the three sons received their share! To help them find a fair solution to their predicament, Beremiz then asks his friend to give his own camel to the brothers, a suggestion to which his friend reluctantly assents. As a result, the division now goes smoother, and the three boys are happier, having respectively received 18, 12 and 4 camels. They are so happy, indeed, that they give the 2 remaining camels to Beremiz and his friend!

I once had to teach young children (and some not so young!) to sum fractions, and it was easy to generate a whole set of problems based on the same approach (a collection of fractions that do not sum to 1). I believe this will give rise to straightforward illustrations for your book, Liz.

Malba Tahan's book " The Man Who Counted [1]" may be claimed to be the way in which any Brazilian younger than 80 has first gotten in touch with the beauty of math. Its romanticized recreational presentation makes it easily accessible and great fun for children already in their early school years. "Splitting 35 camels" is the first mathematical problem which the readers are confronted with, in this book.

[1] http://en.wikipedia.org/wiki/The_Man_Who_Counted

127
[+2] [2015-01-04 16:02:57] Hanno

When I was 11, my math teacher asked us whether we thought that any three points in the plane that do not lie on a single line would lie on a single circle, and I remember being amazed to see that this was true.

Haha, what returning to this problem and using limits to find a circle with center at $(\infty,\infty)$ using point conversion and stuff? - Simply Beautiful Art
128
[+2] [2015-02-22 13:41:43] Fausto Vezzaro

There is a nice and simple theorem that still was not mentioned here (maybe because it is in the beautiful Paul Lockhart "A mathematician's lament" you already read?). Summing the first odd numbers we see a curious regularity: $$1+3=2^2$$ $$1+3+5=3^2$$ $$1+3+5+7=4^2$$ $$1+3+5+7+9=5^2$$ ...and so on. This is charming. Furthermore it is instructive for a kid to see that any numerical calculus like this above can't prove that this rule will work "forever", but the power of a creative proof reach this goal (THE proof is a graphical one, it is sufficient consider the angular "L" of square of pebble to "see the light").

No offense, but I think someone has already put this answer here. And yes, I do think it beautiful. - Simply Beautiful Art
I didn't read scrupulously this beautiful mass of posts, but it looks that the only post that pointed out this theorem is Jose Brox's one, 10 months later than mine. Anyway I think speak about induction looks superfluous in this case: the graphical proof is simple and powerful. - Fausto Vezzaro
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[+2] [2015-06-04 05:56:34] Gil Keidar

This is rather recent (Less than a year ago), but, since I am 14, I suppose it should still apply. I remember that I was bored in some class, and that I took out my calculator and started playing with it, writing "hello" with numbers upside down. Then I saw this button (this was a scientific calculator) that said "log," and so I pressed it. At first I received "error" for log(0) = -infinity (well, close enough), but then I tried other numbers, 1,2 10. Then I saw that at 10 it would blurt out 1, and at 100 2. I then realized that what log did was find the exponent of a number from a base number (of course, I didn't know that terminology then) but it was still pretty amazing. (I also learned later on that all calculators are log base 10)

Edit: is there something wrong with this answer? Why was it down voted?

No idea why this was downvoted; it seems like a perfectly reasonable response to the question. - mweiss
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[+2] [2015-12-09 20:19:32] Jose Brox

Maybe not the first one, but when I was young and experimenting with natural numbers, I astonishingly found that the sum of odd numbers has a formula: they add to a square number!

$1+3+...+(2n-1) = n^2$

It was only much years later that I learnt how to prove it rigorously (by induction), but I could see thinking some (long) time that $(n+1)^2-n=2n+1$, and that was convincing enough for me at the time (and still is! :D).

I also found in my "little investigations" as a boy that the square of a prime number (bigger than $3$) is always one more than a multiple of 24: $5^2 = 24+1, 7^2 = 48+1, 11^2 = 120+1, \ldots$

This had me in awe for like two years, until I was able to give a proof. The process of looking for and finding the proof was for me more beautiful than the result, and maybe that was the first time that this happened to me.

By the way, this is how I arrived to that result: I knew about prime numbers, and I was trying to compose some song at the piano with them, allowing to push only the prime-numbered keys. I was disappointed, because I could play any note if I allowed my scale to be circular, so the primes were no restriction at all for my song. But then I proved with the squares of the primes and... voilà! The same keys kept repeating! I thought why it was so, and saw that it was some relationship between the primes and the number 12, since there are 12 notes in a piano scale. I wrote tons of ordered numbers on rows of 12 columns and you can imagine the rest...

(1) :D Beautiful! Must've been fun - Simply Beautiful Art
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[+1] [2013-03-08 19:25:31] happybuddha

Like most people, my most amazing discovery was tables. How 2+2+2 was six and how 2 times 3 was also six. And then I could count the number of chocolates lying on a table when they were paired. And then, even if chocolates were not grouped, I could mentally take a base of 2 and count 2, 4 6, 8.. chocolates and always be the first one to count the number of chocolates/things on a table. Most recently I was extremely fascinated by a model at display in the science/maths museum in Cambridge. The model was describing accuracy in probability. It was two sheets of glass standing between which there were random rods connected to the sheets in a certain way. On the glass was drawn a graph (like a parabola or a sine wave) which was a prediction of how the end graph would look like and to shape the graph there were little balls dropped over a period of 10 minutes or so between the sheets. What it proved was the 100% accuracy in the probability of a certain shape of a graph being formed with random balls thrown for a certain period of time. It just blew my mind away and I was standing there with little children for 30 minutes watching this over and over and was awed everytime the same graph was formed. I searched a lot on the MIT museums website but am not able to find this exhibit mentioned. It may more have been a physics thing.

Look up "normal distribution." - Akiva Weinberger
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[+1] [2013-04-28 00:40:11] Diego Silvera

The most wonderful thing I've recently seen is this [1] (sorry it's in French) form of the Sieve of Eratosthenes [2] and of course your question too.

[1] http://blogdemaths.wordpress.com/2012/09/30/le-crible-de-matiyasevitch/
[2] https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

This is great! Matiyasevich has a brief English explanation on his website, too: logic.pdmi.ras.ru/~yumat/Journal/Sieve/Sieve.html - Quasicoherent
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[+1] [2013-04-28 01:19:07] Linus

I was hooked on math by a small side note in a kid's book of mathematics about perfect numbers, numbers that are twice the sum of their factors. For example, 6 is the smallest perfect number because 1 + 2 + 3 + 6 = 2 × 6 and 28 is the next one because 1 + 2 + 4 + 7 + 14 + 28 = 2 × 28. The next perfect numbers are 496, 8,128, 33,550,336, and 8,589,869,056.

I was so fascinated by the idea that I proved that those numbers were perfect by listing out all their factors and adding them together. And to this day I wonder: somewhere up there in the vast expanses of integers, could there be an odd perfect number?

I think that James Sylvester stated it eloquently: "...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle." Marcel Danesi, in his book The Liar Paradox and the Towers of Hanoi, stated it significantly less eloquently: "No odd perfect numbers have ever been found. They probably do not exist."

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[+1] [2013-08-20 10:11:07] Mark Hurd

I think one of my early favourite mathematical things was simply "proof by contradiction" -- any of them.

I think its appeal is that you nearly have proof by example, except that you're proving a negative.

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[+1] [2013-09-24 07:54:05] Y Y

When I was a kid, I remember that I used to make 'little discoveries', although after I got all excited I realised other people already knew this and I didn't discover anything new, I was always really proud when I made one of those discoveries, Here are a few that I made and approximately how old I was when I did make the discovery (just saying I am 13 years old right now).

• An odd number plus an odd number is even, an even number plus an even number is even, and odd plus even is odd (approximately age 4-5)
• The rule for the Fibonacci sequence (I remember I saw it on the board or something in my classroom and then I just found the pattern) (age 6, I remember this one accurately bc it was during the first year of my primary education)
• If you rip a piece of paper in half, and then both in half again, and again etc, the number of pieces of paper you have when you ripped it $n$ times is $2^n$ (age 10)
• The fact that you can split a triangle into two right triangles and apply trig ratios to them (I later learned the Sine and Cosine Laws so yeah) (age 11)
• I was learning differential calculus, and I didn't know integral calculus yet. My dad told me that a integral is the inverse of the derivative, Then I figured out the rule that offsets the power rule and I figured out that the integral of nothing is an arbitrary constant (When i officially learnt integral calculus I found the proper formulas and I was just using the wrong symbols) (age 12)
• I proved to myself that the cardinality of the set of Reals and the set of Complex numbers are the same (age 13)

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[+1] [2013-09-25 18:42:53] Alex

I started math reasonably late, so I'm not sure this is a perfect example for a children book. I was totally amazed that the number $\pi$ is encountered in completely unrelated situation. Of course, I knew it's a ratio of circumference of the circle to the diameter, but then I learnt of the Normal probability density function.

So if you stretch a bell-shaped curve (Gaussian function) from $-\infty$ to $\infty$ the area under it $converges$ to $\sqrt{\pi}$?? How is it possible that this area has something to do with the square root of the ratio of circumference of the circle to the diameter? To be honest, even now, after learning the related proofs and derivations I still find it quite baffling.

(1) This recalls the introduction to Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences. - Matt
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[+1] [2015-02-13 18:15:25] მამუკა ჯიბლაძე

My story is not that impressive - as a kid I've observed on several simple examples that $(a-b)(a+b)$ is $a^2-b^2$, and that decided my fate.

The moral is I think that there are many ways one might share joy of math with kids, and showing them its beauty is probably not the universally efficient one. At least in my case, nothing compares to the feeling that accompanied an independently initiated discovery of something out of the material world yet undoubtedly as real as anything material, maybe even more so.

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[+1] [2015-03-08 11:36:36] user217174

In the age four or five i knew: $$2*5=5*2$$
I hope that you understand how this result is wonderful for me on this age, because yet i didn't use to commutativity of multiplication on $\mathbb N$!
In addition i didn't generalize this fact to another numbers!

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[+1] [2015-07-29 03:31:08] Mirko

There is a series of math children books in Russian by
Владимир Артурович Лёвшин [1]. To list some:
Магистр рассеянных наук [2] (translates roughly as Master (as in M.Sci) of the absent-minded
sciences, though google translates it as Master scattered Sciences),
Новые рассказы Рассеянного Магистра [3] (New stories of the absent-minded Master),
Путешествие по Карликании и Аль-Джебре [4] (google transtales it as Travel Karlikanii and Al Gebre),
Черная маска из Аль-Джебры (The Black Mask from al-ğabr(=al-gebra)).
More of them at http://www.koob.ru/levshin/ (in Russian).

I am a Bulgarian (presently working in New York), and as a child (could have been anything between 6 yr old and 9 yr old) read the Bulgarian translation of Путешествие по Карликании и Аль-Джебре (or it might have been one of the other books listed above).

I was fascinated. At hindsight mathematically the book is fairly simple or even routine (goes on to set and solve an equation, must have been a quadratic one, though it might have even been linear), so once you know how to solve such equations it might appear boring. But amazingly it does it in a way that unfailingly keeps the readers attention. It is written like a detective story (the $X$ with the black mask was enchanted and was to be freed by the Master, and its assistant the Нуличка, i.e. the Naught or the Null), with characters to relate to, number system and operations introduced and, thus, developing in parallel, the necessary math background and an intriguing story to follow and enjoy. I motivated myself to understand the math details (it might have been that we had not yet covered that material in school), so I could keep reading. I was also interested in logic-puzzle books at about the same time. I cannot single out a particular math piece that is exciting in this book (generally it is about real numbers or perhaps integers, setting and solving equations), it is the whole process of taking an ignorant (but intrigued) child and making him willing to follow the story, and to eventually learn algebra (at the level of quadratic equations) on their own, and make them feel great about it (and at the same time to not know at all what exactly feat they have done, that is, there was no feeling at all that I was being educated in "accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets", to borrow from Ahmes [5], even if there is no direct relation).

I do not know if an English translation is available of this series (google doesn't seem to know about it, and google knows everything, unless I do not know how to find it). I think this is a great book and would recommend it to anyone who could read it (or, well, would certainly recommend it to children, since adults would be spoiled with what they already know, and might not enjoy it). In my opinion this book has the spirit of adventure, and it might make an interesting reading (it reminds me of another Russian (or Soviet) well-known "adventure" book, The Twelve Chairs, with sequel The Little Golden Calf, though both format and subject are very different, but perhaps one could feel that both represent Russian culture ... don't know what the author(s) would have thought of this alleged affinity though).

And of course, I do appreciate things like Euclid's proof [6] that there are infinitely many primes, or that ( Pythagoras or Hippasus [7]) $\sqrt{2}$ is irrational (these were some of the first things I enjoyed introducing to my students last semester in a History of Math class), but for me these came "later" when I was already a converted mathematician (or I thought of myself so). I can't tell when this conversion happened, but it might have been in early school. I was good at math (so my teachers were happy and my schoolmates sought my help, and for that matter everyone would keep telling me that my grandmother, whom I newer saw, was a famous (or infamous, because she would uncompromisingly fail bad pupils) math teacher in my home-town), but it is not just about being good at it (as I realized, there were better students than me, once I got into university, in particular some of those coming out from so-called matematicheska gimnazia - a high-school emphasizing math, science and languages), so it is not so much about being good at it, as about being attracted by math, willingness to keep working/discovering, and the adventure and meaning that comes with it.

[1] https://ru.wikipedia.org/wiki/%D0%9B%D1%91%D0%B2%D1%88%D0%B8%D0%BD,_%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80_%D0%90%D1%80%D1%82%D1%83%D1%80%D0%BE%D0%B2%D0%B8%D1%87
[2] http://www.koob.ru/levshin/magistr_rassey_nauk
[3] http://www.koob.ru/levshin/new_stories
[4] http://www.koob.ru/levshin/travel
[5] https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus
[6] http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
[7] https://en.wikipedia.org/wiki/Square_root_of_2#History

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[+1] [2016-11-24 01:08:55] Joel Cohen

I believe my first encounter with mathematical beauty was with geometry. This might seem rather trivial, but I found this flower shape beautiful in many ways :

I always loved how it looked, and the fact that we could draw it with just a compass and 7 similar circles. I was amazed by the fact that this construction existed (in other words, I found the fact that the regular hexagon inscribed in a circle of radius $R$ had a side of length $R$ to be a striking coincidence, even though I couldn't formulate it in those terms). Using this basic construction, I would also make several other drawing on my notebooks : equilateral triangles, triangular tilings, regular hexagons, hexagonal tilings, stars of David...

I was comforted in my fascination by the fact that similar patterns appeared in nature and art.

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[0] [2013-03-22 03:44:45] asmeurer

As others have mentioned, kids love $\pi$. Prime numbers are also good, if they have a good handle on division. I think the fundamental theorem of arithmetic is intuitively true once you understand it (at east it was to me).

It would be great to mention some unsolved problems, like the twin prime conjecture or the Collatz conjecture.

For me, one thing that I remember being fascinated about at an early age was the fact that multiplication is commutative. That $3+3+3+3+3=5+5+5$ (or if you want, five baskets with three apples each is the same as three baskets with five apples each) was not immediately obvious to me, and the fact that it worked for any two numbers amazed me. Once you understand the geometric "square of dots" proof it makes sense, but I think that before that it doesn't.

Knuth up arrow notation is worth mentioning. Kids love that multiplication is repeated addition and that powers are repeated multiplication, and would be interested to see that idea taken further.

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[0] [2013-04-29 05:57:50] John Marty

I was always good at maths as a child, and took to reading extension maths books for fun (other kids thought I was weird). When I was about 10 I was completely hooked when I saw Euclid's proof for an infinity of primes. I had been given it as a question in one of the books I was reading. I spent about an hour desperately trying to prove it . . . then I looked at the solution - I was stunned by its elegant simplicity. Another thing I really enjoyed was finding cool facts about numbers in kids maths cartoon books and proving them. I loved to show WHY things always worked, that is perhaps my favorite thing about maths.

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[0] [2013-05-05 07:40:38] Xpast

For me it was Monty Hall problem [1]:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I saw this problem when I was 15 year old. I answered correctly (I probably used some kind of math intuition), but I thought that probability in the second case is $1/2$. Actually it is $2/3$. The proof is beautiful, as well as the answer. This fact amazed me. Even now, at 18, I suppose it is quite a beautiful problem.

[1] http://en.wikipedia.org/wiki/Monty_Hall_problem

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[0] [2013-05-06 04:59:47] Jori Mäntysalo

My first think of infinity was going from one corner of square to opposite corner. Start with route (0,0) -- (0,1) -- (1,1), then take (0,0) -- (0,½) -- (½,½) -- (1,½) -- (1,1) and so on. Path will come visually closer to diagonal, but lenght will stay at 2.

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[0] [2014-04-09 22:27:52] Devon Parsons

Draw any triangle. On each side of the triangle, draw an equilateral triangle such that the new equilateral triangle shares a side with the original triangle. Connect the midpoints of your three new triangles - the result is another equilateral triangle!

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[0] [2014-07-23 09:24:29] b.sahu

A story which I heard when I was in Primary School motivated me to understand the Power of Exponentials. The story goes like this--..........A Brahmin Priest presented the King with the Chess Board and explained to him how to pay War Games on this board . The King was pleased and asked the priest , what he wanted as a reward ..........The priest asked the King , that as he was very poor ,he needed some grains to feed his family .He asked the King to put one grain of rice in the first square of the chess board. , then put two grains in the second square ,four grains in the third square----and continue this way doubling the number of grains in the next square --till he reaches the end--the 64th square ."I will take whatever grains are there on the Chess Board..that will be sufficient for my needs"..said the Brahmin...........The king tried to satisfy the needs of the Brahmin ,but soon found out that all the grains in the Kingdom will still fall short of his needs ....... . The King was pleased with the priest's intelligence and appointed him as the Royal Astronomer & Astrologer .

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[0] [2014-07-23 09:50:02] b.sahu

This is a new interesting 4x4 Magic Sqaure, which I believe will be interesting to School Children. Here each element of the square is a square number. This was provided by Dr. Geoffrey Campbell

509020 is the sum of rows and columns

  29^2 |  191^2 |  673^2 |  137^2 || 509020
-------+--------+--------+--------++--------
71^2 |  647^2 |  139^2 |  257^2 || 509020
-------+--------+--------+--------++--------
277^2 |  211^2 |  163^2 |  601^2 || 509020
-------+--------+--------+--------++--------
653^2 |   97^2 |  101^2 |  251^2 || 509020
=======+========+========+========++--------
509020 | 509020 | 509020 | 509020 || 509020


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[0] [2014-08-21 10:27:51] Antinous

The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."

(1) What's it matter that $i$ doesn't exist? All numbers don't exist. (WARNING: The preceding statement is opinion.) - Akiva Weinberger
Good point. I think at the time I "understood" what numbers were, but these new numbers (complex numbers) were so strange, and were presented with the added "imaginary" verbiage that they were mesmerizing to me! Of course later I realised that complex numbers are not complex at all and that there is absolutely nothing imaginary about the idea of $i$. Complex numbers are a system of numbers which obey a certain set of rules in a consistent way. But I'm pleased to have been captivated :-) - Antinous
By the way—I'm not sure if you've seen this before—compare$$\left(1+\frac i{1000000}\right)^{1000000}$$with$$\cos1+i\sin1$$(radians). They agree to roughly six digits! (It's not a coincidence.) EDIT: Google has a built-in calculator. You can type in (1+i/1000000)^1000000 and cos 1+i sin 1. EDIT EDIT: (1+i/(10^13))^(10^13) gives more accuracy. - Akiva Weinberger
In fact: $e^{ix}=\cos(x)+i\sin(x)$. - Antinous
Yes, I know. But this fact is (one way) to prove it. - Akiva Weinberger
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The symbol:

$$\int_{a}^{b} f(x) dx$$

It was cool. But then came along contour integration!

$$\int_{-\infty}^{\infty} \frac{\sin(x)}{x} dx$$

Using a contour integral

$$\oint_{C} f(z) dz = \int_{-\infty}^{\infty} f(x) dx + \int_{\Gamma} f(z) dz$$

And Residue Theorem,

Complex Analysis and Real analysis proofs.

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[0] [2015-01-21 03:11:21] George Frank

I have spent many decades studying why so many highly intelligent people are so mystified by mathematics. Lockhart's view is very serious and cannot be negated by the personal experiences of mathematically inclined people. My study has clearly shown that the best advice is to be simple and sensible. For example, our place number system is an ingenious solution to the problem of too many different names and shorthand symbols for quantities. The solution is not sensible if the problem is not clear. Addition is immensely useful regardless of how it is done, including by a calculator. So is subtraction. multiplication is a wonderfully ingenious way to count when the items counted come in fixed size packages. Division is also very useful, again completely aside from how to do it. Our conventional emphasis on HOW is terribly off-putting. In this electronic age, "how" is far less important anyway. Mathematics is not a skill and should not be identified as one. Numbers and numerical operations and functions and condition equations and so on, and the properties of all of these, are completely real and sensible and have nothing to do with so-called "reasoning" or "rigor" or "skill" etc. Everything sensible involves reasoning. And rigor is the concern of mathematicians, not lay appreciators and users of mathematics. And "skill" is vastly overrated. It is easy to develop skill if you understand what the subject is about. It is the latter that is missing in our education.

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[0] [2015-01-29 11:15:04] Sufyan Naeem

The first thing for me is the working of an equation. it is, to me, like a stanza of a poem that tells us many things in minimum words. No one would have ever thought of describing a geometrical figure. Every one used to draw it before math's entry in the real world. It's awesome for a mathematician to say that write me a circle, ellipse etc.

In order to tell people that math is not only concerned to problem-solving, I have produced my own quote.

" Practice is hollow without understanding ".

This makes me think of Physics, where practice has lead to revelations that have called for mathematical expressions and intuitive understandings, much like String Theory's origin or Quantum Theory, or, now that I think about it, pretty much all of Physic's history. - Simply Beautiful Art
@SimpleArt I used a phrase "Problem-solving" which clears my intention. For example, in Chess, once you understand the purpose of the move "En passant" , you are never going to forget 1) The way to make it 2) The terms and conditions of the move, otherwise, you may forget how to execute that move. The move is, "If your Pawn is in 5th rank and opponent Pawn make its first move with 2 squares, you can capture it and occupy the square, it skipped" Condition: "You can only make this move in the first immediate opportunity". Purpose is to reduce the freedom of Pawn (In short). - Sufyan Naeem
Well, that's nice. Reading your comment makes me realize how different our interpretations are, yet they can be described in essentially the same way. First time I fell for En passant, I was really confused. Now, I get to use it on other people, and I have never forgotten it. - Simply Beautiful Art
@SimpleArt this is another thing. Of course, when you repeatedly practice something, it takes shelter in your brain. Even an animal can learn something by repeatedly practicing it. BTW, my quote states "Practice is hollow..." and this is different from saying "Practice is useless..." or from saying "Practice is nothing...". - Sufyan Naeem
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[0] [2015-02-12 21:59:55] Joshua Benabou

Telescoping series. Double counting to prove combinatorial identities. All the paterns in Pascal's triangle. The medians of a triangle always intersect at one point. Using roots of unity filter to solve combinatorics problems.

Guage invariance over Floer homologies for conformal Khovanov manifolds in $n$-dimmensional geometries.

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[0] [2015-03-08 11:18:57] Michael

My favourite maths book when I was little was 'Magic House of Numbers' by Irving Adler.

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[0] [2015-11-28 00:01:09] user285523

Personally, I was STUNNED by

$$1+2+3+4\cdots=-1/12$$

This undoubtedly sparked my interest in mathematics. (Although I didn't know it then, this is a zeta-regularized sum)

YES, this is the reason why I love summations (because they don't always make sense, hehe) - Simply Beautiful Art
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[0] [2016-01-24 01:52:22] Simply Beautiful Art

Personally, I thought math was beautiful on a number of occasions:

$$1x+2x=3x$$

$$1zebra+2zebras=3zebras$$

Another time I found mathematics beautiful was when I learned that almost all functions have a writable inverse, written using Lagrange's Inversion Theorem.

Another cool thing for me was big numbers. It started with infinity, then I discovered very large finite numbers, which are studied in Googology.

The discovery of infinity has led me to infinitely summations, which I found interesting that they were calculable and sometimes exerted weird solutions.

The discover of $i=\sqrt{-1}$ was cool, but even cooler was the discovery that $\sqrt{i}=\frac{1+i}{\sqrt{2}}$, making me realize that I could not make new types of imaginary numbers by square rooting further. This lead me to complex analysis and the solution to $x^i$.

By sitting down and writing out the formula for the perimeter of an $n$ sided polygon, I discovered $C=2\pi r$ by taking what I didn't know was a limit to infinity. It required a bit of help though.

My own realization that some of the solutions to $f(f(x))=x$ could be found using $f(x)=x$ and that this could be extended to any amount of iterations of $f$.

The disappointing discovery that one cannot find the inverse of the general quintic polynomial in terms of a finite amount of elementary operations. Of course, you can still approximate with root finding algorithms or Lagrange's Inversion, but they are neither exact nor finite in method of reaching the solution and sometimes they fail.

The discovery that one can find the square root of a number using algorithms was pretty impressive for me.

The discovery of the Lambert W function allowed me to solve soooo many exponential problems, but then a hit an edge, a barrier full of currently invertible exponential problems like $x^{x^x}=y$, given $y$ and trying to find $x$.

The discovery of the factorial is often a fun little thing for young students, it makes them think of the interesting ways that math can work. I personally tried to extend them to all positive reals, but, like some other answers, it appeared to be impossible for my talents.

Then I discovered the Gamma function and learned Calculus.

The definition of the Euler-Mascheroni Constant was truly amazing as it gave me a method for easily approximating the natural logarithm for positive whole numbers, which extends to all positive numbers through logarithmic properties.

And lastly, I would like to point at mathematically rigorous idea in physics where velocity affects air drag, which in turn affects velocity, which will again affect air drag, etc. The sheer confusion in all of this was mind-blowing.

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[0] [2016-01-24 02:08:34] YoTengoUnLCD

One of the biggest awes I experienced was when I could fully understand how you could prove that addition and multiplication of real numbers was commutative: trying to understand this it made me go to the basic construction of the Naturals, Integers, Rationals, and finally the reals (via the dedekind cuts approach).

I just thought that journey was lovely.

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[0] [2016-02-03 03:37:39] frogfanitw

I remember being fascinated by amicable numbers, the subject of my junior high science fair project in the early 1970's. I was using a huge book of factorization tables that I couldn't check out from the public library. I spent hours trying to plug prime numbers in the formulas given by Euler and Erdos.

DEFINITION: A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.

For a list see https://oeis.org/A259180

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[0] [2016-03-03 17:49:45] Yves Daoust

The first equation:

Knowing that the selling price is $220$, and the margin is $10\%$, what is the purchase price ?

At the time I was able to derive the benefit $200\times 10\%=20$ or the selling price from the purchase price $200+200\times10\%=220$ but has no idea how to do the reverse (purchase prince from sale price) as the unknow "had to be known" to compute itself with $?=230-?\times 5\%$.

The rewrite with a symbolic quantity $P+P\times5\%=P\times(1+10\%)=220$ was a revelation !

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[0] [2016-04-16 20:29:35] Shubham Avasthi

The first time I was fascinated by mathematics was when I read Christian Goldbach's conjecture. From that day onwards, I am trying to decode the mystery of primes, which seem to be simple at first sight but are actually very difficult to understand. That's the beauty of mathematics.

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[0] [2016-06-22 12:37:51] Charles Rockafellor

What was the first bit of mathematics that made you realize that math is beautiful?

For me, it was when I was 3 years old (possibly 4), contemplating my hands and fingers. I had the sudden epiphany that 5+5 absolutely had to equal 10 every time that you added them together -- not merely that they had done so repeatedly, mind you, but that they must do so in every event. I was admittedly a little off base there, not yet knowing of quirks such as modulo, but it was so astounding that I ran to the bathroom to tell my mother.

There have been a lot of other wonderful moments in math, for me, but that initial one was like seeing into the mind of god, reading the very fabric of creation, and fully knowing that reality is comprehensible. :-)

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[0] [2016-06-22 13:30:15] tatan

I tried to find the number of ways in which a number can be expressed in term of sum of two numbers and I ended up learning Partitions [1] which showed me how everything can be expressed mathematically....

[1] https://en.wikipedia.org/wiki/Partition_(number_theory)

(2) A little too short to be posted as an answer, in my opinion. Without any elaboration, I don't feel this adds to the discussion. - 6005
(2) Please, elaborate. Be wary of posting answers that are significantly shorter than the Question that is being addressed. - hardmath
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[0] [2016-08-20 01:27:43] heather

These aren't the first things I found beautiful about mathematics, but:

1. A mobius strip. So fun, and I don't know why. Intriguing, as well.
2. Khan Academy has a bunch of math videos, but they have a section that is "for fun and glory" and really, those are the best videos they've got. From Borromean onion rings (a mathy thanksgiving) to an interesting discussion of fractals, and the infinite amount of paint you'd need to paint a dungeon that is designed after the Koch snowflake, and then, of course, the Fibonacci sequence, and so much more, they are amazing videos, and are really kind of mind-blowing. And oh so funny.
3. Binary and other base systems. I don't know when I first realized that base 10 isn't the only way to count, but I know that binary is my favorite now (especially counting on your fingers in binary - that's kind of fun). It is also interesting to me that numbers are equivalent between bases.
4. Solving math problems by programming. I don't know why this is so beautiful to me (I'm not sure beautiful is the right word) but it is. I'm able to write a program, and it solves it. The challenge is fun and it is so rewarding when, after debugging and figuring out you made a silly mistake that made the compiler crash, it works and you get the answer. It's just kind of beautiful, I think, and oh so fun.
5. Pi. Because Pi day. I will always remember when Pi day fell on a day during spring break, and it was my mom's half day and she baked a shoo-fly pie and that's how we celebrated pi day, with pie (of course, this is somewhat normal, but it was so fun). So wonderful. Now, of course, we should've calculated the area of the pie, using, of course, pi...
6. Getting the quadratic formula from completing the square on $ax^2+bx+c = 0$ on my own, which was so fun. Then,to memorize the quadratic formula, our math teacher had the whole class sing it to the tune of pop goes the weasel. We all were red-faced and laughing at ourselves, but we had it memorized without any further ado.

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[-6] [2013-03-18 14:43:38] Doug Spoonwood

I don't find it beautiful, but I still find the idea expressed by the following something of a psychological curiosity:

How can it be that when some algebraists say "AND" and "OR" they mean exactly the same thing?

OR means this that "false or false" is false, "false or true", "true or false" as well as "true or true" are true, or more compactly:

    F  T
F  F  T
T  T  T


AND means this:

    F  T
F  F  F
T  F  T


But, since NOT(x OR y)=(NOT x AND NOT y) and NOT(T)=F and NOT(F)=T, OR and AND, to an algebraist, mean exactly the same thing!

(3) Your answer implies that $\neg ( \perp \lor \top) \iff ( \neg \perp \land \neg \top) \iff ( \top \land \perp ) \iff ( \perp \lor \top)$. Your truth table for $\land$ is wrong. - Andrew Salmon
(1) @AndrewSalmon Thanks, I don't know how I did that. - Doug Spoonwood
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