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.
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:)
- Will
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.
7*3 = 25
, 7*4 = 34
, 7*5 = 43
, 7*6 = 52
, 7*7 = 61
. Beautiful. - Connell Watkins
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.
:)
Do you have a GitHub.com account? That's probably the easiest way of publishing it. - Will
I used to love naughty $37$.
$37 \times 3 = 111;$
$37 \times 6 = 222;$
$37 \times 9 = 333;$
$37 \times 12 = 444;$
$37 \times 15 = 555;$
$37 \times 18 = 666;$
$37 \times 21 = 777;$
$37 \times 24 = 888;$
$37 \times 27 = 999;$
37037*3=111111
, 37037*6=222222
, 37037*9=33333
... - MiniQuark
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.
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:
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
.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_notation12345679 x 8 = 98765432
- jwg
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:
This isn't what did it for me, but it's fairly simple and quite nice:
$$0.9999999999\ldots =1$$
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.
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.
1 5 10 10 5 1
becomes 1 6 1 0 5 1
- Nils Werner
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_numberMy son loved this when he was little - patterns everywhere:
(Copy from http://mathforum.org/library/drmath/view/57919.html)
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".
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.
I always thought cycles in decimal fractions were magic, until I realized I can easily create whichever cycle I wanted:
I failed a number theory exam because the professor did not know this trick and said I needed to prove it.
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
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.
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.
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/CataloniaWho remembers magic squares? Those sparked my interest in mathematics.
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:
A few things come to mind:
Sir Francis Galton's bean machine ^{[1]}, which demonstrates the Central Limit Theorem ^{[2]}, is quite remarkable.
Source: Wikipedia's Entry on the Bean Machine ^{[3]} Here's a video of this device in action: http://youtu.be/xDIyAOBa_yU
I mentioned this in comments elsewhere, but factorization diagrams are fascinating.
Source: The Math Less Traveled ( Post 1 ^{[4]}, Post 2 ^{[5]})
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_machineI 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.
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]}.
[1] https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_HotelEverybody 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]}
[1] http://www.youtube.com/watch?v=wCyC-K_PnRYThe Golden ratio ^{[1]}
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_ratioI 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.
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.
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:
You can download them from Wikipedia ^{[1]}.
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/FractalsFor 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/
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!"
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 :)
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.
Thanks to @FacebookAnswers for suggesting Conway's Game of Life ^{[1]}, a cellular automaton devised by John Conway in 1970.
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.
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_LifeEuclidean 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.
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 ;)
orig http://imageshack.us/scaled/thumb/221/heartwl.png ^{[1]} white http://imageshack.us/scaled/thumb/23/heart01q.jpg ^{[2]} black http://imageshack.us/scaled/thumb/268/heart21.bmp ^{[3]}
[1] http://imageshack.us/f/221/heartwl.png/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.
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?
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/TopologyTwo instances where I thought math was amazing:
:)
That was very fun for me.:)
I had the "grid" epiphany, too! - Akiva Weinberger
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_HornIn elementary school, my math teacher taught us this trick for the 9 multiplication values:
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...
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/When I was maybe 8 or 9, the following trick was showed to me as a sanity check for calculation by hand.
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!
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
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".
1+
and 0
, then you can write all numbers! (Might take a while for a few of them.) - Akiva Weinberger
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.)
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.
Solving for an unknown. 2x = 4 so x = 2. Beautiful.
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_HotelWhen I was young I found a riddle:
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.
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 $7^2+24^2=25^2$.
When I saw my first list of mathematical axioms (algebraic in this case). This was when I was 11.
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.
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%29The 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.
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!!!
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.
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/088385757XIt'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.
Complex numbers was awesome to me
i² = - 1
Pi has always fascinated me. The notion that perimeter of every possible circle imaginable divided by its diameter always results Pi is astonishing.
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 $3^2$
$3^2 = 2^2 +2 +3 $
$3^2 = 4+2 +3$
$3^2 = 9$
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.
I discover that math is beautiful again and again, I suspect there is no end to this. Some of things that blew my mind:
Cheers!
[1] http://en.wikipedia.org/wiki/Desargues%27_theoremAs 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)
$$\sqrt{\sqrt{\dotsb\sqrt{x}}} = 1$$
(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).
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):
Parallel lines. I was amazed to find out that they would never, ever meet.
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.
Pythagorean theorem
If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for any positive integer $k$.
$3^2 + 4^2 = 5^2$
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.
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)
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_numeralsI 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.
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.
$\epsilon$
), 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
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.
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.htmlI 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.
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.
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/20893If 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.)
For me it was when I realized that with sine and cosine I could draw a circle!
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 :)
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_EinsteinWhen 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.
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!
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.)
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.
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 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.
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.
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.A4ngeThe 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 :-)
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
cryptarithmetic
puzzles. - Will
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.
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:
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.
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.
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.htmThe 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.
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.
A Fibonacci spiral and the way that at large enough scales it converges on the golden ratio.
Also the golden ratio.
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.
From an interview with Vladimir Arnold ^{[1]} (NOTICES OF THE AMS, Vol. 44, No. 4):
[1] http://hans.math.upenn.edu/Arnold/Arnold-interview1997.pdfPlease 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.
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.
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.
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.
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 !
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 costed $0.5? 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...
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:
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!
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
The realization that you can go on counting forever.
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.
At about 10 or 11 I discovered that the area of a circle was half the circumference multiplied by the radius.
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.
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.
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.
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.
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%29I 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_geometryNot 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])}
[1] http://rads.stackoverflow.com/amzn/click/0140135219I 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 $8$).
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.
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_CountedWhen 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.
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").
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?
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...
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.
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/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."
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.
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).
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.
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.
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!
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).
I also remember that as a child I had a problem understanding infinity (безкрайност in Bulgarian, literally endlessness), and kept thinking about it. It is not clear if I understood it (it looked like a winding road that kept going no matter what), but at some point I stopped thinking about it. Nowadays I presumably know that there are different related notions: limitless/ boundless / infinite, and different parts of math might have use of one or the other (e.g. manifold without a boundary like the circle, vs the real line which extends both ways, or transfinite numbers which go just one way but could be used for counting). I could not quite accept that infinity exists (and strictly speaking nobody could prove that it does, but anyway it is an accepted convention), my point is that I forced myself to imagine that winding road never-ending, to try to get an idea of infinity, but I don't think I ever convinced myself. I could not see the whole thing, even if any time I approached the end, it kept extending (as I would just generate one more piece of it in my mind and put it there for the sake of the argument), so for me infinity was something that I cannot exhaust by way of observation, but cannot comprehend either. I could not rule out its existence, so I live with that, but I never saw it, so I can't vouch for it. (Much later at university I had a dream, almost a nightmare when I was supposed to pick a rifle from what seemed an endless field of identical rifles, and I couldn't make a choice. Eventually I picked one, its virtue was that is was exactly the same as all the others, but fulfilled the task of picking one. Somehow I tend to relate this to the Axiom of Choice, though strictly speaking you do not need AC to pick just one element of just one set. And I have no idea why it was rifles, and not, say, apples. Also, that was a multiset, not a set, so I don't know what it had to do with AC, I guess I had to come up with something familiar when I woke up, and we had already studied AC. Or perhaps someone could indeed relate this to AC in a meaningful interpretation.)
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.
Please let us know when your book is published :)
[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%87I 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.
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.
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.
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_problemMy 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.
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!
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 .
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
The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."
(1+i/1000000)^1000000
and cos 1+i sin 1
. EDIT EDIT: (1+i/(10^13))^(10^13)
gives more accuracy. - Akiva Weinberger
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.
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.
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 ".
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.
My favourite maths book when I was little was 'Magic House of Numbers' by Irving Adler.
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)
Personally, I thought math was beautiful on a number of occasions:
$$1x+2x=3x$$
$$1zebra+2zebras=3zebras$$
Applying words can really help young children understand mathematics better.
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.
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.
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
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 !
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.
You asked:
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. :-)
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)These aren't the first things I found beautiful about mathematics, but:
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!
:)
- apnorton:)
- Will111111111^2
after seeing the pattern in long-multiplication of11*11
and111*111
- wim