MathematicsHow many digits of $\pi$ are currently known?
[-7] [3] Yatin K
[2012-05-31 08:46:35]
[ reference-request pi ]
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How many digits of $\pi$ are currently known?

(1) Purged comments as they mostly refer to a previous version of the question. After Gerry's edit it is a completely different question and actually has an (albeit non-definitive and can stand to change with time) answer. - Willie Wong
(1) Re-opening as in the newly edited form it is not a duplicate of the chosen duplicate target. - Willie Wong
(1) You may also want to read-up on the BBP extraction algorithm which allows one to compute the $n$th binary bit expansion of $\pi$ without computing the previous ones. - Willie Wong
(5) What do you mean by "known"? Nobody remembers the ten trillion. And they are not written down on paper. So we are talking about a computer beeing able to reproduce them. So all its digits are known. - TROLLHUNTER
(3) @Xnyyrznaa: It is good that they are not all written down on paper, as this page points out. - user1729
(9) Old comments from downvoters (me being one of them) are purged since OP was fixed to a normal shape, and these comments don't reflect the current situation. Well, why the downvote? 1. I think the first version of OP is something the questions should be judged for as well. 2. You plagiarized another use to answer your own question. That answer you accepted. 3. Seems like you used another account of yours "Yatin K" to support yourself, and grant yourself a bounty. I think it is enough for to downvote. It is even too much - Ilya
:P No its the other account - Yatin K
[+28] [2012-05-31 09:40:35] user 726941

The current record is ten trillion and fifty digits. [1]

The last known digit is 1 .


(1) @YAK Try reading this. - user 726941
[+22] [2012-06-01 09:46:29] user1729

How many digits of $\pi$ are currently known? Well, all of them!

It is possible to compute the $n^{\text{th}}$ digit of $\pi$ without computing the preceeding $n-1$ digits first! See, for example, Wikipedia [1].

The problem with the question as stated, I believe, comes down to defining "known". @Xnyyrznaa makes an excellent point in the comments above when he says "nobody remembers the ten trillion...they are not written down on paper." So, in some ways, skullpatrol's answer is somewhat unsavoury. We "know" that the ten-trillion-and-fiftieth digit is a one, but we do not "know" the rest; not by any reasonable metric (that is, according to his answer).

The concept of "know" corresponds to a function. In life, because everything is finite, this function is simple - it is the "look it up on a list" function (and is the function skullpatrol is getting at). In mathematics, this function can be pretty exotic. My point is that a much better function exists for finding a specific digit of $\pi$.

For instance, Wikipedia [2] tells us that the five-trillionth, 40-trillionth and quadrillionth digits of $\pi$ are all zero.


(5) In fairness, one might argue that not all digits are known, since for truly vast $n$, well, even just inputting and storing $n$ is hard enough, let alone running that formula on it... - Ben Millwood
(6) Well, yes, but you will forgive me the use of exaggeration to get my point across? (If an algorithm existed which was $O(n)$ then I might try defending myself. However, the algorithm is $O(n^2)$, I believe, so really I don't have a leg to stand on!) - user1729
[-1] [2013-10-18 07:54:03] user101454

Also I think that this is sort of misconception that how many digits oh $\pi$ are known. $\pi$ is a number in itself, whose digits need not be known as it is an irrational number, a type of undefined abstraction which is used to model certain things in our nature, like the length of hypotenuse of a unit isoceles right triangle. But what can be known is a rational number of the form $x/10^n$ where x,n belong to the set of integers that is nearer to $\pi$ compared to all other rational numbers of that form.

What do you mean by "undefined abstraction"? Is this referring to irrational numbers? As these are rigorously defined using Dedekind cuts. - user1729