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I want to know: what is the last digit of $\pi$?

Some people say there are no such thing, but they fail to mention why.

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Sometimes even rational numbers don't have a "last digit", think of 1/3=0.33333... –  Vhailor Jul 21 '11 at 22:12
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Nice Poem. ${}$ –  jspecter Jul 21 '11 at 22:13
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If $\pi$ has a last digit, then $0.999\ldots\neq 1$. –  Asaf Karagila Jul 21 '11 at 22:13
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This question should specify "base 10". The "no last digit" phenomenon depends on how $\pi$ is represented. To take a contrived setting, base-$\pi$ numbers, then $\pi$ is written as $1$. I'm not trying to be pedantic here: representation is a fundamental part of this question. –  Fixee Jul 22 '11 at 6:09
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@jspecter: A slightly enhanced version: $$ \text{I desperately want to know}\\ \text{The last digit of $\pi$}\\ \text{Some people say there's no such thing}\\ \text{They fail to mention why} $$ –  joriki Dec 22 '12 at 3:49
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6 Answers

up vote 13 down vote accepted

There is no "last" digit of $\pi$. If there was a last digit, then there could only be finitely many digits in front so that $\pi$ would be a rational number. However $\pi$ was shown to be irrational in the 18th century by Lambert.

(This Meta.StackExchange post is a joke based on the impossibility of finding such a last digit)

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Thanks for the link to the Meta.StackExchange post: hilarious. I've got my laughs in for the day. We all need a good dose of humor now and then! –  amWhy Jul 21 '11 at 22:16
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Since you may have never seen the topics in my colleagues' answers, I'll try to explain them in some detail.

Suppose for the sake of argument that when $\pi$ is written as a decimal expansion ($3.1415 \dots$) it does have a final digit. This would clearly imply that there is a finite number of terms in the expansion. All real numbers with finite decimal expansions can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers (whole numbers).

By this reasoning we conclude that $\pi = \frac{a}{b}$ for some positive integers $a$ and $b$, i.e., that $\pi$ is rational. This is the starting point for this short proof given by I. Niven in 1946, which is especially easy to follow if you've had a little trigonometry and even less differential calculus. The proof concludes with an absurdity like the existence of an integer between $0$ and $1$, which implies that $a$ and $b$ do not exist and $\pi$ is irrational (and has an infinite decimal expansion). It should be noted that the irrationality of $\pi$ was first established by Lambert in 1761 by studying the continued fraction expansion of the tangent function and using the identity $\tan \frac{\pi}{4} = 1$. More generally, he proved that if $x$ is rational, then $\tan x$ is irrational.

In short, there is no final digit in the decimal expansion of $\pi$ because it is irrational.

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Sadly, your link to Niven's proof doesn't work. –  jprete Jul 22 '11 at 4:53
    
Ok, try it now. –  user02138 Jul 22 '11 at 5:25
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Proving that $\pi$ is irrational is more difficult than proving that $e$ or $\sqrt{2}$ or $\log_2 3$ is irrational. See http://en.wikipedia.org/wiki/Proof_that_pi_is_irrational .

Proving that an irrational number has no last digit is easier than that: http://en.wikipedia.org/wiki/Irrational_number

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Even rational numbers usually have no "last digit": what is the last digit of $$ 0.1313131\dots = 0.\overline{13} = \frac{13}{99} ? $$

So what sort of numbers have a last digit?

One, numbers with a terminating decimal expansion: numbers like $\displaystyle 2.23627 = \frac{223627}{100000}$. As you can see, all such numbers can be written as a fraction with denominator being a power of $10$.

Two, depending on your definition of "last digit", numbers like $0.4677777\dots = \frac1{100}46.77777$ = $\displaystyle \frac1{100}\left(46 + \frac79\right) = \frac1{100}\frac{421}{9}$. These numbers can be written as $\displaystyle \frac1{10^k} \frac{n}9$ for some integers $k$ and $n$.

So a number $x$ has a "last digit" if and only if $(9\cdot 10^k)x$ is an integer for some $k$. Only very special numbers are of this form, and it should be no surprise that $\pi$ is not. (Admittedly, I don't actually see how to prove this without invoking $\pi$'s irrationality, but it's a much weaker property.)

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HINT $\rm\ \pi = 3.1415\ \Rightarrow\ 10^4\: \pi = 31415\ \Rightarrow\ \pi = 31415/10^4\ $ is rational, contra Lambert's proof.

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The last known digit of $\pi$ is 1, which is the ten trillion and fiftieth digit.

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I think that this is incorrect and potentially misleading. E.g. see this answer that mentioned that the quadrillionth digit is $0$. Perhaps more relevant to your answer, computing specifically the ten trillion and fifty-first digit is not a major feat. –  Jonas Meyer Jun 20 '12 at 14:48
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It is oddly arbitrary to talk about the last "known" digit of $\pi$ since a subsequent and presumably unrelated digit can be found with little effort using the results of the previous computation, i.e. BBP. In that sense it is sort of like saying the largest known number is a googleplex. –  Dan Brumleve Nov 2 '12 at 10:02
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protected by Qiaochu Yuan Jul 21 '11 at 22:17

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