TITLE: What is the last digit of $\pi$?
QUESTION [10 upvotes]: 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.

REPLY [8 votes]: 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.)