Why does pi has so many digits?

Question

Pi is used in calculating circles, spheres and round things. In calculate, pi is written down as 3.14. But the full pi is incredibly long (over a TRILLION digits). So, why there are so many digits in pi?

Answer 1

Terminating decimals (ones with only finitely many non-zero digits after the decimal point) can be written as fractions $\cfrac ab$ where $a$ and $b$ have no common factors, and $b$ has only the prime factors $2$ and/or $5$ - so $b$ could be $625$ or $200$ or $1024$, for example.

Any other fraction written in lowest terms will have a decimal expansion which eventually recurs (repeats). But we can all write down decimal expansions with patterns which don't repeat - these cannot be written as fractions - we say they are not rational numbers.

Georg Cantor famously proved that there are strictly more decimal expansions than there are rational numbers. There are so many decimal expansions possible that the number is uncountable, while the rational numbers (with expressions which eventually terminate or recur) are countable. The vast majority of "real" numbers are not "rational" numbers.

For numbers which emerge not as fractions but from geometry or algebra it is sometimes difficult to determine whether they are rational or not. It was Lindemann who eventually resolved the status of $\pi$ as a transcendental number - showing that it is not rational, and that it is also not the root of a polynomial which has rational coefficients (not an "algebraic" number). So the decimal extension of $\pi$ will never terminate or recur.

There is another property which most "real" numbers have - given any sequence of digits, that sequence appears in the decimal expansion infinitely many times with the frequency you would expect if the digits in the expansion were chosen at random. Being a normal number is a tricky property and it is not known whether $\pi$ is normal or not.

This is material worth exploring yourself, though some of it can be quite challenging. Hardy and Wright's "Introduction to the Theory of Numbers" covers decimal expansions in some detail - some of which is elementary and some requires more mathematical background.

I trust this brief survey shows you how much there is to uncover in the fascinating world of numbers.

Answer 2

$\pi$ is very long indeed. In fact, it has way more than a trillion digits when you try to write it down in decimal: it has infinitely many digits. That means that, no matter how many digits you write down, there are still more left over.

Why is this true? Well, I won't repeat the proof here for you, but here's an answer to a related question: actually, almost all numbers are like this! "Nice" numbers like $17$ and $\frac{2}{5}$ are comparatively rare. Even a number like $\frac{1}{3}$ has a decimal expansion that goes on forever, though $\frac{1}{3}$ is still kind of nice, because its decimal expansion repeats forever ($0.333333\dots$). Most numbers are numbers like $\sqrt{2}$ and $\pi$, which can't be written down either as finite terminating decimals or as repeating decimals, but have "infinitely many" digits.

It's usually also hard to prove that a specific, given number has this property, which is why I don't want to prove it for $\pi$. But there are some easy special cases. For example, it's quite easy to see that the number $\sqrt{2}$ must be "irrational", and therefore that its decimal expansion can't either terminate or repeat. You can look up this proof yourself.