Is it possible that pi is finite, but we haven't calculated it in full?

My first reaction to this question, and many others like it, is exasperation. Really? Again? But let me start with a few straight up answers, before I indulge in some measured ranting.

It’s not possible that the decimal expansion of \pi is finite and we just weren’t patient enough. \pi is an irrational number, and irrational numbers have non-terminating, non-repeating decimal expansions. This is fact, not belief or hope (try this Wikipedia article, or How do you prove that \pi is an irrational number?)

It is possible that whoever taught you about the decimal expansion of \pi incorrectly presented it as “we’ve never managed to reach the end” or “we’ve never found a pattern” or “we haven’t been able to calculate it in full.” Such statements are nonsense. We know what \pi is with complete precision, and we can prove that it is irrational, just like we can prove that \sqrt{17} is.

This has been known for 250 years now.

Also, you should realize that decimal expansions are usually terrible at representing numbers “in full”. They can’t even represent 1/7 in full. There are many other ways of representing numbers, and \pi has plenty of perfectly finite and clear representations.

Finally, do not be misled to think that “\pi is infinite.” It’s not. It’s a finite number, less than 4. Once again, the decimal expansion is just confusing you. You wouldn’t say that 1/7 is infinite, right? Or \sqrt{2}?

And now… what I’d really like to know is: Why? Why does the question keep coming up, over and over and over and over and over again? Why are people so worried about the decimal expansion of \pi? What’s the problem with it being infinite? Is it some kind of blemish on our existence? A hurdle we need to work around? A problem? A concern? Why?

Can \pi be a finite number in a non-Euclidean space?

Is there a base where pi is a rational number?

Could we create a math that forces certain non-real numbers like Pi to be finite?

What would the consequences be if Pi had finite number of digits?

If we cannot calculate \pi exactly, how can we claim with confidence that it requires infinitely many decimal places to represent?

Is it possible to fully compute all the digits in the decimal representation of pi?

Why haven't we found the exact value of \pi yet?

How many digits would PI have if it ended?

Will we be able to complete \pi?

Is pi a finite or an infinite number?

The question, in hundreds of variations, has been answered hundreds of times. \pi is a finite number. It is irrational. It is computable. It is finitely definable. It has “a pattern”. Some of its representations (the poor ones) require infinitely many characters. Plenty of other representations (the much better ones) only require finitely many.

The digits in the decimal expansion of \pi don’t matter. Seriously, please, don’t worry about them. Many, many numbers have infinitely long decimal expansions. Some are repeating. Most are not. It’s ok. It’s not an issue. It’s not a problem we need to fix, it’s not a problem we can fix, and in fact it’s not a problem at all.

Decimal representations are often the worst possible way to think about a number.