eli5 Is there a reason that the decimals of pi go on forever (or at least appear to)? Or do it just be like that?

Question

Edit: Thanks for the answers everyone! From what I can gather, pi just do be like that, and other irrational numbers be like that too.

In: 5168

Answer 1

It’s hard to answer the “is there a reason” question, because it is something of a mistaken way of looking at it. You might ask, is there a reason why 4 is between 5 and 3 on the natural number line? That is just what it is to be the number 4. There is not much sense to be made of the question of why it is the the number between 5 and 3. There is no causal or historical explanation that can be appealed to.

There are, however, adjacent questions that can be asked, which might help you better understand what is going on. The number line consisting in whole numbers in sequence, like 1,2,3,4… and so on, is the natural numbers. When we add decimals into the mix, we get numbers between the natural numbers. The line of numbers called “The Reals” includes all of the numbers that are between the whole numbers, like 1,2,3,4. The Reals have a property that is sometimes called “density.” What this means is that for every two numbers, there is another number in between them. Between 1 and 2, there is 1.1. Between 1 and 1.1, there is 1.01. Between 1 and 1.01, there is 1.001. Etc. If you do this enough, you’ll realize very quickly that this process can go on forever, creating an infinite expansion of decimal places for each new number you generate using this method. So, it is just a property of real numbers that they go on forever like this, with an infinite decimal expansion that is the result of this density property. Pi is just one of the numbers on the line of the Reals and it goes on forever.

What makes Pi unique among numbers that go on forever is that it is an irrational number. This means that it cannot be expressed as a fraction. This also means that the numbers in the line don’t repeat. Contrast this with a number like 1/3. This is .33333…(3’s repeating infinitely). Pi and the number picked out by 1/3 both go on forever. But Pi does not repeat.

So, what is the reason why Pi goes on forever? Because it is a real number and Real numbers have infinite decimal expansions creating a dense number line. Why are they like that? That’s just what it is to be a Real number. Might as well ask why does a square have 4 sides. That’s just part of what it is to be a square.

In short, it just be like that.