This is a request for help, not an attempt to challenge anything.
Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance accross.
That boggles my mind, but maybe it should not.
I think crazy thoughts like: "this means that a the path of a circle around a unit lenth line segment has a non existant length".
Is there a way to accept that the number is irrational and not break from reason?
That $\pi$ is irrational does not mean that it's impossible to express the distance around a circle in terms of the distance across. It only means it can't be expressed as a ratio of integers.
BTW here you can read some proofs that $\pi$ is irrational.
It is quite easy to prove that $\sqrt{2}$ is irrational. It's even easier to prove that $\log_2 3$ is irrational. But it's hard to prove $\pi$ is irrational.
Here is a related question that may be of interest, where you will read about how to prove some numbers are irrational.
Some numbers are indeed impossible to express. Those are the numbers that are not definable. Easier to deal with, but still difficult, are numbers that are not computable.
However, it is not so difficult to provide a clear definition of the quotient of the circle's circumference by its diameter. We can let the matter stand there and simply name the quotient "pi", or we can do like Archimedes and define that quotient to be the limit of the regular polygon's perimeter to its diameter (using either choice of the inner or outer diameter) as the number of sides approaches infinity. We have the epsilon-and-delta definition of limits to help convince us that this is a valid definition.
Well, contrarywise, is there any reason why $\pi$ should be a rational number? A rational number just means it's expressible as a ratio between two whole numbers. Is there a reason that $\pi$ (or any other arbitrary value) need to be expressible a ratio of two whole numbers?
Well, I can think of two intuitive (but wrong) reasons. But I want to express that, in a way, irrational numbers intuitively make more sense than rational numbers.
Consider distance and space. Surely it is a continuum and every possible distance is going to exist and they are infinite. There aren't any holes and jumps between points. So let's say you put up a sign here and say "This is here!" and one mile you put up a sign saying "this is one mile" and you put sign post each exactly splitting the mile into 6 parts so you have posts every 1/6 mile.
Now take a pea-shooter and shoot it anywhere randomly to the mile line. Where does the pea land. Well, if it hit exactly one of the sign posts that'd be kind of unlikely. So says someone (there's always someone)we can split the distance into smaller pieces, into 60ths of miles 120ths miles, thousands of miles, 573th of a mile. But is there any reason why the pea should match up perfectly with any of those? In fact doesn't it seem unlikely?
Okay, here come Pythagoras just walking down the street (what's he doing here? Don't ask) and he says "Everything is whole numbers; there must be some precise number that cuts this mile to precisely that point." And you say to him "Why? Where did you get that idea? Why should that be?" And he basically says it'd make life perfect and nice and it'd be religiously beautiful if it were so.
Well, okay, but what about $\sqrt 2$ you say. He gets a dirty look and suddenly you are very, very glad you are not on a boat.
Okay, that was a fantasy. I think in a naive and simplistic way it's intuitive to think that because we can split an apple into rational number parts, it should work the other way, and we should be able to take any value and find the parts that broke it away from the whole. But we shouldn't assume naive ideas and if we view numbers as continuum distance, instead of discrete apples and distinct apple slices. It shouldn't be intuitive anymore.
Consider decimals. To get from 3.7 to 3.8 you have to pass through 3.75 first and to get to 3.75 you must go through 3.726 and each precision deeper there are infinite degrees of precision. As we swim past them we are going from point to point continually. The Pythagorases of the world would have as jumping from descrete whole number slice to whole number slice. There isn't any reason to think that this is the correct way to view this. And, it turns out, it isn't the correct way to view.
What is reality? Is it swimming through a continuum, or is it jumping from precise knife cut to knife cut?
So numbers are a continuum and there exist infinitely many numbers that simply aren't a discrete j/k amount for a precise whole number of exactly $1/k$ slices. And why shouldn't there be?
Irrational numbers can and do correspond to length
Yes. An irrational number is a number that cannot be expressed as a ratio between two integers. It does exist as a length. For example, $\sqrt{2}$ is famously irrational, but it is extremely easy to show it as a length: just draw the diagonal of a square of side $1$.
Conceptualisation of irrational numbers
The whole rational vs irrational business started in ancient Greece. The Pythagoreans (the members of a secret group of mathematicians) believed that the whole universe could be described in terms of whole numbers: $1,2,3,4,...$
They were doing geometry, and they realised that some lengths were not integer multiples of other ones (more than $2$ times as long, but not quite $3$ times).
That's when they invented ratios: relationships between numbers. For example, $2$ is to $3$ just like $4$ is to $6$, in the sense that if you were to draw a rectangle with sides $2$ and $3$ and, next to it, one with sides $4$ and $6$, they would look the same (try it!). We express that relationship, by saying: \begin{equation} \frac{2}{3} = \frac{4}{6} \end{equation}
Since they have faith in the integers, they believed that any length could be understood this way, two lengths coming from the simplest shapes: the circumference of the circle of diameter $1$ and the diagonal of the square of side $1$.
Now you know that it turns out that both $\pi$ and $\sqrt{2}$ are irrationals. So what does it mean? It just means that it cannot be expressed as a ratio!
Check out this video for a brilliant exploration of all of this!
