If a number is irrational (meaning "not a ratio", not "crazy"), it goes on forever. That's because if it didn't go on forever, we could express it as a fraction of the number up to where it ends, divided by 1 followed by the number of decimal places in the fraction. In 1768 it was proved that pi is irrational. I don't know what proof was used then, but here is a more modern proof. It might well make your brain hurt a bit, so to give you a picture of how such things are proved, here is a simple proof that the square root of 2 is irrational:

First a couple of basics:
An odd number times an odd number is always an odd number.
An even number times an even number is divisible by 4: that is to say, if you divide it by 2 the answer is still even.

To be rational, we can write the square root of 2 in the form A/B with whole numbers A and B such that it is one of

ODD/ODD,
ODD/EVEN or
EVEN/ODD

We can ignore EVEN/EVEN because we could divide through by 2 without affecting the fraction until we got one of the two forms above.

So A/B = sqrt(2)

Square both sides:

A*A/(B*B) = 2

and rearrange it to

A*A = 2*(B*B)

Now,if A is odd, A*A is odd... but twice anything is even, so it can't be equal to the other side of the equation whether B is odd or even.

If A is even, A*A is divisible by 4, so we can cancel the 2 on the other side, and (A*A)/2 is still even. But B is now odd, so B*B is odd, and the two sides still don't match up.

So we must conclude that to express the square root of 2 as a ratio of whole numbers, one of the numbers must be neither odd nor even, and there aren't any such numbers, so the suare root of 2 is irrational.