Why can't $\pi$ be expressed as a fraction?

Question

Why can't $\pi$ be expressed as a fraction?

If pi is the ratio of a circle's circumference to its diameter, why can't we simply take a circle, measure its circumference and diameter, and derive the fraction?

Say we have a string of some length and we place it such that it forms a circle. Then we will know the circumference and we can measure the diameter. The diameter might be difficult to measure but its length surely is some fixed number. If it's not possible to do this, does it that mean that the limit to determining the exact value of pi is only technological and not mathemetical?

Answer 1

"The diameter might be difficult to measure but its length surely is some fixed number." Yes, it is some real number, and as such is an infinite decimal to begin with, and the same is true for the circumference. Only if you are very lucky, or if you have made your circle to measure, you can have, e.g., the diameter $d$ with a terminating decimal, say $d=1$. Therefore the ratio between circumference and diameter is the quotient of two infinite decimals before we even think about it in mathematical terms.

Above all you have to be aware of the following: At math stackexchange we are not talking about tape measures and circular disks made up of atoms, but about mathematical circles, which only exist in our brains. These circles are objects of various mathematical theories (euclidean geometry, calculus, etc.), and only in such a "spiritual world" we can think about the ratio between circumference and diameter of a circle in general terms. Doing this, among other things we shall find out that for circles drawn on the spherical surface of the earth the ratio in question is not a constant.

Answer 2

Most real numbers are not rational, where "most" can be made precise in a number of ways. For example we may cover the rational numbers with a collection of open intervals such that the sum of the lengths of the intervals used to cover the rationals is smaller than any $\epsilon>0$ (this just comes from the fact that the rational numers are countable), meaning that that rationals take up an extremely small part of the real line. Another way to state this in terms of measure/probability theory is that if you pick a real number at random the probability of getting a rational number is zero. Thus it seems to make sense that most numbers we encounter in nature should be irrational ($\pi$, $e$, the golden ratio $\tau$, etc.), as there are a bunch more of them to choose from than rational ones.

Answer 3

If you tried to express it as a fraction you could put 22/7 but this is only an approximation, the more accurate you get the larger and more abstract the fractions will get but as pi is irrational you will never be able to fully express it as a fraction.