# Why can't $\pi$ be expressed as a fraction? [duplicate]

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?

• Because the notion of a "fraction" is when both, numerator and denominator, are integers, which in a circle cant happen. Mar 26, 2013 at 7:57
• In some sense, it can be
– user45099
Mar 26, 2013 at 8:02
• Pi can be expressed as the ratio $\pi/1$. It cannot be expressed as ratio of two integers (i.e. it is not rational). Being expressible as a ratio of integers has nothing to do with technology.
– anon
Mar 26, 2013 at 8:07
• Here are some reasons: en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational I suspect that you may not find these particularly helpful. If I can think of a more accessible explanation I will post it. Mar 26, 2013 at 8:11
• If you construct a circle with a radius of the visible universe out of hydrogen atoms, then you can only measure $\pi$ to an accuracy of about $2\times 10^{-37}$. $\pi$ was known to about this accuracy back in 1630. Jul 8, 2014 at 1:59

"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.
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.
• On the other hand, there are only countably many computable real numbers, and only these I would consider to arise in "nature". I wonder what is the density of $\mathbb{Q}$ in them? Mar 26, 2013 at 8:40
• @arbautjc: The analogy is slightly shaky in that in a precise measure-theoretical sense the fraction of real numbers that are rational is $0$, whereas the fraction of all humans that are currently alive is roughly 6% :-). Mar 26, 2013 at 9:14