I have been looking at the popular proofs that rational numbers have limitations when trying to define real-world lengths.
For instance, there does not exist a rational $c$ such that $c^2=2$. Basically, proofs focus upon the fact that all square roots of natural numbers, other than of perfect squares, are irrational.
I would like to see other illustrations of proofs that show existence of irrational numbers without relying upon square roots of natural numbers. But on other areas of mathematics.
A rational number's decimal representation is terminating or eventually periodic. So, for example, consider $0.1101001000100001\ldots$, where the numbers of $0$'s between successive $1$'s are $0,1,2,3,4,\ldots$. This is neither terminating nor eventually periodic, so this number is irrational.
[EDIT] BTW, this particular example happens to be $$\frac{10^{\frac{1}{8}} \vartheta_{2}\! \left(0, \frac{1}{\sqrt{10}}\right)}{20}$$ where $\vartheta_2$ is a Jacobi theta function.