Good example. It's amazing how quickly you leave the rationals, even with a square, you almost immediately need numbers that can't be expressed as the ratio of two integers... historically, was that the first encounter with an "irrational" number (Pythagorean theorem applied to a right isosceles triangle)? I do remember the proof from number theory about 20 years ago, though I could never recreate it now from memory.

And algebra itself is one of those real things that calculus helps to explain!

Take, for instance, the Fundamental Theorem of Algebra, where the proof in terms of elementary complex analysis and/or topology closely related to complex analysis make the truth of the theorem geometrically obvious, the (mostly) algebraic proofs I've seen leave me with little more than the desire to re-check the proof, because I'm not at all certain that something equivalent to the F.T.A. hasn't been implicitly assumed at some point in the proof.

Now it may be "just me" — I've always had an easier time following analytical proofs than abstract algebraic ones — but just thinking of the necessary prerequisites — homotopy between maps defined by complex polynomials vs. what? Galois theory? — I don't think it's just me.

Incidentally, complex numbers are another case where, as with the irrationals, a poorly-chosen name has made simple and quite generally useful ideas seem esoteric to those not already familiar with the subject.

> Try to cut that square in half diagonally and things get irrational really fast!

In theory. In reality you have an integer number of atoms in one half and another integer number in the other.

There are no irrational numbers to measure, and squares made from "continuum material" do not exist in reality.