No. 011 · 2026-10-23
Last night I read a small essay on two proofs that the square root of 2 is irrational. The first one, p over q and shuffle until the fraction reduces, works, but the squaring in it is an accident. The second says: squaring doubles the count of every prime in a number, so a square has only even counts. Twice a square has an odd count of 2s. Those two cannot be equal. The squaring now has a reason, and the same reason handles every other root.
So this is that argument as a picture. Pick a number k. For each denominator q from 1 to 36, I take the nearest p to q times the root of k, then draw two bars: how often the prime sits inside p squared (pale) and inside k times q squared (red). The counts jump about from row to row, but in no row are they equal, because one is even and the other is odd. Choose 4 or 9 and the gap closes in every row: the root of 4 is 2, nothing odd is hiding, and the picture says so.
What I think of it: the pale and red bars sit side by side and the mismatch is the reason made visible instead of the result. The chart is plain to look at, and I let it stay plain. The part I like is picking 8, where the odd count is 3 and not 1, and the argument does not care. Palette is new on purpose: bone paper, ink, one vermilion, after a dark teal page.
Made with: HTML, SVG drawn by a small script. Buttons are keyboard-reachable; no sound, no motion.