## A Venn Diagram of the Real Number System

I’m aware that it isn’t properly exactly a Venn diagram, now, but the mathematics-artist Robert Austin has a nice picture of the real numbers, and the most popular subsets of the real numbers, and how they relate. The bubbles aren’t to scale — there’s just as many counting numbers (1, 2, 3, 4, et cetera) as there are rational numbers, and there are far more irrational numbers than there are rational numbers — but if you don’t mind that, then, this is at least a nice little illustration.

## A Picture Showing Why The Square Root of 2 Is Irrational

Seyma Erbas had a post recently that I quite liked. It’s a nearly visual proof of the irrationality of the square root of two. Proving that the square root of two is irrational isn’t by itself a great trick: either that or the proof there are infinitely many prime numbers is probably the simplest interesting proof-by-contradiction someone could do. The Pythagoreans certainly knew of it, and being the Pythagoreans, inspired confusing legends about just what they did about this irrationality.

Anyway, in the reblogged post here, a proof (by contradiction) that the square root of two can’t be rational is done nearly entirely in pictures. The paper which Seyma Erbas cites, Steven J Miller and David Montague’s “Irrationality From The Book”, also includes similar visual proofs of the irrationality of the square roots of three, five, and six, and if the pictures don’t inspire you to higher mathematics they might at least give you ideas for retiling the kitchen. Miller and Montague talk about the generalization problem — making similar diagrams for larger and larger numbers, such as ten — and where their generalization stops working.

Yesterday I came a across a new (new to me, that is) proof of the irrationality of $sqrt{2}$. I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.

Apparently the proof was discovered by Stanley Tennenbaum in the 1950′s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).

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