My knowledge about the foundation history of irrational numbers was challenged today, and I’m pretty happy about it.

I had recently tweeted a Vi Hart video that gave a fun, geometric proof of the classic first proof of irrationality: [latex]\sqrt{2}[/latex] is irrational. If it weren’t, that would mean you could build a square that had integer sides and an integer diagonal, and that would allow you to build a smaller square with the same process. To get a contradiction, repeat until you run out of integers.

After I tweeted the video, I got a response claiming that…

I was somewhat taken aback. In source after source, I’ve seen [latex]\sqrt{2}[/latex] named as the first number ever proved irrational. Variations on the same proof abound. And here was a claim that [latex]\phi[/latex], the golden ratio, actually holds the rightful place in history as humankind’s first brush with “the unnameable.” There seems to be a historical argument; but how complicated is the proof?

In fact, it’s so wonderfully simple that there’s a pedagogical argument to be made for teaching that [latex]\phi[/latex] is irrational before we even mention the Pythagorean Theorem or square roots. You need to know how to find angles in regular polygons and chase them around diagrams, and know how Isosceles triangles work. The thrust of the proof is the same as for [latex]\sqrt{2}[/latex], but it sidesteps the parity argument that can sometimes feel less tangible.

Let’s imagine, as the Pythagoreans might have, that every number is rational. An equivalent way to state this is there is always some scaling of any pair of lengths that allows them both to be positive integers. (To the Pythagoreans, the relationship between any two lengths was identical to the relationship between two whole numbers, axiomatically.) So suppose we have a regular pentagon with integer side length a and integer diagonal b.

The ratio of b to a, is precisely the golden ratio, by the way. But we don’t even need to know what it is. We’ll just try to show that a and b can’t both be integers.

First off, chase some angles around and you’ll see pretty much every angle is either 36, 72, or 108 degrees. This gives a bunch of Isosceles triangles. It follows quickly that

[latex]x = 2a – b[/latex] [latex]y = b – a[/latex] [I’ll leave that piece as an exercise. It’s pretty satisfying to chase angles around and have everything come out nicely.] This implies that x and y are positive integers. But they are the side and diagonal of a regular pentagon again, so the argument repeats! And this is the crux of the problem: positive integers can get smaller for only so long before they run less than 1. (Just like the infinite regress that was hinted at in that error-laden but inspiring work, Donald in Mathemagic Land.)Conclusion? The original pentagon couldn’t have been drawn with integer sides to start with. And that means it couldn’t have been drawn with two rational sides, or else we would have scaled them up to whole numbers. And that means the relationship between the side and diagonal of the regular pentagon is irrational.

And there we have it. Irrational numbers without actually dealing with numbers at all. Or evenness and oddness of numerators and denominators.

A delightful discovery. We’ll likely never know for sure what length the Pythagoreans proved irrational first, but that’s a strong claim for [latex]/phi[/latex] over [latex]/sqrt{2}[/latex].

Especially because, as Donald found out, the Pythagoreans were all over pentagons and the golden ratio.