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This article is adapted from Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else by Jordan Ellenberg. © Penguin Press 2021.
One of the great charms of number theory is the existence of irrational numbers—numbers like the square root of 2 or π that can’t be expressed as the ratio of any two whole numbers, no matter how large by Jordan Ellenberg, professor of mathematics at the University of Wisconsin and the author of Shape and How Not to Be Wrong. He blogs at Quomodocumque.
Photo: Getty Images Plus
The legend goes—probably false, but hey, it makes a point—that the discovery of the irrationality of√2 was so disconcerting to the Pythagoreans, who wanted all numbers to be rational, that they threw the discoverer into the ocean.
Among the mysteries of the irrationals, one number holds a special place: the so-called golden ratio. The golden ratio’s value is about 1.618 (but not exactly 1.618, since then it would be the ratio 1,618/1,000, and therefore not irrational) and it’s also referred to by the Greek letter φ, which is pronounced “fee” if you’re a mathematician and “fie” if you are in a fraternity. If you want an exact description, the golden ratio can be expressed as (1/2)(1+√5.)...
The golden ratio doesn’t arise only in geometry; in the Fibonacci sequence, where each number is the sum of the two previous ones (1, 1, 2, 3, 5, 8, 13, 21, 34, …), the ratios between consecutive terms approach φ more and more closely as the terms get larger and larger. (But of course those ratios never arrive at φ, because, again, irrational!)...
But my favorite thing about the golden ratio has nothing to do with pentagons or Pepsi. It’s that the golden ratio, among all irrational numbers, is the most irrational one.
What can that mean? Either a number is the ratio of two whole numbers or it isn’t*.
Source: Slate Magazine
