18 February 2021

Golden Maths — Advanced arithmetic (Hurwitz's Theorem)

Among the countless golden number myths, there is a very intimidating one: it is supposedly the most irrational number. Impressive, no? All the more so because to the uninitiated 'irrational' sounds like 'too subtle for the human ratio' — while in fact 'ratio' is here simply referring to the 'fractions' of primary school, ratios of integers. To any mathematician it is clear that numbers like 𝜋 or e are incomparably 'more irrational' than golden 𝜑, whose sole complexity is a square root. In a way, 𝜑 is even the simplest irrational number, because it has the simplest possible continued fraction: 𝜑 = [0;1,1,1,...]. What people mean by the strange expression the most irrational number is in fact: the hardest to approximate by rational numbers. It is true that the golden number belongs to the family of numbers that are the hardest to approximate by rationals, but among its relatives it is not the biggest and not the smallest; they come as big or small as you want. But in a certain way it can be said that 𝜑 is the simplest among the irrational numbers allowing the least number of approximating fractions. 

Read all about it here. The paper is self-contained, and doesn't rely on external references. It is indebted to

Ivan Niven, Herbert S. Zuckerman  and Hugh L. Montgomery, An introduction to the theory of numbers, Fifth Edition, 1991, Chapter 7.