The moment when humans made the abstraction from a set of objects to its cardinality and thus discovered counting is lost in history. There are other moments of similar impact that we know more about. Today, we are so familiar with numbers that we often forget that they are used to measure quantities and even ignore units, confusing distances and durations.
For the early pre-Aristotle Greek mathematicians, lengths were not numbers at all. Numbers occurred as proportions, as ratios of lengths (or durations). One segment could be say twice as long as another segment. More generally, the Greeks called two lengths commensurable if their ratio is, in modern terms, a rational number. They would detect this by fitting one number, like 30, as often as possible (once) into another number (43), take the remainder (13), fit that in the second number (30) as often as possible (twice), take the remainder (4) etc. etc. What emerges is the continued fraction above, terminating eventually, because the denominators get smaller and smaller.
If the lengths are not commensurable, the continued fraction becomes infinite, like the one above. For the Greeks, this expression was essentially an algorithm. An infinite fraction is a mind boggling thing. How does one even compute it?
Geometrically, this can infinite continued fraction arises by comparing the side length 𝝋 of a regular pentagon to a segment of length 1 of its diagonals. Simple similarity of triangles tells us that 𝝋=1+1/𝝋. Rewriting this once leads to 𝝋=1+1/(1+1/𝝋), and if we keep going a little while longer, we arrive at the infinite continued fraction above. This reproduces how the Greeks proved that the Golden ratio 𝝋 is irrational (if it was rational, the continued fraction would be finite).
Similarly, the above dissection of a unit square into a rectangle shows that (√2-1)(√2+1)=1. This is arithmetically easy, but the concept of a root of a number didn’t make sense to the Greeks in the early days. This equation is the essential ingredient to prove the continued fraction expansion of √2 (and thus its irrationality).
Of course, the standard proof by contradiction that is taught today (and which probably goes back to Aristotle) makes the cumbersome process of finding continued fractions cumbersome. We will see next week that they still serve higher purposes.