Mathematical Stories 3 – Irrationality and Infinity

The Ancient Greek Pythagoreans discovered the existence of irrational numbers in the fifth century BCE, to the legendary demise of one of their number, Hippasus, and were perturbed by them since they contradicted the firmly-held belief that everything related to number and geometry came back to natural numbers and their ratios (rational numbers).

The Greeks were rather late to the party, though. It’s thought that Indian mathematicians were thinking about irrational numbers a whole three hundred years earlier – Manava (c. 720 BCE) thought that certain square roots could not be determined – but, as with many things in the history of mathematics, it is not always clear where, or from whom, an idea originated. Much like any academic history, mathematics is a mix of human thought across time and space.

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Mathematical Stories 2 (or Shakespeare never saw a decimal point)

I wrote a blogpost on some interesting stories in the history of maths a couple of years ago which I was reminded of chatting to some historians on Twitter yesterday. The history of maths is not something you really meet until you’re an undergraduate mathematician, and that tends to be only an introduction. It’s something you have to choose to pursue to find much out.

Over the years I have read more and more and the thing that strikes me repeatedly is the way our subject has evolved over time. Take any one topic and its current incarnation seems to be the result of thousands of years of both disparate and conjoined thought across many people on almost every continent, which makes tracing their history quite difficult. Over a few posts I am going to take some topics and look at where they have come from.

Mathematical Stories

Everyone loves a good story. Stories transport us to another time or place and make us think outside of ourselves, question our status quo. The mathematics classroom is not one of the more predictable places to find a good story, but that doesn’t mean there aren’t any.  One of my favourites goes something like this:

The Pythagoreans were an ancient mystical sect, a group of men who wore white robes and spent their days measuring and wondering; wondering at the beauty of nature and the expanse of the cosmos, resolute in their belief that the universe could be explained and deciphered using mathematics. They were masters of geometry and sought to understand the workings of the world by analysing numbers and shapes and the intersection of the two. One of the Pythagoreans’ core beliefs was that any number could be written as the ratio of two others in the way that 6 is 12/2, or 2.333333… is 7/3. This was the gift of the gods, that any number could be expressed through any other. It felt complete, it felt perfect. But for the Pythagoreans, perfection was about to be shattered.