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.

Let’s start with our decimal numeral system. It always amazes me that ancient civilisations were able to perform highly complex mathematics without the obvious benefits of a place value system. The ancient Greeks had a system that used letters for 1-9, different letters for 10-90 and yet more for 100-900, which got problematic as you ran out of letters. Imagine learning your addition facts when there were 27 different symbols just to deal with 1-999. The Roman system was a little easier to deal with – only seven symbols to write 1-1000 – but both systems could quickly get cumbersome.

The ancient Chinese had a similar system to the Greeks, with symbols for 1-9, 10-90, etc, but these symbols were independent of the alphabet and were a little more logical in their form.

It was the ancient Indians who were ahead of the game. There is evidence of powers of 10 as far back as 1200BC in Indian writing and the decimal system developed from there. The *Bakhshali Manuscript* from the third century includes a symbol for zero and the system, now called the Hindu-Arabic numeral system, was written down and described by Brahmagupta sometime in the early seventh century.

Brahmagupta’s writing includes positive and negative integers and, perhaps most importantly, the number 0. Previous number systems had no need for a symbol for 0. In Roman numerals, for instance, you combine the symbol for 100 and 1 to make 101: CI. This is perhaps why many of the ancients didn’t really consider zero a number. The Hindu-Arabic system, however, is a *positional* system – the position of a digit determines its value. That means that a 0 digit is essential to maintain the place value of other digits. The number 502 has no tens, without the digit 0 it becomes 52 and the value of the 5 and the 2 is not as it should be.

‘Zero’ was a highly controversial idea that took a long time to catch on in Europe (see Charles Seife’s *Zero: The Biography of a Dangerous Idea* for the story here). It was Leonardo of Pisa, more commonly known as Fibonacci, who popularised the system which he encountered on travels to Arabia with his merchant father. Arab merchants and travellers had brought the system across from India and added to it. In the 10th century, al-Uqlidisi wrote about decimal fractions, which extended the system to parts of a whole, and Fibonacci documented the system in his 1202 treatise *Liber Abaci*, the same one that detailed the Fibonacci sequence.

At this point the system still wasn’t entirely as we would recognise now. It was a Persian mathematician, al-Kashi, who introduced the decimal point in around 1400 and in 1585 the Dutch Simon Stevin wrote a book called *De Theinde*, which introduced decimal fractions (decimals) as we know them to Europe. Just think about that for a moment – William Shakespeare probably never met a decimal. He certainly never met the multiplication symbol, since that wasn’t introduced until two years after his death, by William Oughtred in 1618. (The sixteenth and seventeenth centuries began the formalisation of symbols we use today – 1525 saw the addition, subtraction and square root symbols printed in Cristoff Rudolff’s *Die Coss*, an algebra textbook, while in 1659 Johann Rahn introduced the division symbol, or obelus).

So there you have it, Shakespeare never saw a decimal point (I think!). Now to decide what topic to look at next.

*The ideas here form part of a book I have written for Oxford University Press, called “How to Enhance Your Maths Subject Knowledge: Number and Algebra for Secondary Teachers” which should be available at the end of June and which you can order here or here.*

There are four posts in this series. The others can be found here:

Mathematical Stories 3 – Irrationality and Infinity

Mathematical Stories 4 – An historical and international endeavour

If you cover the well-known story about the young Gauss being asked to add the integers from 1 to 100 I hope you’ll look beyond the canard that his teacher was a bullying oaf. As far as I can tell, there’s no evidence whatsoever for this. He was a trained professional, and got his teaching assistant (!) to devise a programme specifically for Gauss. In fact the TA became a professor of mathematics in his own right. All this half a century before England had anything resembling a national network of schools. If Gauss had been English we’d never have heard of him. See:

https://established1962.wordpress.com/2017/01/03/give-a-dog-a-bad-name-johann-georg-buttner/

Alan Parr

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I had never viewed his teacher as a bully. Any reading I’ve seen on Gauss suggests his teacher was astounded. Yours is an interesting post, thanks for sharing it.

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