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.
Of course, perhaps the greatest discovery of the Pythagoreans was what we now call Pythagoras’ Theorem, where the area of the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
In a cruel twist of fate it was this very discovery that eroded the foundations of the Pythagoreans’ tower of mathematical knowledge. Imagine you have a right-angled isosceles triangle where the two perpendicular sides have length 1. Then, by the theorem, the hypotenuse is equal to √2. So, the square root of two can be drawn and it surely must be a line of finite length if it’s possible to draw it. The problem comes, so legend goes, that one of the sect’s members, a man called Hippasus, was able to prove that cannot be written as the ratio of two other numbers, so not only is it not of finite length, it has no repeating patterns. In modern parlance, it is not a rational number but an irrational one. His probable proof can be read here. What a blow! The world as they believed it to be, in all its predictable glory, was shattered. So Hippasus, who was on a ship at the time, was unceremoniously thrown overboard. Imagine that: murdered for discovering a new type of number!
It turns out that the world of mathematics, as a subset of the world at large, has its fair share of intrigue, deception and drama. How about Isaac Newton, who had a longstanding feud (or beef, as I often tell my students in as serious a voice as I can. It never works, I sound ridiculous!) with Gottfried Liebniz over who discovered calculus first, a feud which was settled at the time by a report in Newton’s favour, produced by the Royal Society but written by Newton himself. Not exactly an independent review there! Scholars are still undecided as to who first discovered what and how much each man’s work influenced the other.
Then there’s Hypatia, an ancient Greek mathematician who was the head of the Neoplatonic school at Alexandria and had the ear of important men with her wise counsel, yet was ever-so-gruesomely murdered in a conflict between Christians and Jews. Another dramatic death belongs to the impetuous Evariste Galois, who made massive contributions to solving polynomial equations and was one of the founders of group theory, an area of pure mathematics (you know, the bit of maths that has no relevance to the real world) that has applications in physics, chemistry, biology, statistical mechanics and more. Galois was arrested numerous times for trying to overthrow the King, reacted with violent pride when Siméon Poisson (he of probability fame) refused to publish his work due to its apparent poor quality, and died from a gunshot wound to the stomach received in a duel, the suspicious reasons for which have been obscured. Was the duel over an illicit romantic relationship or was it instigated by the French authorities, fed up of the young man’s political activism?
Of course the majority of the passage of mathematical time has been rather less sensational but fascinating nonetheless. For instance, were you aware that the Cartesian grid we use today was initiated by René Descartes and developed by his commentators around the mid-seventeenth century, yet the solution to the quadratic equation (which we view inextricably from its graphical representation) has been in development since Babylonian times, through the Indian Brahmagupta in the seventh century, Spanish Abraham bar Hiyya in the twelfth century, Italian Gerolamo Cardano in 1545 until the Flemish Simon Stevin published a complete formula in 1594? Descartes, in his seminal work La Géométrie, wrote the formula in the way we know it today, without yet seeing the importance of the solution as represented in a graph.
While we’re on equations, Cardano also wrote methods of how to solve cubic and quartic equations yet did so without the number zero. It had no place in his mathematics. Even though Brahmagupta was one of the first to set out the idea of zero it did not gain widespread acceptance until the seventeenth century – Fibonacci, da Vinci, al-Khwarizmi, all practically zero-less! Brahmagupta tried to explain what happens when you add, subtract, multiply and divide by zero, but was confounded by the latter – how can you have a number that cannot be divided by? The idea of a mathematical operation being undefined is a fascinating consideration in its own right, plenty a discussion in my Further Maths classes has arisen over the years in response to this idea, and younger students can engage in it too, through arithmetic and asymptotes on graphs.
Circle geometry is another area that has developed over continents and centuries. Our earliest evidence of an understanding of π is from the Babylonians, around 2000 years BC, who approximated it to . Indian mathematicians in 600BC used the value 3.16 while Archimedes, 350 years later, showed that its value was between and and proved that the area of a circle was πr2. Fast forward eight hundred years and Chinese mathematician Zhu Chongzhi finds π to seven decimal places, a feat which is not trumped until the fifteenth century when Madhava of Sangamagrama publishes a beautiful series, of which the first twenty-one terms give the value of π to eleven decimal places:
Interestingly, in the same century, the Persian Jamshid al-Kashi used a geometric method (the perimeter of a regular polygon with 805,306,368 sides), as opposed to a strictly numerical one, in order to calculate π to sixteen decimal places. From the seventeenth century, the accuracy increases dramatically, reaching one hundred d.p. by 1706 with the British astronomer John Machin, but it wasn’t until Swiss Johann Lambert in 1761 that we even knew that we were dealing with an irrational number! Of course, computing power in the twentieth century has brought about extreme rapid growth in our known accuracy of π: in 1949 we had 2037 d.p. and by 2014 we had 13.3 trillion digits! Do we need that many digits? Practically, no. NASA only uses sixteen for its space navigation systems and we are told that the circumference of the known universe can be calculated to within the radius of a hydrogen atom using only thirty-nine digits! So the continuing search for more digits is a just search for more knowledge, but knowledge for its own sake is a good enough reason to continue.
Much of the mathematics discovered over the centuries has been done so through nothing more than the pursuit of knowledge, and much of it has had later applications that the first discoverers could never have foreseen. Indeed, Cambridge mathematician G.H. Hardy believed that mathematics should be studied purely for its beauty, and prided himself on the inapplicability of his mathematics. He could never have known that nearly one century later internet encryption, upon which we all daily rely, would be built on the foundations of the number theory he studied and deemed to be so very useless.
In uncovering the story of mathematics you cannot escape the reality of our wonderful subject, an amalgamation of thousands of years of work by people all around the world who are united by a common language. For students in the mathematics classroom, an injection of time and place into their pursuit is a great tool in the teacher’s box. It may be that they gain a deeper understanding of an aspect of their work as it is placed in a timeline and they see its development. It may be that they are drawn out of their small world and given a glimpse of a genuinely international pursuit. It may be that they realise that mathematics is not merely a set of rules but an evolving discipline that hasn’t been finished yet. Or it may just be that they come away with an interesting story to tell their parents, about geometry and proof and life and death.
UPDATE: June 2018
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 are here: