By Kenneth Falconer
“This is not maths – maths is about doing calculations, not proving theorems!” So wrote a disaffected student at the end of my recent pure maths lecture course. Theorems, along with their proofs, have gotten a bad name.
The first (and often only) theorem most people encounter is Pythagoras Theorem, discovered over 2500 years ago; that if you square the lengths of the two perpendicular sides of a right-angled triangle and add these numbers together then you get the square of the length of the third side. To many, the name Pythagoras conjures up memories of eccentric maths teachers enthusing over spiders webs of lines. Yet, if the writer of the software underlying your computer had not known their Pythagoras and other such theorems, you would not now be viewing this neatly aligned text or navigating around your screen at the touch of a mouse.
A theorem is the name for an incontrovertible mathematical fact, a statement that is an unavoidable consequence of precisely defined terms or facts that have already been established. Pythagoras’ Theorem follows inexorably from the notions of a straight line, a right-angle and length. A couple of hundred years later, Euclid formulated his theorems or ‘Propositions’ of geometry which became the foundation of western mathematical education for the next 2000 years. My favourite is the Intersecting Chord Theorem: if you draw two intersecting straight lines across a circle and multiply together the lengths of the parts of the chords on either side of the intersection point then you get the same answer for both chords (see diagram). This is a remarkable statement: there seems no obvious reason why it should be so. Yet it is an inevitable consequence of the definition of a circle. Sadly, learning the formal propositions of Euclid by rote, as they were often taught in the past, may have hidden their substance and elegance and turned off many budding mathematicians.
Many further geometrical theorems have been established since Euclid’s days, some with evocative names. The Ham Sandwich Theorem says that given three objects there is always a plane that simultaneously divides each object into two parts of equal volume; thus a sandwich can always be divided by a straight slice so that the bread, butter, and ham are all equally divided between the two portions. Then, according to the Hairy Ball Theorem, it is impossible to comb a sphere covered with hair or fur in such a way that the hairs lie down smoothly everywhere on the sphere. One consequence, perhaps reassuring at times of extreme weather, is that at any instant there is somewhere on the earth’s surface where there is no wind.
The Mandelbrot set has become an icon recognised by many with little or no mathematical knowledge but who have been fascinated by its intriguing beauty. The Fundamental Theorem of the Mandelbrot Set, as it is sometimes called, relates geometrical aspects of this extraordinarily complicated object to the simple formula z2 + c. The theorem was contained in the writings of Pierre Fatou and Gaston Julia back in 1919, but was virtually forgotten until in the mid-1970s Mandelbrot’s computer images revealed the set’s intricate detail. A picture can bring a theorem to life!
Of course, not all theorems are about geometry. Some concern properties of numbers; perhaps the most famous is Fermat’s Last Theorem, that the equation xn + yn = zn has no solutions with x, y, z and n positive whole numbers with n greater than 2. This elegant statement was enunciated by Pierre de Fermat in 1637, but was only proved conclusively by Andrew Wiles less than 20 years ago, with a proof running to well over a hundred pages that only a very few professional mathematicians are in a position to understand. I am not aware of any practical applications of Fermat’s Last Theorem outside pure maths. On the other hand, Fermat’s Little Theorem, proposed in 1640, is a valuable tool for calculation. For example, it tells us immediately that the enormous number obtained by multiplying 2013 by itself 3000 times (written 20133000 and having almost 10,000 digits), leaves a remainder of 2013 when divided by 3001. Fermat’s Little Theorem can be proved in a few lines but has hugely important consequences, indeed it underpins many of the cryptographic methods that are used to keep computer and bank data secure.
Theorems are the pillars of mathematics. New theorems, often building on the foundations of earlier ones, are continually being proved. Yes, some may be esoteric, but others have been fundamental in the development of things that we take for granted, such as Stokes’ Theorem for electronic communication and fluid flow. And, though I obviously failed to convince my student, they are the basis for many of the calculations undertaken daily by scientists and engineers.
Kenneth Falconer is author of Fractals: A Very Short Introduction and Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2014). He has been Professor of Pure Mathematics at the University of St Andrews since 1993.
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Image credits: 1) Figure drawn by author; 2) Image computed by Ben Falconer