by Wells, David
Figure 18.1 Parabola cutting both axes of graph
Figure 18.2 Parabola cutting one axis of graph
However, by algebra, the second equation has the two roots 3+ and 3 − which involve ‘the square root of −1’. What should be done with them? Mathematicians who first came across these exotic objects were naturally intrigued, but also wary. They were labelled imaginary in contrast to real and there was a temptation to regard them as absurd. What indeed could they possibly mean? The short answer is that the question itself is not quite right. A better question is, ‘What ought they to mean, what should we decide to make them mean in order to make maximum sense of the situation?’
Looking at elementary algebra from our perspective, as a game, our first thought is to add as just another piece in the game but to do so with confidence we need to know that this will never get us into trouble, that the square roots of negative numbers will always be consistent with the rest of the game. A first step is to ‘suck it and see’. Does basic algebra work with the addition of – not forgetting the roots of other negative numbers? Yes, seems to be the answer. We can add and subtract with no difficulty and multiplication and division and even powers seems to ‘work’ also. For example,
This suggests that (1+)3 = −8 and so the three cube roots of −8, if it has three roots when complex numbers are included, should be −2, (1+) and one other. However, the product of the three roots should be 8, and so the third root should be, 8/2(1+) or so it looks as if the roots of −8 are −2, and 1 . Sure enough,
This kind of playing around with complex numbers seems never to lead to contradictions, which is encouraging, even convincing, but proves nothing.
What about another possible fly in the ointment? There might be two or more ways to define a number such that i2 = −1. Let's suppose that there are two different square roots of −1 and that they obey all the usual rules. Call them i and j, so that i2 = −1 and j2 = −1.
Then i2 − j2 = 0 so, factoring, (i − j)(i + j) = 0.
So either i − j = 0 and i = j, or i + j = 0 and i = −j.
The first possibility is that there is only one root of −1. The second tells us that if i is a root then so is −i, but that's OK because we already know that, for example, 4 has two roots when negative numbers are included, +2 and −2. So if i2 = −1 then we expect that (−i)2 = −1 also. Going back to the usual identity,
if i = , so i2 = −1 then,
which certainly is strange, because 2 is usually thought of as a prime number, but if we introduce it seems to have integer factors after all! In the factor 1+i, 1 is an integer and i = is a complex integer, so 1+i counts as a complex integer.
So it seems that in order to accommodate this new discovery we shall have to change our idea of what an integer factor is, but this is not a serious problem to the mathematician. Changing definitions is something that mathematicians have often done as they explore their mathematical worlds more and more deeply.
Proving consistency
Jean Robert Argand (1768–1822) solved the problem of consistency by a method, then original, but now familiar: he built a model showing that complex numbers can be represented geometrically in the plane (Figure 18.3).
Figure 18.3 Vertices of a rectangle in the complex plane
To the horizontal number line we add a vertical and independent complex number line. The graph shows the numbers 1 + 3i, 4 − i and their sum, 5 + 2i. Addition and subtraction become (to use modern language), the addition and subtraction of vectors but this poses no problems at all. Multiplication is more tricky but Argand can handle it. Each complex number, thought of as a vector, can be described by its length and the angle it makes with the real axis.
1+3i has length , by Pythagoras, and it makes an angle tan−13 or 71.565° with the real axis. 4 − i has length and angle −14.036° with the real axis.
We are now ready to calculate (1+3i)(4 − i) by algebra and by using Argand's diagram. By algebra, (1+3i)(4 − i) = 4+12i − i+3 = 7+11i which has length and makes an angle tan−111/7 or 57.529°.
On Argand's diagram, we multiply the lengths and add the angles: × = , and 71.565 − 14.036 = 57.529. The match is perfect, even allowing for errors in rounding off the three angles (Figure 18.4).
Figure 18.4 Argand diagram showing product of two numbers
Argand proved that the match is always perfect and it follows that elementary algebraic operations with complex numbers are equivalent to familiar operations with angles and similar triangles from Euclidean geometry and so we can indeed have complete confidence that if we add to everyday algebra we will not run into trouble.
Complex numbers give us a new perspective on the very idea of a number, on the roots of equations, on the graphs which represent them, and so many other topics. Our next example is less powerful but it does show how apparently difficult problems can be almost miraculously solved by transformations that give us novel and powerful perspectives.
Transforming structure, transforming perception
Inversion is an ingenious means of transforming any plane geometrical figure into another using just a reference circle, with centre O, the centre of inversion, and radius r (Figure 18.5).
Figure 18.5 Inversion in a circle
Given a point, A, its inverse A′ is constructed by joining A to O and then finding A′ on AO such that AO·A′O = r2. (For inversion in 3-D, you use a sphere.)
Inversion was discovered by Jakob Steiner (1796–1863). It is very powerful because the basic properties of the transformation are so simple: • A line not through O is inverted into a circle through O, and conversely.
• A circle not through O is inverted into a circle not through O.
• A circle in unchanged by inversion if and only if it is orthogonal to the reference circle.
• The centre of a circle and its inverse circle are in line with O.
• Angles are unchanged by inversion.
• Any pair of non-concentric circles can be jointly inverted into a pair of concentric circles.
• Inversion is its own inverse: the inverse of an inverse, in the same circle, is the original figure.
• Tangents remain tangents, and tangent circles remain tangent circles.
The power of inversion is illustrated by Steiner's porism (Figure 18.6).
Figure 18.6 Starting diagram for Steiner's porism
In this figure, P and Q are two non-concentric circles. Circle A has been drawn to touch both, and B drawn to touch A, P and Q. Then C touches B, P and Q, and so on, forming a chain of circles. Steiner's porism says that if and only if the final circle touches A, then it will do so whatever the initial position of A.
Figure 18.7 shows an example. The space between the original two circles P and Q has been filled with a chain of six circles touching in sequence. (This is a perfect subject for animation using Java and a standard geometry package, and many programs will be found on the web in which the entire chain of circles can be slid smoothly between the original circles.)
Figure 18.7 Steiner's porism
Steiner's porism claims that in this chain the last circle will touch the first wherever the first circle is placed. How can this beautiful theorem be proved? Very simply, we choose a point of inversion that will invert the circles P and Q into two concentric circles, as in Figure 18.8. The chain of six different circles has becomes a chain of six identical circles between P and Q.
Fig 18.8 Steiner's porism and proof by inversion
This transformation is always possible – it follows from the basic properties of inversion. Steiner's porism now says that you can rotate the chain of six circles, as if they were identical balls in a mechanical bearing, and indeed this is quite obvious! So Steiner's original porism is true also.
As a footnote, the Steiner's porism diagrams for N circles all have extra properties: in Figure 18.9, eight points of contact have been joined in four pairs: they concur, recalling the Seven Circles Theorem (p. 129).
Figure 18.9 Ring of four ci
rcles and concurrency
The four common tangents to consecutive pairs of touching circles in the chain, also concur, at the same point.
In Figure 18.10 another three lines through pairs of points of contact, concur.
Figure 18.10 Ring of four circles and another concurrency
These patterns are typical and there are many more which we have not shown. Readers may like to search for them.
Jean Victor Poncelet (1788–1867) discovered several very different porisms. In Figure 18.11, the triangle ABC has its vertices on circle P and its sides touching circle Q. Poncelet's porism says that however vertex A is moved round the circle (e.g. Figure 18.12), the triangle will still be completed when BC touches the inner circle.
Figure 18.11 Poncelet's porism – first position
Figure 18.12 Poncelet's porism – second position
The ‘same’ theorem applies to conics in general: in Figure 18.13 the triangle touches the circle and has its vertices on a parabola. The vertices A, B and C can be moved at will.
Figure 18.13 Porism of triangle, circle and parabola [steiner.math.nthu.edu.tw/disk3/cabrijava/poncele t-porism.html]
Once again, a hidden structure appears and the resulting figure has many of the features that we expect to find in mathematical beauty including simplicity, mystery – or mystery-revealed – and surprise, as well suggesting further questions, such as: where must the circle be placed in relation to the parabola for Poncelet's porism to work?
19 Mathematics and beauty
Pure mathematicians have written a stream of encomiums on the beauty of mathematics. Reading them, anyone might conclude that mathematics is an art like painting or music or poetry. The greater the mathematician, the more vividly, it seems, they appreciate its beauty. Fermat wrote to a friend, ‘I have found a great number of exceedingly beautiful theorems.’ No doubt one of the theorems he had in mind was that every prime number of the form 4n + 1 is the sum of two integral squares in exactly one way. For example,
and so on. Primes of the form 4n + 3, such as 7 or 19, are never the sum of two squares. This is extremely surprising, because the primes are so irregular and mysterious and the squares so regular and simple – so almost all mathematicians find his theorem extraordinarily beautiful.
Newton once wrote to Henry Oldenburg, the Secretary of the Royal Society in London,
I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz's method for obtaining convergent series is certainly very elegant…
[Newton 1676]
We have already met Leibniz's series,
What could be simpler or more elegant and, once again, surprising? Who would have thought that the ratio of the circumference of a circle to its diameter should be connected to the reciprocals of the integers?
Mathematics is beautiful but mathematicians not only enjoy beauty, they use it as a criterion of success and as a guide in their work – but all guides, of course, occasionally fail. As Henri Poincaré explained,
When a sudden illumination invades the mathematician's mind…it sometimes happens…that it will not stand the test of verification…it is to be observed that almost always this false idea, if it had been correct, would have flattered our natural instincts for mathematical elegance.
[Poincaré 1914]
Poincaré is saying that their aesthetic sense is so strong that it can even lead mathematicians astray! Note also Poincaré's reference to verification which, to the mathematician means proof. Mathematicians continually speculate but only proof certifies their conclusions.
How can aesthetic judgement, such a powerful psychological force, be anyone's guide to successful mathematics, despite strong individual differences of opinion? That is the subject of our speculations in the next two chapters.
Hardy on mathematics and chess
G. H. Hardy was a purer-than-pure mathematician who hated war, and a non-conformist who hated religion and regretted even working within the sound of a Roman Catholic church [Hardy 1941: Intro.]. He claimed, famously, that,
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
[Hardy 1941/1969: 84]
He scorned the application of mathematics to anything at all and once expressed the hope that nothing he had ever discovered would have any practical use. He turned out to be mistaken though to be fair his one deliberate contribution to applied mathematics, the Hardy–Weinberg law in genetics, was not published as a mathematical paper but sent as a letter to Science, which started by saying that, ‘I should have expected the very simple point which I wish to make to have been familiar to biologists’ and went on to refer to the mathematics involved as ‘A little mathematics of the multiplication-table type’ thus putting readers of Science in their place [Hardy 1908].
Our concern is that Hardy also made an aesthetic link between maths and chess, and mathematical puzzles:
The fact is that there are few more ‘popular’ subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune…There are masses of chess-players in every civilised country…and every chess-player can recognise and appreciate a ‘beautiful’ game or problem. Yet a chess problem is simply an exercise in pure mathematics…and everyone calling a problem ‘beautiful’ is applauding mathematical beauty…We may learn the same lesson…from the puzzle columns of the popular newspapers. Nearly all their immense popularity is a tribute to the drawing power of rudimentary mathematics, and the better makers of puzzles…use very little else. They know their business; what the public wants is a little intellectual ‘kick’, and nothing else has quite the kick of mathematics.
[Hardy 1941/1969: 86–88]
Hardy is no doubt correct but his claim begs as many questions as it answers. Millions do get a kick out of maths and puzzles – but millions don't. Why not? And how is this ‘kick’ connected to success at maths?
Experience and expectations
As human beings we get satisfaction both when our expectations are met, and when they are not – when we are surprised! When we look at a visual pattern, or listen to music in which a rhythm repeats and a line of melody returns again and again, our expectations are met and we feel an aesthetic pleasure: but we also get our ‘kick’ when the pattern fails to continue because a different – and maybe deeper – pattern exists.
School pupils who have only graphed straight lines and parabolas are easily impressed when they first meet a curve with asymptotes: instead of heading steadily towards the edge of the graph paper the asymptote races off the page. An asymptote also has the strange feature that the curve gets closer and closer to it, but never meets it – an example of limits, another counter-intuitive topic which easily intrigues children.
This double effect of expectations met and expectations surprised is especially strong in mathematics just because our confidence in maths is so strong, so our expectations are perfectly met – except when they are amazingly confounded.
This creates a potential double bind for teachers: their pupils need experience before they can develop expectations and then have them confounded: so how can teachers give their pupils beautiful experiences to motivate them from the start? One answer is through the use of mathematical recreations that are already more or less ‘familiar’.
Mathematical objects and proofs give the same effect: a ‘kick’ out of structure and pattern and a ‘kick’ out of surprises, for example when a brilliant and beautiful proof introduces an unexpected idea.
Beauty and Brilliancies in chess and mathematics
On the Continent what we call chess Brilliancy Prizes are, significantly, called Beauty Prizes. They are not won for merely calculating many moves ahead. Tactical brilliance helps but strategical depth is just as important and the greatest masterpieces display both. Alekhine's famous game against Reti [Reti-Alekhine, Baden-Baden 1925]
does involve astonishing look-ahead but that's only a small part of its beauty. As the brilliant tactician Rudolph Spielmann exclaimed, ‘I can see the combinations as well as Alekhine, but I can't reach his positions!’ Strategical depth is the desideratum. Botwinnik's famous win against then world champion Capablanca in the 1938 AVRO tournament was just such a strategical masterpiece, rather than a display of fireworks.
In maths, as in chess, mere calculation is not enough. Imagination and insight are necessary to create beauty and, plausibly, strategical or structural depth.
Beauty, analogy and structure
The rules of the game create structures that have a mysterious beauty simply because they are so hard to discover. Chess players of today appreciate the beautiful balance between the weakness of backward pawns and their dynamic possibilities, but this feature took centuries to discover.
