An EFG of Mathematics

Adam McBride, Department of Mathematics, University of Strathclyde

Introduction

Lots of books have been written with titles of the form "An ABC of ...". I have often felt that A, B and (to a lesser extent) C get far too much attention at the expense of other equally worthy characters. Admittedly X, Y and Z are kept usefully employed in the realm of equations but what about all those in between? Here we shall try to redress the balance by thinking about E, F and G. This will give us an excuse for talking about three of the greatest mathematicians of all time. In each case we start with a few salient biographical details before moving on to marvel at some celebrated mathematical results. The selection is highly personal. There is an emphasis on Geometry and Number Theory, two of the oldest and most elegant branches of Mathematics. Indeed the whole exercise is an excuse to wallow in sheer unadulterated beauty. What a shame that the majority of students at school (and university) never see these results. Perhaps those reading this article will find the time to incorporate some parts into their teaching. It would be well worth the effort.

E is for Euler

Leonhard Euler was born in Basel in 1707. At the age of 20 he applied for a professorship. When his application was turned down, he was a bit miffed and headed off to St. Petersburg to join some of the younger members of the Bernoulli clan. The severity of the climate and the strain of working led to Euler losing the sight of one eye. During the period 1741-1766 he lived in Berlin before returning to St. Petersburg for the rest of his days. He eventually lost the sight of his other eye. In later life his children (13 in all!) and his pupils copied his work as it was dictated. For what it was worth, he could recite Virgil's Aeneid from end to end. He died in 1783.

Throughout his life he was a prolific writer of books and papers. Among his contemporaries were Taylor, Maclaurin and de Moivre. Rather than giving a long list of publications, let us concentrate on a few specific results. One of the most famous concerns the Konigsberg bridges.

Figure 1

The problem was to decide whether it was possible to go for a Sunday afternoon stroll and to arrive back at the starting point having crossed each of the seven bridges exactly once. In solving the problem, Euler laid the foundations for the area of Mathematics we now know as Graph Theory. The four areas of town were represented by dots (vertices) and the bridges by lines joining the vertices (edges), leading to the following diagram:

Figure 2

Suppose it were possible to go for a walk as required. Then every time we go into a vertex by one edge we must come out again by another edge. We are only in business, therefore, if at every vertex there is an even number of edges. Since this is not the case a walk under the stated conditions is impossible.

From what might loosely be called a circular tour, let's turn to a different type of tour.

1	48	31	50	33	16	63	18
30	51	46	3	62	19	14	35
47	2	49	32	15	34	17	64
52	29	4	45	20	61	36	13
5	44	25	56	9	40	21	60
28	53	8	41	24	57	12	37
43	6	55	26	39	10	59	22
54	27	42	7	58	23	38	11

Figure 3

The numbers 1, 2, ..., 64 have been placed in the squares of a standard 8 ´ 8 chessboard in a particular way. The sum of the 8 numbers in any row or column is always the same, namely 260. This is not true for the two main diagonals so that we do not have a strict magic square. However, we have something else instead. To get from each number to the next, we perform a knight's move, as in chess. When we reach 64, we might want to get back to where we started with our final move. Although this is not possible with a single board, put another board on the right of the original one. Then a knight's move gets us from 64 on the first board to 1 on the second and we can start all over again. Euler is credited with finding this Knight's Tour, which has been copied many times since (even being further embellished for the purposes of The Listener Crossword).

Now for some geometry. I never cease to be amazed by the beauty lying hidden in a humble triangle. Students nowadays seem to be in big trouble when it comes to the standard concurrences. Most students know that the medians of a triangle are concurrent at the centroid, G. (Some of the best students may even be able to provide a proof for a general triangle via vectors.) After that, the fun starts. Some students have a vague notion about altitudes. It is a nice exercise on the scalar product to prove that the three altitudes are concurrent at the orthocentre, H. (Of course, this can be proved by "pure" geometry, too.) When it comes to perpendicular bisectors of the sides, we are into uncharted territory for almost all of today's students. Surprise, surprise! The three perpendicular bisectors are concurrent at a point, K say, which is the centre of the circumcircle of the original triangle. OK, so far? The best is still to come!

You might like to try this as an investigation with your students.

1. Draw a large scalene triangle on a sheet of paper.

The triangle must be large as the diagram will get quite cluttered.

It is vital that the triangle is scalene, i.e. all three sides must have different lengths.

(If the triangle is isosceles or equilateral, degeneracy sets in and the excitement is lost.)

Finally, it might make things easier if the triangle is also taken to be acute-angled (although this is not essential).

2. Construct the 3 medians, concurrent at G.

3. Construct the 3 altitudes, concurrent at H.

4. Construct the 3 perpendicular bisectors, concurrent at K.

(Each of 2., 3. and 4. can be a sub-investigation for students who have not seen the results before.)

5. Look carefully at K, G and H. What is going on?
There should be 2 conjectures here!

What we are now looking at is the Euler Line of the given triangle. Apparently the Greeks missed this line and Euler is credited with its discovery almost 2000 years later. If you haven't met this before, I hope you will be impressed by the beauty of it all. And there's more!

6. Call the original triangle ABC. Let the altitudes be AD, BE and CF.

With H as the orthocentre, let P, Q and R be the mid-points of AH, BH and CH, respectively.
Let U, V and W be the mid-points of BC, CA and AB, respectively.
Look at the 9 points ; ; .
What is going on?

It turns out that these points all lie on a circle called (surprise, surprise!) the 9-point circle of . Given that 3 non-collinear points determine a circle, what is the chance of 9 points (all related in a natural way to ) all lying on one circle?

7. Where is the centre of this circle?
(Hint: It is related to the Euler line.)

Only a complete and utter philistine could fail to appreciate all this, yet most students never have the opportunity to savour the wonder and excitement. Of course, there remains

8. Prove all the conjectures made above!

However, we shall leave that as an exercise for the reader.

Here's another question. What do you regard as the most elegant formula in Mathematics? In a poll organised by a journal called The Mathematical Intelligencer there was a clear winner. To set the scene, let us go back to Euler who found the exponential form

related to de Moivre's Theorem. On putting we arrive at

This has been described as the "epitome of what Euler achieved". The formula contains 5 of the most basic constants in Mathematics, namely, 0, 1, p, e and i. In the words of Felix Klein

"Is not all analysis centred here?".

F is for Fermat

A century earlier than Euler, there lived Pierre de Fermat (1601-1665). Fermat is often described as the last great "amateur" mathematician or the "Prince of Amateurs". By profession Fermat was a civil servant who rose to become a Supreme Judge in Toulouse. In this capacity he occasionally sentenced criminals to death or had priests burned at the stake. Mathematics was a hobby to which he dedicated all his spare time. He was fond of setting challenges and making observations or conjectures, for which he rarely supplied proofs. The most notorious instance concerns Fermat's Last Theorem. For those who have been on another planet and have missed all the recent excitement, it is as well to recall the statement one more time:

For any integer , the equation

has no solutions in which x, y and z are all positive integers.

As is well known, Fermat scribbled in the margin of a book that "I have found a truly marvellous proof of this result but the margin is not big enough to contain it." Over 350 years later, in 1994, the theorem was finally proved by the English mathematician Andrew Wiles. You can read all about it in the best-seller [3]. Did Fermat have a proof? I doubt it.

Fermat contributed much more than his Last Theorem. For example, there is Fermat's Little Theorem which says that

if is prime then is divisible by for every positive integer .

Mention of primes gets us right into Number Theory. For centuries, people have tried to come up with formulae that generate prime numbers (not necessarily all primes, just some). Fermat threw his hat into the ring with the formula

for what is now called the th Fermat number. For n = 0, 1, 2, 3 and 4 the resulting Fermat number is prime but is not prime. Indeed

as can be checked directly. This was discovered by Euler in 1732.

(Incidentally this example can be used to show the difference between "proof" in the science lab. and proof in Mathematics, where tools such as mathematical induction are needed.)

The Fermat numbers have attracted a lot of attention. (Why? Stay tuned and you'll find out shortly!) It turns out that apart from the first five primes, every other Fermat number that has been checked has turned out to be composite (not prime). No one knows if there are any more Fermat primes. Of course, the numbers increase very rapidly with . To get the general idea, think of i.e. . In [1], Beiler offers the following information about this number (very handy if the conversation is flagging at a party!).

(i) If each (decimal) digit of was written on a tile 1mm square, the tiles would cover the surface of the Earth 6 times over.

(ii) Alternatively, assume that an average book contains 1000 pages, each page has 100 lines and each line contains 100 digits. (These are generous estimates.) Writing out would then use up all the books in all the libraries in the world 250 times over.

Yes, is rather big and yet we know that it is divisible by . (This is done via modular arithmetic, i.e. congruences.) We shall come back to shortly.

G is for Gauss

Carl Friedrich Gauss is often called "The Prince of Mathematicians". He was born in Braunschweig (Brunswick), Germany in 1777. He soon displayed a precocious talent, finding a mistake in his father's accounts at the age of 3. A few years later, his teacher told him to add up the first 100 integers in the hope that it would keep him quiet for a while. The answer came back almost instantly. By the age of 14, Gauss was apparently conjecturing a result now known as the Prime Number Theorem:-

Let denote the number of primes .

For example, p(14) = 6 (primes being 2, 3, 5, 7, 11 and 13).

Then, as , .

Here "ln" denotes the natural logarithm while "~" means "is asymptotic to" or (very loosely) "behaves like". The Prime Number Theorem was not proved rigorously until 1896.

At the age of 19, Gauss discovered a remarkable connection between Number Theory and Geometry which changed his life. It finally convinced him to pursue a scientific career (as opposed to studying things like philology). What is this remarkable connection? Consider the following

Question For what positive integers is it possible to construct a regular polygon with sides using only a straight edge and compasses?

Sometimes "ruler" is used instead of "straight edge" but the whole point is that the ruler has no scale on it. Thus, it is not possible to measure any lengths (or angles for that matter). For N = 1 and 2 there is nothing to do. Accordingly we shall start with N = 3. There is indeed a fairly simple way of constructing an equilateral triangle according to the rules. By successive bisection, we can then handle N = 6, 12, 24,... . (Try it. We're back to perpendicular bisectors and circumcircles again!) Things soon become much harder. What about N = 5 and N = 7, say?

The answer to the Question, as given by Gauss, is stunning.

The regular N-gon can be constructed if and only if

(a product of distinct Fermat primes).

Here can be 0 (i.e. the power of 2, corresponding to successive bisection, may be absent). The crux of the matter is the bracket involving Fermat numbers which are prime. Now you can see why there is an interest in ! According to Gauss we can construct an -gon for values of such as

but we cannot construct a regular 7-gon, for instance. As might be imagined, the proof is highly non-trivial. For a readable account of some of the ideas involved, see [2, Chapter 5].

Gauss was so thrilled by his discovery that he requested that a regular 17-gon should appear on his tombstone. In the event he had to make do with a plaque on a wall! Remember that Gauss was aged 19 when he proved this result. Three years later he completed his doctoral thesis which contained the first correct proofs (four in all) of the Fundamental Theorem of Algebra. This states that any polynomial P of degree has a zero in the complex plane, i.e. there is a complex number such that . He went on to prove many important results in pure and applied mathematics and his fame was assured.

Much more could be said about our three giants. However, we have probably provided more than enough evidence of their brilliance and flair. As mentioned at the start, the choice of material was a personal one. One criterion was that all the basic ideas are simple and free of technical jargon. I hope that, despite time pressure to get through syllabuses for external examinations, teachers might be able to let their students see some of this extra-curricular material and catch a glimpse of the wonder and beauty of Mathematics.

References

1. A. H. Beiler, Recreations in the Theory of Numbers, Dover, 1964

2. M. Sewell (ed.), Mathematics Masterclasses, O.U.P., 1997, 0-19-851493-X

3. S. Singh, Fermat's Last Theorem, Fourth Estate, 1997, 1-85702-669-1