Mathematics |
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| Vital Numbers Our civilisation hurtles down digital information highways - but without the science of numbers it would crash |
For most people, numbers are useful - but no more than a tool. Look closer and you'll find a puzzling and exciting parallel world where the impossible exists and the obvious can't be proved
Numbers are everywhere. In a sense, we are what we can count, and our
computerised civilisation counts almost everything. Without numbers, it would
simply cease to exist.
Sometimes the numbers that surround us are simple records, obvious to all:
figures that indicate taxes due, the stock market index or the balance in
our bank accounts for example. But other numbers - usually hidden in the
ceaseless flow of computer data - interact and control. The numbers that
quietly manage fuel economy in a car engine save us money; the hundreds of
thousands of numbers involved in the navigation of a jetliner, guide it through
crowded skies to land safely at journey's end.
Numbers can even create a world of their own: when you put on a virtual reality
headset and gallop off on your virtual horse to rescue a virtual damsel from
a virtual dragon, you are playing out a drama of numbers. Every item of virtual
scenery exists as a list of numbers stored in a computer. As you move, the
program performs thousands of calculations, from which it decides what images
to send each eye to maintain the illusion.
Numbers probably arose when ancient civilisations were becoming organised
thousands of years ago. One theory is that people developed symbols before
they learnt to count. A king would keep tabs on what he owned - his livestock
and riches - by using clay tokens.
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Babylonian Tablet: 600 BC.These marks,made
in clay before baking,told how much water a water clock needed for how
long Napiers Bones: this device,made with ivory rods,was an aid for multiplication and division.The scotsman John Napier built it before inventing natural base (base e) logarithms in the 17th century -which did the job much better A tally stick:the earliest evidence of counting - each notch could have represented a unit of livestock.This one is 2,500 years old:some date back to 30,000 BC |
Officials marching the king's sheep through a gate would have dropped one
token into a bowl for each sheep that passed. They would then seal all the
tokens inside a clay envelope, stored in the treasury for safekeeping. If
later the king wanted to know whether any sheep had been stolen, the officials
would break open the envelope, walk the sheep through the gate again, and
take out one token for each sheep that went by. Spare tokens meant missing
sheep.
After a while the officials may have grown tired of breaking open envelopes
and sealing them up afterwards, so they scratched symbols on the outside
of the envelope to correspond to the tokens inside - perhaps using tally
strokes. Finally some bright spark realised that the symbols on the envelope
were all you needed. and that the tokens inside were redundant.
Although Babylonians and Egyptians used arithmetic as early as the 3rd millennium
BC, it was not until the ancient Greeks that the almost magical power of
numbers was appreciated. In the 4th century BC, Euclid founded the theory
of numbers, which set out the abstract laws and axioms by which numbers can
be manipulated.
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| Digital Futures Traders in a stock exchange shuffle whole economies as a blur of figures whizzes past on elevated screens |
Among the many remarkable things Euclid proved was that there are infinitely
many prime numbers (see the box on special numbers, next page). His studies
form the foundation of the modern discipline of number theory.
The Greeks could get quite excited about the special properties of some numbers.
The cult of Pythagoreans after Pythagoras, who gave the world the square
on the hypotenuse - saw the number one as the primordial unity from which
all else is created. Two was the symbol for the female, three for the male.
Therefore five (two plus three) symbolised marriage.
The number four was symbolic of harmony, because two is even, so four (two
times two) is "evenly even". Four also symbolised the four elements out of
which everything in the universe was made (earth, air, fire, and water).
Thirteen was a lucky number in ancient Mexico, and in Italy seventeen is unlucky
Six symbolised health and was "perfect" because its divisors - one, two and
three - add up to six again. Some numbers retain mystical meanings today.
A lot of us still think of seven as lucky and thirteen as unlucky. There
are numerous explanations. Seven is the sum of the spiritual three (the holy
trinity) and the material four (the elements) and thus embraces all aspects
of the universe. The Babylonians recognised the existence of seven celestial
bodies: the Sun, the Moon, Mercury. Mars, Venus, Jupiter and Saturn. The
God of the Bible created the world in six days (a perfect number) and rested
on the seventh.
In the Christian tradition 13 is unlucky because there were 13 people - Jesus
and his 12 disciples - at the Last Supper, and it was the thirteenth person,
Judas Iscariot, who betrayed Jesus. But the unlucky reputation of 13 goes
back much earlier. In the pre-Christian Middle East 13 had long been long
considered unlucky just because it wasn't 12, the number of signs in the
all-powerful zodiac.
However, in some cultures 13 is considered lucky. For example. in ancient
Mexico there were held to be 13 heavenly spheres and 13 gods. And in Italy
17 is considered bad luck.
For mathematicians number theory offers a different kind of magic: an
intellectual play ground full of wonders. Some of the most celebrated problems
of mathematics are easy to state but formidably hard to solve. A puzzle that
had been posed in 1640 was proved only last year - and it took an extraordinary
range of modern mathematical techniques to do it (see
Fermat's Last
Theorem, next page).
For most people. the calculations involved remain utterly alien. But you
don't need a degree in maths to see their richness and beauty.
Ian Stewart
[ Ian Stewart's latest book, "The Collapse of Chaos: Discovering Simplicity
in a Complex World", with Jack Cohen, is published in June (Viking
£20)]
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The mysticism of numbers A 19th-century diagram containing the magical trinity.The pseudo-science of numerology has links with black magic and alchemy - some numbers are considered auspicious for the casting of spells or mixing of potions.Serious mathematicians find number theory quite magical enough without invoking the dark arts |
Special Numbers |
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| All men and women are born equal,
but the same does not apply to numbers. Some have magical qualities that
are revered by mathematicians almost as guiding forces of nature. The rest
are just, well, numbers. Zero For a long time people didn't think of zero as a number. Numbers are used to count things, and you can't count no things. But the decimal system - which evolved between 3000 BC and AD 1000 - needed a symbol for "no tens", "no hundreds" and so on. It was natural to ask what that 0 on its own meant. Zero is the only number for which the operation of division makes no sense. Pi (p) The question "How long is the circumference of a circle of one unit diameter?" looks simple, but the answer led to a new kind of number - p, or 3.141592653689... It has been proved that the digits, which are known to billions of decimal places, never repeat the same pattern. Nor can p be represented by a fraction or expressed in simple algebraic form. That is why p is known as a transcendental number. e ip + 1 = 0
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The square root of minus
one In around 1500 mathematicians began to wonder what would happen if negative numbers were allowed a square root (the problem being that any number when multiplied by itself gives a positive number). They introduced a new kind of number, called an "imaginary" number, to show that it was something different from conventional, "real" numbers. By 1750 the symbol I had been introduced to denote the square root of minus one. Numbers like 2 + 5i were called complex numbers - meaning that they had two kinds of numbers, and not that they were incredibly complicated. Just as there had been with 0, there was a huge row about i. Only when it was clear I had importance in relation to fluids and electricity did everyone agree it was valid. Prime numbers Primes are intriguing because they show no obvious pattern. A non-prime number (like six) is said to be composite; it has more than one factor (two and three). A prime - 2, 3, 5, 7, 11 - can only be cleanly divided by one and itself. In 1640 Pierre de Fermat said he'd found a way of predicting prime numbers, with 2n+1, where n is a power of two. For the first five values of n, the outcomes - 3, 5, 17, 257, 65537 - are all primes. But the sixth (264+1) is not: it equals 641 x 6700417. No further prime Fermat numbers have been found. e - the natural number Suppose you start with £1 and invest it at an annual interest rate of 100 per cent for a year. At the end of the year you will have £2-your original £1 plus £1 interest. If the interest is 50 per cent every six months, compound, your total rate of interest is still 100 per cent, but you get £2.25 (£1 + 50p + 75p). If the same total rate of interest is compounded over ever-shorter periods, the amount you end up with after a year gets closer and closer to £2.7182818... This number - called e - is the base of natural logarithms. Like p it is not an exact fraction. |
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Golden mean The Renaissance architects of the Santa Maria Novella in Florence knew all about proportion. "Perfection" was based on the Greek ideal of the golden mean: the ratio - between two sides of a rectangle when this also equals the ratio between the sum of the two sides and the longer side. The number 1.618034... is also seen at work in the great Renaissance art, regular pentagons and in the sequence of Fibonacci numbers |
Great mathematical mysteries |
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There's no limit to theoretical
puzzles that mathematicians would like to solve. Here are three of the most
famous. Fermat's Last Theorem This 356-year-old problem concerns an extension of the idea of Pythagorean triples. These are numbers that can be represented on the sides of a right - angled triangle. Remember "the square of the hypotenuse is equal to the sum of the squares of the other two sides"? (The hypotenuse is the longest side, opposite the right angle.) Three, four and five form a right-angled triangle, since 32 + 42 = 52. Fermat wondered if there were similar numbers for cubes too. He got nowhere and decided there must be a reason for this. In the margin of his copy of an ancient Greek text, the Arithmetica by Diophantus, he made the most famous note in the history of mathematics: "To resolve a cube into the sum of two cubes, a fourth power into two fourth powers, or any power higher than the second into two of the same kind, is impossible; have found a remarkable proof of this. The margin is too small to contain it." His "remarkable proof" has never been found and experts generally believe that whatever he had in mind must have contained an error. A British mathematician, Andrew Wiles, tackled the problem in a series of lectures in Cambridge last year. He kept secret the fact that he had a proof until the last lecture. When he announced the proof, there was a sudden ,silence; then the entire room burst into spontaneous applause. |
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| Fermat in space A 3-D computer visualisation of the Last Theorem Right: Professor Andrew Wiles: the man who cracked the centuries old conundrum |
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| In fact, despite the excitement
when Wiles made his announcement last year, an examination of his proof has
since turned up a few errors Most of these have been repaired and only one
still causing concern Wiles remains confident that his ideas will work. Goldbach's conjecture There are many problems concerning prime numbers. One of the most famous is -whether every even number bigger than two is a sum of two primes. Christian Goldbach was an amateur mathematician of the 18th century. He asked his friend the Swiss mathematician Leonhard Euler this question. Euler couldn't solve it and nor has anybody since. Riemann's hypothesis This is one of the most outstanding problems - if not the problem - in mathematics. In the 18th century Bernhard Riemann came up with the infinite sum 1/1s + 1/2s + 1/3s.... He hypothesised that it equals zero for certain values of S when S is a complex number (one with an imaginary component), only when the real part is 1/2. ("Trivial" zeros also arise when S is a negative even number.) This idea has deep connections with the distribution of prime numbers; it is thought that a solution would unlock a new world of mathematical secrets. |
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Magic Sequences |
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In 1202 Leonardo of Pisa (later
dubbed Fibonacci) started the trend in number theory for spotting strange
sequences. Fibonacci numbers: a pair of rabbits produce two young a year. The next year the same thing. The year after that the same pair and its first two young (now mature) produce a pair each (two pairs). The number of pairs of rabbits follows the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34... where each number is the sum of the two before it. Fibonacci numbers have curious patterns, which have been found repeated in nature. Of three consecutive numbers - 5, 8, 13, - the product of the outer two differs from the square of the inner one by one (5 x 13 = 65; 82 = 64). "Lucky" numbers are obtained by a process of repeated "sieving". First you remove every second number to give the odd numbers. That sieving was based on two; the next is based on three. Every third number is removed to get 1, 3, 7, 9, 13, 15... In this evolving sequence, the next number is seven, so you remove every seventh number, and so on. The remaining numbers (1, 3, 7, 9, 13, 15, 21...) are called lucky. Their main mathematical significance is that they appear to share several properties with prime numbers: they come along about as often and as irregularly. Mystery sequence: one of the most frustrating problems in number theory concerns a different kind of sequence. Think of a number: say seven. As it's odd multiply by three and add one; 22 is even, so divide by two (11). Repeat indefinitely. The sequence starts 7, 22, 11, 34, 17, 52... then settles down: 8, 4, 2, 1, 4, 2, 1... It looks like you always end up with a repeating cycle, but nobody knows for sure. If you think it's obvious that such a sequence will get down to one and then repeat, try a variation in which you treble odd numbers and then subtract one. Start with 17 and see what happens. |
| The logic of petals Most flowers have a Fibonacci number of petals - it was through mimicking nature that they were discovered |
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The evolution of numbers |
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The evolution of numbers Most
early number symbols started as variations on I, II, III. Babylonian numbers
(circa 200 BC) were made on pieces of wet clay with the end of a stick. For
larger numbers they invented a shape for the number ten, and used multiples
of that for 20, 30 and soon, till 60, which was represented by the symbol
for 1, and 120 by 2, etc. Modern numerical notation is quite different. Instead of repeating the same stroke to denote larger numbers, we use a whole series of different symbols. And instead of having a distinct symbol for ten and multiples of ten, we use those same symbols (1 to 9) plus a new one (0). It is position that denotes whether a digit is a unit, ten, hundred or thousand, and so on. This is how the so-called "base ten" or decimal system works. The Mayans, who lived in South America around AD 1000, worked to base 20. In their system the symbols equivalent to our 525 would mean (5 x 20 x 20) + (2 x 20) + (5 x 1), which is 2,045 in our notation. The numerical base a society uses affects which numbers are regarded significant. Cricket fans always get upset when a batsman scores 49 and then is out, because he has just missed a half-century. But this is a decimalist way of viewing the situation. If the Mayans had played cricket, that number of runs would be represented by 29. For aliens on the planet Silimidon, where they use base seven, an innings of 49 is a century: (1 x 7 x 7) + (0 x 7) + (0 x 1) = 49. |
| Evolving numbers In early Indian civilisations, 4 was the lowest number that required a symbol of its own. Horizontal strokes can be seen evolving into 2 and 3 |
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May94
