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
Almost all of us are curious and want to know more about the meaning of numbers.
The date of our birth, the number plate of our car, the street number that we live in and so on are all made up of numbers and they all have a distinct meaning.
Numbers are everywhere. In a sense, we are what we can count, and our computerized civilization 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 civilizations were becoming organized thousands of years ago. One theory is that people developed symbols before they learn to count. A king would keep tabs on what he owned - his livestock and riches - by using clay tokens.
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 realized 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.
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) symbolized marriage.
The number four was symbolic of harmony, because two is even, so four (two times two) is "evenly even". Four also symbolized the four elements out of which everything in the universe was made (earth, air, fire, and water).
Special Numbers 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 (Π)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 - Π, 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 Π be represented by a fraction or expressed in simple algebraic form. That is why Π is known as a transcendental number.
e ip + 1 = 0
This is one of the most awe inspiring equations there is. It elegantly demonstrates the connection between those five most important numbers 1,0,e,i and Π.
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 + 50c + 75c). 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 Π it is not an exact fraction.
Magic Sequences
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 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.
Round numbers An exhaustive empirical study of Golan Levin, with the aid of custom software, public search engines and powerful statistical techniques, in order to determine the relative popularity of every integer between 0 and one million. The resulting information exhibits an extraordinary variety of patterns which reflect and refract our culture, our minds, and our bodies.
For example, certain numbers, such as 212, 333, 486, 911, 1040, 1492, 1776, 68040, or 90210, occur more frequently than their neighbors because they are used to denominate the phone numbers, tax forms, computer chips, famous dates, or television programs that figure prominently in our culture. Regular periodicities in the data, located at multiples and powers of ten, mirror our cognitive preference for round numbers in our biologically-driven base-10 numbering system. Certain numbers, such as 12345 or 8888, appear to be more popular simply because they are easier to remember.
- Secrets of Numbers by Ian Stewart mathematician http://en.wikipedia.org/wiki/Ian_Stewart_(mathematician)
- The Secret Life of Numbers by Golan Levin, et. al. (February 2002) is a commission of New Radio and Performing Arts, Inc., for its Turbulence web site. It was made possible with funding from The Greenwall Foundation.
- Numerology by Lubomir Dimitrov MSc,
