The Secret Of Numbers
>> Sunday, May 31, 2009
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.
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 (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
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 p.
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.
Great mathematical mysteries
There's no limit to theoretical puzzles that mathematicians would like to solve. Here are three of the most famous.
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
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 p.
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.
Great mathematical mysteries
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. 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.
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
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.
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. 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.
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
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.
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