Read Online Fermat's Last Theorem by Simon Singh - Free Book Online Page B
Authors: Simon Singh
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repeats itself forever means that the decimal has a very simple and regular pattern. This regularity, despite the fact that it continues to infinity, means that the decimal can be rewritten as a fraction. However, if you attempt to express an irrational number as a decimal you end up with a number which continues forever with no regular or consistent pattern.
The concept of an irrational number was a tremendous breakthrough. Mathematicians were looking beyond the whole numbers and fractions around them, and discovering, or perhaps inventing, new ones. The nineteenth-century mathematician Leopold Kronecker said, âGod made the integers; all the rest is the work of man.â
The most famous irrational number is Ï. In schools it is sometimes approximated by 3 1 â 7 or 3.14; however, the true value of Ï is nearer 3.14159265358979323846, but even this is only an approximation. In fact, Ï can never be written down exactly because the decimal places go on forever without any pattern. A beautifulfeature of this random pattern is that it can be computed using an equation which is supremely regular:
By calculating the first few terms, you can obtain a very rough value for Ï, but by calculating more and more terms an increasingly accurate value is achieved. Although knowing Ï to 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom, this has not prevented computer scientists from calculating Ï to as many decimal places as possible. The current record is held by Yasumasa Kanada of the University of Tokyo who calculated Ï to six billion decimal places in 1996. Recently rumours have suggested that the Russian Chudnovsky brothers in New York have calculated Ï to eight billion decimal places and that they are aiming to reach a trillion decimal places. However, even if Kanada or the Chudnovsky brothers carried on calculating until their computers sapped all the energy in the universe, they would still not have found the exact value of Ï. It is easy to appreciate why Pythagoras conspired to hide the existence of these mathematical beasts.
Â
The value of Ï to over 1500 decimal places
When Euclid dared to confront the issue of irrationality in the tenth volume of the
Elements
the goal was to prove that there could be a number which could never be written as a fraction. Instead of trying to prove that Ï is irrational, he examined the square root of two, â2 â the number which when multiplied by itself is equal to two. In order to prove that â2 could not be written as a fraction Euclid used
reductio ad absurdum
and began by assuming that it could be written as a fraction. He then demonstrated that this hypothetical fraction could be simplified. Simplification of a fraction means, for example, that the fraction 8 â 12 can be simplified to 4 â 6 by dividing top and bottom by 2. In turn 4 â 6 can be simplified to 2 â 3 , which cannot be simplified any further and therefore the fraction is then said to be in its simplest form. However, Euclid showed that his hypothetical fraction, which was supposed to represent â2, could be simplified not just once, but over and over again an infinite number of times without ever reducing to its simplest form. This is absurd because all fractions must eventually have a simplest form, and therefore the hypothetical fraction cannot exist. Therefore â2 cannot be written as a fraction and is irrational. An outline of Euclidâs proof is given in Appendix 2 .
By using proof by contradiction Euclid was able to prove the existence of irrational numbers. For the first time numbers had taken on a new and more abstract quality. Until this point in history all numbers could be expressed as whole numbers or fractions, but Euclidâs irrational numbers defied representation in the traditional manner. There is no other way to describe the number equal to the
The concept of an irrational number was a tremendous breakthrough. Mathematicians were looking beyond the whole numbers and fractions around them, and discovering, or perhaps inventing, new ones. The nineteenth-century mathematician Leopold Kronecker said, âGod made the integers; all the rest is the work of man.â
The most famous irrational number is Ï. In schools it is sometimes approximated by 3 1 â 7 or 3.14; however, the true value of Ï is nearer 3.14159265358979323846, but even this is only an approximation. In fact, Ï can never be written down exactly because the decimal places go on forever without any pattern. A beautifulfeature of this random pattern is that it can be computed using an equation which is supremely regular:
By calculating the first few terms, you can obtain a very rough value for Ï, but by calculating more and more terms an increasingly accurate value is achieved. Although knowing Ï to 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom, this has not prevented computer scientists from calculating Ï to as many decimal places as possible. The current record is held by Yasumasa Kanada of the University of Tokyo who calculated Ï to six billion decimal places in 1996. Recently rumours have suggested that the Russian Chudnovsky brothers in New York have calculated Ï to eight billion decimal places and that they are aiming to reach a trillion decimal places. However, even if Kanada or the Chudnovsky brothers carried on calculating until their computers sapped all the energy in the universe, they would still not have found the exact value of Ï. It is easy to appreciate why Pythagoras conspired to hide the existence of these mathematical beasts.
Â
The value of Ï to over 1500 decimal places
When Euclid dared to confront the issue of irrationality in the tenth volume of the
Elements
the goal was to prove that there could be a number which could never be written as a fraction. Instead of trying to prove that Ï is irrational, he examined the square root of two, â2 â the number which when multiplied by itself is equal to two. In order to prove that â2 could not be written as a fraction Euclid used
reductio ad absurdum
and began by assuming that it could be written as a fraction. He then demonstrated that this hypothetical fraction could be simplified. Simplification of a fraction means, for example, that the fraction 8 â 12 can be simplified to 4 â 6 by dividing top and bottom by 2. In turn 4 â 6 can be simplified to 2 â 3 , which cannot be simplified any further and therefore the fraction is then said to be in its simplest form. However, Euclid showed that his hypothetical fraction, which was supposed to represent â2, could be simplified not just once, but over and over again an infinite number of times without ever reducing to its simplest form. This is absurd because all fractions must eventually have a simplest form, and therefore the hypothetical fraction cannot exist. Therefore â2 cannot be written as a fraction and is irrational. An outline of Euclidâs proof is given in Appendix 2 .
By using proof by contradiction Euclid was able to prove the existence of irrational numbers. For the first time numbers had taken on a new and more abstract quality. Until this point in history all numbers could be expressed as whole numbers or fractions, but Euclidâs irrational numbers defied representation in the traditional manner. There is no other way to describe the number equal to the
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