Free Fermat's Last Theorem by Simon Singh
Authors: Simon Singh
square root of two other than by expressing it as â2, because it cannot be written as a fraction and any attempt to write it as a decimal could only ever be an approximation, e.g. 1.414213562373 â¦
For Pythagoras, the beauty of mathematics was the idea thatrational numbers (whole numbers and fractions) could explain all natural phenomena. This guiding philosophy blinded Pythagoras to the existence of irrational numbers and may even have led to the execution of one of his pupils. One story claims that a young student by the name of Hippasus was idly toying with the number â2, attempting to find the equivalent fraction. Eventually he came to realise that no such fraction existed, i.e. that â2 is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasusâ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasusâ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.
The father of logic and the mathematical method had resorted to force rather than admit he was wrong. Pythagorasâ denial of irrational numbers is his most disgraceful act and perhaps the greatest tragedy of Greek mathematics. It was only after his death that irrationals could be safely resurrected.
Although Euclid clearly had an interest in the theory of numbers, it was not his greatest contribution to mathematics. Euclidâs true passion was geometry, and of the thirteen volumes that make up the
Elements
, books I to VI concentrate on plane (two-dimensional) geometry and books XI to XIII deal with solid (three-dimensional) geometry. It is such a complete body of knowledge that the contents of the
Elements
would form the geometry syllabus in schools and universities for the next two thousand years.
The mathematician who compiled the equivalent text fornumber theory was Diophantus of Alexandria, the last champion of the Greek mathematical tradition. Although Diophantusâ achievements in number theory are well documented in his books, virtually nothing else is known about this formidable mathematician. His place of birth is unknown and his arrival in Alexandria could have been any time within a five-century window. In his writings Diophantus quotes Hypsicles and therefore he must have lived after 150 BC ; on the other hand his own work is quoted by Theon of Alexandria and therefore he must have lived before AD 364. A date around AD 250 is generally accepted as being a sensible estimate. Appropriately for a problem-solver, the one detail of Diophantusâ life that has survived is in the form of a riddle said to have been carved on his tomb:
God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! late-born wretched child; after attaining the measure of half his fatherâs full life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.
The challenge is to calculate Diophantusâ life span. The answer can be found in Appendix 3 .
This riddle is an example of the sort of problem that Diophantus relished. His speciality was to tackle questions which required whole number solutions, and today such questions are referred to as Diophantine problems. He spent his career in Alexandria collecting well-understood problems and inventing new ones, and then compiled them all into a major treatise entitled
Arithmetica.
Of the thirteen
For Pythagoras, the beauty of mathematics was the idea thatrational numbers (whole numbers and fractions) could explain all natural phenomena. This guiding philosophy blinded Pythagoras to the existence of irrational numbers and may even have led to the execution of one of his pupils. One story claims that a young student by the name of Hippasus was idly toying with the number â2, attempting to find the equivalent fraction. Eventually he came to realise that no such fraction existed, i.e. that â2 is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasusâ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasusâ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.
The father of logic and the mathematical method had resorted to force rather than admit he was wrong. Pythagorasâ denial of irrational numbers is his most disgraceful act and perhaps the greatest tragedy of Greek mathematics. It was only after his death that irrationals could be safely resurrected.
Although Euclid clearly had an interest in the theory of numbers, it was not his greatest contribution to mathematics. Euclidâs true passion was geometry, and of the thirteen volumes that make up the
Elements
, books I to VI concentrate on plane (two-dimensional) geometry and books XI to XIII deal with solid (three-dimensional) geometry. It is such a complete body of knowledge that the contents of the
Elements
would form the geometry syllabus in schools and universities for the next two thousand years.
The mathematician who compiled the equivalent text fornumber theory was Diophantus of Alexandria, the last champion of the Greek mathematical tradition. Although Diophantusâ achievements in number theory are well documented in his books, virtually nothing else is known about this formidable mathematician. His place of birth is unknown and his arrival in Alexandria could have been any time within a five-century window. In his writings Diophantus quotes Hypsicles and therefore he must have lived after 150 BC ; on the other hand his own work is quoted by Theon of Alexandria and therefore he must have lived before AD 364. A date around AD 250 is generally accepted as being a sensible estimate. Appropriately for a problem-solver, the one detail of Diophantusâ life that has survived is in the form of a riddle said to have been carved on his tomb:
God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! late-born wretched child; after attaining the measure of half his fatherâs full life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.
The challenge is to calculate Diophantusâ life span. The answer can be found in Appendix 3 .
This riddle is an example of the sort of problem that Diophantus relished. His speciality was to tackle questions which required whole number solutions, and today such questions are referred to as Diophantine problems. He spent his career in Alexandria collecting well-understood problems and inventing new ones, and then compiled them all into a major treatise entitled
Arithmetica.
Of the thirteen