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Wed November 21, 2012 By:

# i want a project on the early history of mathematics

Thu November 22, 2012

Discovery of Irrational Numbers.

Introduction: An irrational number is any real number which cannot be expressed as a fraction p/q, where p and q are integers. They are precisely those real numbers that cannot be represented as terminating or repeating decimals.

Who Discovered Irrational Numbers?

The combination of vague ancient reports and modern guesswork has sometimes evolved into a much more emphatic and colourful tale.

Hippasus of Metapontum in Magna Graecia, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. He may have lived in the late 5th century BC. Iamblichus states that he was the founder of a sect of the Pythagoreans called the Mathematici in opposition to the Acusmatici.

Apparently Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved that we couldn't write the square root of 2 as a fraction and so it was irrational and hence  is the first irrational number discovered.

Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. Pythagoras of Samos and his secret society elevated mathematics to the realms of philosophy and looked at it as the one true language of nature. In that sense, the Pythagoreans can be regarded as the true founders of modern mathematics.

Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Â‘Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

Another version says that Pythagoras declared the irrationality of the square root of 2 to be a secret, not to be revealed to anyone outside the Brotherhood. When one of the member, Hippasus dared to defy Pythagoras' decree and inform the outside world of the discovery, he was killed.

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean who probably discovered them while identifying sides of the pentagram.

Pythagoras had previously discovered the famous Pythagorean theorem, which states that in a right triangle, the sum of the square powers of the lengths of the legs is equal to the square power of the length of the hypotenuse. The diagonal of a square divides it into two right triangles. Suppose that the sides of the square are one unit long, and let the length of the diagonal be d. The Pythagorean theorem says that 12 + 12 = d2, or d2 = 2. Thus, d is a number whose square is equal to 2. According to Pythagoras' credo, this number should be rational. But if it is a rational number, which one is it?

The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

1. Take a right triangle whose sides are 1 unit in length
2. By the Pythagorean theorem, the diagonal is ?2
3. Suppose that ?2 is a natural number, or that it is the ratio of two natural numbers. In both cases, it can be represented as the ratio of two natural numbers, ?2=m/n
4. Suppose that m/n has been reduced to its lowest common form by division
5. It follows that either m and n are both odd, or that m is odd and n is even, or that m is even and n is odd (if not, we could reduce m/n even further by dividing both numbers by 2)
6. Square both sides of ?2 = m/n, so that 2 = m2/n2
7. Then 2n2 = m2, so that m2 is even, and therefore m is even
8. If m is even, then m = 2x, where x is some other natural number
9. Squaring this, it follows that m2 = 4x2 = 2n2
10. It follows that n2 = 2x2, and therefore n2 is even, which means that n, being a natural number, must be even
11. So we've reached a contradiction: although we assumed that m and n cannot both be even, it now turns out they both are. It therefore follows that ?2 cannot be expressed as the ratio of two natural numbers, and must therefore be in another class of numbers.

So either way irrational numbers were discovered in the School of Pythagoras (founded by Pythagoras), a Brotherhood of mathematicians and philosophers in the Italian port town of Cortona in the 6th century B.C.

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