Eudoxus and Dedekind Irrational Numbers and Mathematical Development Essay

Eudoxus and Dedekind Irrational Numbers and Mathematical Development - Essay Example The theory, as stated, was very oblique and difficult. It was pondered by mathematicians until it was superseded in the nineteenth century. His definition of proportions in Euclid's work exemplifies the struggle taking place in the Greek mind to get a handle on this problem. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples taken in corresponding order. What could such an inscrutable statement possibly mean It seems that Eudoxus (through Euclid) must have sat up nights trying to write something that no one could comprehend. To understand this statement we must remember two things about Greek mathematics. First, Eudoxus was not talking about numbers, but magnitudes. The two were not the same and could not be related to each other. Second, the Greeks did not have fractions, so they spoke of the ratios of numbers and ratios of magnitudes. Hence, our fraction 2/3 was for them the ratio 2:3. For their geometry, they also needed to talk about ratios, not of numbers, but of geometric magnitudes. For example, they knew that the ratio of the areas of two circles is equal to the ratio of the squares of the diameters of the circles. We can show this as (Flegg, 1983) (area of circle A):(area of circle B) (radius of circle A)2:(radius of circle B)2 The Greeks had to be sure that when these ratios of magnitudes involved incommensurable lengths, the order relationships held. In other words, would their geometric proofs be valid when such proofs involved ratios of incommensurable lengths The definition developed by Eudoxus was an attempt to guarantee that they would. The magnitudes in the ratios have the following labels: first: second = third: fourth. Eudoxus said that the first and second magnitudes have the same ratio as the third and fourth if, when we multiply the first and third by the same magnitude, and multiply the second and fourth both by another magnitude, then whatever order we get between first and second will be preserved between the third and fourth. This explanation, simple as it is, can be rather confusing. An example will clarify the matter. We will assign the following lengths to the four magnitudes: 3:6 = 7:14. From this we get the following inequalities: 3 A3:B6 = A7:B14 or 15:12 = 35:28. Now clearly 15 > 12 and 35 > 28. Hence, multiplying by 5 and 2 preserved the order of the two ratios. Eudoxus' definition says that for two ratios to be equal, all values of A and B will preserve the order between the corresponding magnitudes. This gave Greek geometry the definition of magnitudes of ratios it needed to carry out the various proofs relying on proportion. However, magnitudes are not numbers, and the requirement that all values of A and B satisfy the definition introduced, through the back door, the notion of infinity. While Eudoxus' work satisfied the needs of geometers, it was

Are You Living With A Psychopath Essay Example | Topics and Well Written Essays - 1000 words

Are You Living With A Psychopath - Essay Example In recognition of those who have been abused, physically, morally or financially, to all of them, especially to Alicia C. Cussi, who lived brutally exploited, stripped and tormented by Teresa. Her own daughter, Alicia, inspired the creation of NEW ERA ELDERLY FOUNDATION, which will open to help to improve the elderly living conditions, preventing any kinds of elderly abuse caused by "family blindness" and excessive confidence to someone who would take advantage and betray the confidence, abusing us them with impunity. A reporter went to Ensenada, Mexico. Once he gets there, he goes to a supermarket. Upon arrival, a vagabond is being handcuffed by the police, being arrested because, being so hungry, he ate a piece of bread in the store. The reporter offers to pay for the bread, only 30 cents, and the guards let the indigent free. The vagabond was sorry and wants to pay the reported back. He pulls from his pocket a Rolex watch that he had tried to pawn only moments earlier; however, from the look of his appearance, the pawn manager had not accepted. The reporter, too, doesn’t accept the watch and he is pushed away by his friends to continue his trip. He keeps an eye on the news and returns many times to Ensenada, but he does not find the man; months after, he finds the vagabond in California. The reporter decides to lead the homeless man through the history and motivation that he has had to fight to build a foundation to prevent the abuse of older people who, like him and his mother, have been abused by a trusted family member.