History Of The Mathematics: The Pythagoreans

Mathematics History Name Institution Affiliation Mathematics History QA1 1 In Egypt geometry was used as a practical tool to meet the Egyptians building demands; nobody properly understood the theory. It was Thales who visited Egypt and turned their practical knowledge into theory, that is, he introduced abstract geometry. He formulated geometrical theories and used them to solve practical examples, making many people believe his work. Through Thales, the knowledge of geometry quickly spread into Greece and the world and he is considered the father of geometry. 2 The Pythagorean discovered that numbers are quantities that can be represented as a ratio of two integers, what is today known as rational numbers. They used a musical example to prove their discovery; where they showed that beautiful harmonies of vibrating springs corresponded to ratios of whole numbers. They also discovered the figurate numbers, numbers resulting from placing dots in regular patterns. They used squares and triangles to represent the figurate numbers, which is, they concluded that numbers such as n2 can be represented by uniform dots on a square and also from equilateral triangles triangular numbers can be found by placing uniform dots along the lengths of the triangles. 3 Zeno believed that distances and time are undivided whole numbers; using this believe he came up with a paradox that a swifter runner can never overtake a slower runner, an example of Achilles and the tortoise was used. Though he knew Achilles would pass the tortoise he insisted that space and time are not divisible. This problem could have been solved using geometrical series. 4 The Pythagoreans believed that all numbers could be expressed as a ratio of two integers. This belief was later changed when they incommensurable numbers (irrational numbers) which could not be expressed as ratios of two integers. Also, the discovery challenged their philosophical belief that numbers were the essence of all things. The discovery shattered all the work that the Pythagoreans had done in mathematics. Worse still, the discovery went against the religious belief of the people. In order not to jeopardize their earlier work and to avoid the wrath of the people, the Pythagoreans kept the discovery a secret. 5 (a). The first problem was the quadrature of a circle; known as the squaring of a circle – in this problem a square of an area equal to the area of a given circle was to be constructed. The second problem was the duplication of the cube; where a cube twice the volume of a given cube was to be constructed. The third problem was the trisection of a general angle; here a method of dividing an angle into 3 equal parts was to be designed. These problems were solved using a straight age and a compass. (b). Hippocrates made progress in attempting to solve the quadrature of a circle problem and duplication of a cube. Pierre Wantzel made progress on trisecting an arbitrary angle. (c) Yes, using algebra 6 Plato made it possible for mathematics...