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Pythagoras, a Greek philosopher and religious leader, was responsible for important developments in the history of mathematics, astronomy, and in the theory of music. He inspired many later mathematicians and philosophers such as Aristotle. He was born around 500 B.C. on the island of Samos, which is in the Aegean Sea. Most of his early years were spent traveling and searching for wisdom. Around 530 B.C. he finally decided to settle in Crotona, a Greek colony in southern Italy and founded a philosophical and religious school there that attracted many followers. There, he founded the Pythagorean brotherhood. They were a group of his followers who were inspired by his teachings, and whose beliefs and ideas were rediscovered during the Renaissance and contributed to the development of mathematics and Western rational philosophy. The group was strongly religious and devoted to reformation of political, moral, and social life. The group was influential in the region, but eventually its involvement in politics resulted in suppression of the brotherhood. Due to this, Pythagoras was forced to retire and leave the area. He then moved to Metapontum, a Greek city in southern Italy, where he died in about 500 BC.
Throughout the years, all of the works and writings of Pythagoras have been lost. This makes it difficult to distinguish his teachings from those of his disciples. Among the basic beliefs of the Pythagoreans are “the beliefs that reality, at its deepest level, is mathematical in nature; that philosophy can be used for spiritual purification; that the soul can rise to union with the divine; and that certain symbols have a mystical significance.” Although Pythagoras is generally given the credit for the theory of the functional significance of numbers in the objective world and in music, his followers are given the credit for the development of the Pythagorean theorem in geometry and the application of number relationships to music theory, acoustics, and astronomy.
Pythagoreans believed that all relations could be reduced to number relations. They believed that in some way “all things are numbers.” This generalization came from certain observations in music, mathematics, and astronomy. The Pythagoreans noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. They knew, as did the Egyptians before them, that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. Pythagoras, or perhaps one of his students, proved that if triangle ABC is a right triangle with a right angle at C, then c(2) = a(2) + b(2). This Pythagorean theorem, that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides (A squared + B squared = C squared), may have been known in Babylonia, where Pythagoras traveled while he was still young. The converse theorem (If c(2) = a(2) + b(2) in a triangle ABC, then the angle at C is a right angle) appears to have been used much earlier. For example, early Egyptians used knotted ropes to form triangles with sides 3, 4, and 5 units long. Because 5(2) = 3(2) + 4(2), the angle opposite the side of length 5 was assumed to be a right angle.
In astronomy, the Pythagoreans were aware of the periodic numerical relations of heavenly bodies. The celestial spheres of the planets were thought to produce a harmony called the music of the spheres. Pythagoreans believed that the earth was always in motion. The most important discovery of this school, which upset Greek mathematics, as well as the Pythagoreans’ own belief that whole numbers and their ratios could account for geometrical properties, was the irregularity of the diagonal of a square with its side. This result showed the existence of irrational numbers.
In Euclidean geometry the theorem of Pythagoras provides a basis for the definitions of distance and similar equations hold in spaces of higher dimensions.
Pythagoras was one of the most important and influential people in the field of math. To this day, his teachings are used all over the world.
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