Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a merchant from Tyre). As a young man, he left his native city for Croton in Southern Italy, to escape the tyrannical government of Polycrates. Many writers credit him with visiting the sages of Egypt and Babylon before going west; such travels feature in the biographies of many Greek sages. Upon his migration from Samos to Croton, Pythagoras established a secret religious society very similar to (and possibly ...

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Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training. In astronomy, the Pythagoreans were well aware of the periodic numerical relations of the planets, moon, and sun. The celestial spheres of the planets were thought to prod ...

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Pythagoras (approximately 569 BCE – 475 BCE, Greek: Πυθαγόρας) was an Ionian mathematician and philosopher, founder of the mysterious religious and scientific society called Pythagoreans, known best for the Pythagorean theorem which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence ...

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Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspiration source to Plato and platonism. One main subject that is part of pythagoreanism is musica universalis, the music of the spheres. Some Surat Shabda Yoga, Satgurus considered the music of the spheres to be a term synonymous with the Shabda or the Audible Life Stream in that tradition, becaus ...

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The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. The first proof of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction (proof below). However Pythagoras believed in the absoluteness of numbers, and could not accep ...

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The word 'vegetarian' was coined in 1847 when the British Vegetarian Society was formed. Before this, vegetarians were known as Pythagoreans. The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans. It was called "health". ...

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Pocket calculators typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of x using the identity The same identity is exploited when computing square roots with logarithm tables or slide rules. There are numerous methods to compute square roots. See the ar ...

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If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define √A = B. More generally, to every normal matrix or operator A there exist normal operators B such that B2 = A. In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in a satisfactory manner. Positive definite oper ...

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Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. On the other hand it is impossible to determine the origin of the Pythagorean account of substance. It seems that the Pythagorean account begins with Anaximander's account of the ultimate substance of things a ...

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If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define √A = B. More generally, to every normal matrix or operator A there exist normal operators B such that B2 = A. In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in ...

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To every non-zero complex number z there exist precisely two numbers w such that w2 = z. The usual definition of √z is as follows: if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then we set √z = √r exp(iφ/2). Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous). The above Taylor series for √(1+x) remains valid for complex numbers x with |x| < 1. When the number is in rectangul ...

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The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers. The equation likewise produces the continued square root form: Also These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of ...

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Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if where a is the larger part and b is the smaller part. Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are se ...

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Note O: - For more information about this person's contribution to philosophy, see his/her entry in The Oxford Companion to Philosophy. Oxford University Press; 1995. ISBN 0198661320 Note R: - For more information about this person's contribution to philosophy, see his/her entry in the Concise Routledge Encyclopedia of Philosophy. Routledge; 2000. ISBN 0415223644 ...

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Under certain conditions infinitely nested radicals such as represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation If we solve this equation, we find that x = 2. More generally, we find that Beware, however, of the discontinuity for n=0. The infinitely nested square root for n=0 does not equal one, as the "general" solution would indicate. Rather, it is (obviously) zero.

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Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that This gives a startlingly quick proof that the golden ratio is an irrational number. An irrational number is one that cannot be written as a/b where a and b are integers. If a/b is such a fraction, in lowest terms, then b/(a − b) is in even lower terms — a contradict ...

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Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum that log23 is irrational: Assume log23 is rational. For some positive integers m and n, we have log23 = m/n. It follows that 2m/n = 3. Raise each side to the n power, find 2m = 3n. But 2 to any power greater than 0 is even (because at least one of its prime factors is 2) and 3 to any power greater than 0 is odd (because none of its prime factors is ...

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The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers. The equation likewise produces the continued square root form: Also: These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of ...

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The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles. The explicit expression for the Fibonacci sequence involves the golden ratio: The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the F ...

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