Posted: May 22nd, 2023

To date, Pythagoras theorem is arguably the sole primary theorem in all mathematical genres. Even though it has geometric origins, the theory generally ascribed to Pythagoras has been applied in almost all platforms of science disciplines. Numerous proofs have been developed in concurrence and affirmation of the original seminal Pythagorean theorems; some of these verifications were discovered by the most unpredictable persons in utter simplicity, including the twelve-year-old lad (Albert Einstein), an American president among others.

Although, currently we best know the theorem in its algebraic notation, a^{2}Â +b^{2}Â = c^{2Â }â€“ where from we can determine magnitude of one side of a right angled triangle given the other two, Pythagoras visualized it with a geometric perspective in which he related the areas of the resultant squares generated by the sides of a right angled triangle. It was only the convenient tool of algebra, brought to the fore around 1600 CE that has bequeathed us with this simplified algebraic version of Pythagoras theorem (Maor, 2007, p. 22). The elementary components of the theory are believed to have been discovered and utilized by the ancient Babylonians, and probably the Chinese approximately 1000 years before Pythagoras proved the theorem in about 500 BC. Old Babylonian tablets dating back to roughly 1800 -1600 BC, were unearthed whose inscriptions upon close scrutiny revealed a list of Pythagorean triples (that is, positive integersÂ *a, b and c*Â such that a^{2}Â + b^{2}Â = c^{2}) for example (3, 4, 5) and (4800, 4601, 6649). A good example of these Babylonian tablets is the Plimpton 322 collection at Columbia University. The mechanism employed by the Babylonians to arrive at these large Pythagorean triples is yet to be established, though Euclidâ€™s algorithmic analysis formulated 1,500 years later concretized the Babylonian findings. Euclid considered a set of two positive integers, say u and v, where u > v; and did show that the three Pythagorean triples a, b and c can be deduced as;

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Pythagoras theorem is arguably the sole primary theorem in all mathematical genres

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a = 2uv, b = u^{2}Â â€“ v^{2}Â and c = u^{2}Â + v^{2}

It follows then that;

a^{2}Â + b^{2}Â = (2uv)^{2}Â + (u^{2}Â â€“ v^{2})^{2}

=4u^{2}v^{2}Â + u^{4}Â â€“ 2u^{2}v^{2}Â + v^{4}

=u^{4Â }+ 2u^{2}v^{2}Â + v^{4}

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