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Algebra goes back to the Ancient Babylonians, where it was fist developed. www.storyofmathematics.com/sumerian.html
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Chinese scholars have created "The Nine Chapter on the Mathematical Art". Chapter 8 is where algebra is shown.
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Baudhayana, an Indian mathematician, uses algebra to derive Pythogoras' Theorem.
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Diophantus is concidered a father of algebra. He wrote a series of books called "Arithmetica" that were filled with math equations that we now call algebra.
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Important contributions were made by the Hindus as well. Scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji, made these adjustments to our everyday algebra. (400-1600) https://en.wikipedia.org/wiki/Indian_mathematics
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The name "algebra" came from the Arabic word "al-jabr", which means "reunion of broken parts". Which Muhammad ibn Mūsā al-Khwārizmī used in his treatise 'The Compendious Book on Calculation by Completion and Balancing'
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Nicolas Chuquet used the first letters of operations like p for plus and m for minus. His algebra was developed as a series of general methods and in this work negative numbers appear as coefficients, exponents and solutions to problems. His rules for solving arithmetic problems also used zero and negative numbers. Unfortunately his work was little known, and not published until 1880, but his ideas reappear in the early 17th century. nrich.maths.org/6546
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Although many people already used algebra, the first "x" wasn't even introduced until a Greek philosopher, Réne Descartes, first published in his treatise (a written work dealing formally and systematically with a subject) 'La Géométrie'. https://en.wikipedia.org/wiki/History_of_algebra
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Carl Friedrich Gauss proves the fundamental Theorem of Algebra.
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The woman that created modern algebra was Emmy Noether. She modified the old algebra, so that it would be easier. She is very important to the history of algebra. /cosmosmagazine.com/physics/woman-who-invented-abstract-algebra