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The first algebra was on a Babylonian stone tablet. It was first known kind of math, as elementary as it was, it was developed for their time, and similar to that of Egypt
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Jain mathemeticians in India write the "Sthananga Sutra", a piece that contains theory of numbers, fractions, operations, and equations. It is from the very early Jain religion and is the first part of eleven Angas.
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Greek mathematician Diophantus writes his piece, Arithmetica which contains 130 algebraic problems giving numerical solutions to many types of certain equations.
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Wang Xiaotong, a Chinese mathematician, solved specific cubic equations, but his main contribution to the history of algebra was his book, "Jigu suanjing", which is translated to "Continuation of Ancient Mathematics". This was so impressive at the time he presented it to the emperor.
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Al-Samawal gave the first definition of algebra, "it's concerned with operating on the unknown using arithmetic tools, in the same way an arithmetician would operate on the known". He also wrote a mathematical treatise at the age of 19
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Leonardo Fibonacci of Pisa publishes the book Liber Abaci, a work on algebra that introduces Arabic numerals to Europe. Liber Abaci is sometimes considered the origin of European mathematics.
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Niccolò Fontana Tartaglia independently solves the general cubic equation in Italy. He is usually credited with the first solution and the winner of the cut-throat race in the 1600s to see who would solve it first.
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Thomas Harriot is the first to use < and > for greater than and less than. Harriot also introduced a simple notation for algebra, and founded the English School of Algebra.
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Descartes institutes the use of variables x,y, and z as unknown and introduces raised numbers as exponents. This was a large advancement in algebra greatly used today. He was also responsible for many other achievements and was a great mathematician of his time.
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Emmy Noether combines work on theories of algebra into a single algebraic theory used for a foundation for modern algebra. She also extended David Hilbert’s theory on the finite basis problem.