Muhammad ibn MÅ«sÄ al-KhwÄÂrizmÄ«, a Persian mathematician.
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Algebra was supposedly started by the thinking of Abu Abdullah Mohammad Ibn Musa al-Khawarizmi, possibly one of the greatest mathematicians who ever lived. He came up with the idea of "Al-Jabr wa-al-Muqabilah", which involved the use of alalytical solutions in linear, and quadratic forms. The very word "Algebra" is derived from the expression.
15th Century by Johannes Widmann
It was a cool looking greek letter.
It is Arithmetica, by Diophantus of Alexandria, who is generally considered the father of algebra, that may be the first writing on algebra. But there is a bit of debate over who the father of algebra is. Use the link below to the related question.