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Mathematics LibreTexts

3.0: Prelude to the Fundamentals of Algebra

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    As his name portends, Abu Jafr Muhammad ibn Musa al-Khwarizmi was one of the greatest Arab mathematicians of his time. While living in Baghdad during the ninth century AD he became the Chief Librarian at the House of Wisdom, a library and major center of intellectual study. In the year 820 CE, al-Khwarizmi wrote Al-Kitab al-mukhtasar ti Hisab al-jabr w’al-muqabala, translated to, The Compendious Book on Calculation by Restoration and Reduction, the first book to generalize solving equations using the principles of equality. In fact, the word algebra itself comes from al-jabr, meaning reunion or completion.

    Figure \(\PageIndex{1}\): Pages from a 14th-century Arabic copy of the book, showing geometric solutions to two quadratic equations. (Public Domain; Muḥammad ibn Mūsā al-Khwārizmī via The Bodleian Library, University of Oxford)

    Many earlier cultures had employed what we might call algebra in the service of business, land management, inheritance, and trade. The Bablyonians were solving quadratic equations in 2000 CE, but only specific equations, with specific numbers. Hindus on the Indian continent were also developing algebra along side their invention of the symbol for zero 0. But like al-Khwarizmi and the Moslem Arabs of the first millenium, to write equations these early cultures did not use symbols like x or y, or even equal signs = that we use today. al-Khwarizmi wrote absolutely everything in words! 42 would be forty-two!

    Early algebra written all with words is called rhetorical algebra, and a thousand years ago, each mathematician had their own way of expressing it. Algebra was a language with many different dialects, and communicating it from one mathematician to another was difficult as they had to explain themselves with words. It was not until well after the Gutenberg printing press was invented in 1436 and print became standardized, that Rene Descartes, a Frenchman began to develop a modern symbolic algebra.

    In this section we’ll learn how to manipulate symbols in order to al-muqabalah (combine like terms) and al-jabr (restore and balance equations). But we’ll use modern notation to make it easier!

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