Muhammad ibn Mūsā al-Khwārizmī
Muhammad ibn Mūsā al-Khwārizmī was an ancient Persian astronomer and mathematician who made major contributions to mathematics, astronomy, and geography. He is also credited with developing a systematic and logical approach to solving linear and quadratic equations, which gave rise to the discipline of algebra.
His book On the Calculation with Hindu Numerals written about 825 A.D., was principally responsible for the dissemination of the Hindu-Arabic numeral system in the Middle-East and then into Europe. His name is also the source of the word algorithm, a procedure for solving a mathematical problem. Such procedures must be followed very strictly and have a finite number of steps each followed in the correct order.
The term algebra is derived from the title of a later book, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (Arabic الكتاب المختصر في حساب الجبر والمقابلة) written in 830 A.D. The title translates as:The Compendious Book on Calculation by Completion and Balancing.
In Rosen's translation of this book  Al-Khwarizmi describes its purpose as intending to teach :-
- ... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.
Al-Khwarizmi starts off by introducing the natural numbers (1,2,3 etc.), and then moves on to the main topic, the solution of equations. These equations are linear or quadratic, being composed of units, roots and squares. In Al-Khwarizmi's terms, a unit was a number, a root was x, and a square was . However, all of Al-Khwarizmi's mathematics was done entirely in words with no symbols being used.
Some of his contributions were based on earlier Babylonian, Greek and Indian sources. 
Notes & References
- F Rosen (trs.), Muhammad ibn Musa Al-Khwarizmi : Algebra (London, 1831).
- [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Khwarizmi.html A longer account of his life