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 Muhammad ibn Musa al-Khwarizmi/ahl-khwahr-IZ-mee/

780–c. 850
Persian Mathematician
Mathematics Ranking 6th of 46
Al-Kwarizmi - BOOK - USSR - Portrait.jpg

Issued by the former Soviet Union in 1983 on the 1,200th                     anniversary of the birth of Al-Khwarizmi.

Al-Khwarizmi was perhaps the greatest medieval mathematician.(1) He should rightly be known as the father of algebra. Al-Khwarizmi established algebra as a distinct branch of mathematics, and translations of his writings later brought the idea to the West (Europe) along with Hindu-Arabic numerals (i.e., 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the number symbols we take for granted). The word “algebra” is derived from al-jabr, part of the title of his book on the subject, Hisab Al Jabr Wa’l Muqabala (Transposition and Reduction). As Ptolemy before him, al-Khwarizmi contributed effectively to mathematics, astronomy, and geography.

Mathematics–Algebra, Hindu-Arabic Number System, Trigonometry

Early mathematicians lacked symbols for representing mathematical expressions and operations (such as addition and subtraction), which hindered progress significantly. With the introduction of symbols, algebra developed as a separate branch of mathematics and also led to major advances in other areas of the subject. Ancient Babylonian, Egyptian, and Greek mathematicians managed to describe and solve mathematical problems by expressing them in words and numbers (called rhetorical algebra), but it was a longwinded process. A system was needed in which the answer being sought–the unknown–could be expressed in terms of the information known about it.


This remained a verbal process until the ninth century, when al-Khwarizmi laid the foundations of what we now call algebra by setting out systematic and logical rules for solving equations. Later mathematicians built on his work by introducing symbols for operations, such as +, −, and =, and using the shorthand x to represent an unknown, and other letters for variables. The invention of specific mathematical signs and symbols simplified solving problems and established algebra as a field in its own right. It also enabled algebra to be used to solve problems in other disciplines, such as astronomy, physics, and economics, and to solve everyday problems too. Many problems can be expressed algebraically in the form of an equation, and unknown quantities can be calculated by applying the relevant rules of algebra.


In al-Khwarizmi’s book al-Kitab al mukhtasar fi hisab al-jabr wa’l-muqabala (The Compendendious Book on Calculation by Completing and Balancing), he gave analytical (algebraic) and geometrical solutions of linear and quadratic equations. Linear and quadratic equations involve either the first-degree or power (linear) or second-degree or power (quadratic) of an unknown quantity or variable. This book was used as a principal textbook in European universities until the sixteenth century, and introduced to the West the word algebra (al-jabr–“transposition,” “reduction”).(2) “So systematic and exhaustive was al-Khwarizmi’s exposition ... that ... (he) is entitled to be known as ‘the father of algebra.(3) One of the most important pieces of information in al-Khwarizmi’s book on algebra was the quadratic formula or equation. Most students first encounter this formula in second-year algebra. It is probably the most important equation in algebra because it appears so often in various physical problems.(4)


For the quadratic equation (called quadratic because of the x²), ax² + bx + c = 0, the solutions are:


x = −b ± √b² – 4ac




For example, the solutions for the equation x² – 5x + 6 = 0 using the quadratic formula are:


Substituting 1 for a; –5 for b; and 6 for c into the quadratic formula gives:


x = –(–5) ± √(–5)² – 4(1)(6)

       __________________   then



x = 5 ± √25– 24

       _________   finally



x = 3 or x = 2


It is accepted today that a zero at the end of a number makes it 10 times bigger, e.g., 20 is 10 times bigger than 2. But early number systems were rather vague about using zero in this way. They used it to show that the tens column, for example, had nothing in it, but rarely used it in the units column. The first mathematician to use zero in today’s systematic way was al-Khwarizmi, in about 820 C.E. His ideas reached the West through the efforts of the French scholar Gerbert of Aurillac, who became Pope in 999 C.E.(5)


Al-Khwarizmi’s book Algoritmi de numero Indorum (Al-Khwarizmi Concerning the Hindu Art of Reckoning) introduced Hindu-Arabic numerals and their arithmetic to the West. It is preserved only in a Latin translation. From the name of the author, rendered in Latin as Algoritmi, originated the term algorithm. The word algorism originally referred only to the rules of performing arithmetic using Arabic numerals but evolved into algorithm by the eighteenth century. The word has now evolved to include all definite procedures for solving problems or performing tasks. A computer program is essentially an algorithm that tells the computer what specific steps to perform in order to carry out a specified task such as printing students’ report cards.



Al-Khwarizmi compiled astronomical tables which, as revised in Muslim Spain, were for centuries standard among astronomers from Cordova, Spain, to Chang’an, China.




(1) Will Durant, The Age of Faith – The Story of Civilization: Volume 4 (New York, 1950), p. 241.

(2) Ibid., p. 241.

(3) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991), p. 230.

(4) Lloyd Motz and Jefferson Hane Weaver, The Story of Mathematics (New York, 1993), p. 68.

(5) Roger Bridgman, 1,000 Inventions and Discoveries (New York, 2002), p. 62-63.


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