Manasiya 1
Al-Khwarizmi and his Contributions to Mathematics
When I took mathematics courses in grade school and at the university level, I heard about many famous mathematicians. There was Euclid, Euler, Fermat, Gauss, Pythagoras,
Riemann, and many more mathematicians whose work we used in our math. But many of these mathematicians I learned about were European. I wondered if there were mathematicians outside of Europe until I discovered Muhammad ibn Musa Al-Khwarizmi. Al-Khwarizmi was a Middle
Eastern mathematician who made many contributions to algebra and talked about the Hindu-
Arabic numeral system.
Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was a famous Islamic mathematician and astronomer. He was born “in Persia of that time around 780 [CE]”1. There is not much information about his early childhood, but the many contributions he made to math are widely known and accessible. Al-Khwarizmi “lived in Baghdad, where he worked at the “House of
Wisdom” (Dār al-Ḥikma)”2. Here, Al-Khwarizmi and other scholars were “involved the translation of Greek scientific manuscripts”3, allowing them to have copies of major mathematical works created before them. The scholars used this information to advance Arab mathematics. Al-Khwarizmi wrote many treatises at the House of Wisdom, including two famous mathematical treatises.
The Compendious Book on Calculation by Completion and Balancing, or Al-kitāb al- mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala in Arabic, was one of Al-Khwarizmi’s most famous and important works in mathematics. This treatise is a “compilation of rules, together with demonstrations, for finding solutions of linear and quadratic equations based on intuitive
1 Stewart, Doug. "Muhammad Ibn Musa Al-Khwarizmi." 2 The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." 3 O'Conner, J. J., and E. F. Robertson. "Abu Ja'far Muhammad Ibn Musa Al-Khwarizmi." Manasiya 2 geometric arguments, rather than the abstract notation now associated with the subject”4. Al-
Khwarizmi starts the treatise by introducing the reason why he wrote the book. He wanted to teach “what is easiest and most useful in arithmetic”5; for example, Al-Khwarizmi wanted people to use algebra in calculating “inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, …, measuring of lands, the digging of canals, geometrical computations”6 and many more tasks that people came across in society and in their daily lives.
Al-Khwarizmi next introduced natural numbers, which may seem weird because we know what the natural numbers are, but he was helping the people from that time understand numbers.
The first section of treatise of algebra discusses solving equations using algebraic and geometric methods. Al-Khwarizmi’s equations were linear or quadratic and “were composed of units, roots, and squares”7. The six forms of equations Al-Khwarizmi discussed were: “(1) squares equal to roots, (2) squares equal to numbers, (3) roots equal to numbers, (4) squares and roots equal to numbers, (5) squares and numbers equal to roots, and (6) roots and numbers equal to squares”8. To solve these equations, al-ğabr and al-muqābala were used. In Arabic, al-ğabr means to “restore” or “complete”, while al-muqābala means to “complete” or “balance”. To go from x – 5 = 1 to x = 6 is an example of al-jabr, while to go from x2 + 4x = 4x + 16 to x2 = 16 uses al-muqābala. Al-Khwarizmi algebraic solutions “are expressed by words rather than numerals”9, unlike now, where we use numbers and variables to solve equations. Al-Khwarizmi also used squares to solve his equations geometrically. In Al-Khwarizmi’s problem x2 + 10 x =
39, he first put a square to represent x2. Now, Al-Khwarizmi “extended this square to a second
4 The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." 5 Al-Khwarizmi, Muhammad Ibn Musa. The Algebra of Mohammed Ben Musa. Edited and Translated by F. Rosen 6 Al-Khwarizmi, Muhammad Ibn Musa. The Algebra of Mohammed Ben Musa. Edited and Translated by F. Rosen 7 O'Conner, J. J., and E. F. Robertson. "Abu Ja'far Muhammad Ibn Musa Al-Khwarizmi." 8 Overbay, Shawn, Jimmy Schorer, and Heather Conger. "Al-Khwarizmi." 9 Arndt, A. B. "Al-Khwarizmi." Manasiya 3 square whose area is x2 + 10x + 25 with the side x + 5”10. Then the second square’s area, which is 39, plus 25, equals to 64, which is a square with a side length of 8. This means that the third square has a side length of both x + 5 and 8, and they must be equal, since x2 + 10x + 25 = 64.
Thus, x + 5 = 8 and x = 3.
The second portion of Al-Khwarizmi’s treatise of algebra discusses applied problems and examples he worked out. He worked out problems regarding “mercantile transactions of people”11, meaning the buying and selling of items and calculating the price or quantity of the items. Mensuration, or the area, volume, or length of shapes, was a topic Al-Khwarizmi also worked on. The final portion of the treatise talks about legacies, which is the “use of algebra to solve inheritance problems according to proportions prescribed by Islamic law”12.
Al-Khwarizmi on the Hindu Art of Reckoning was another important mathematical treatise which discussed Hindu-Arabic numerals. Although “no copies of Al-Khwarizmi's original Arabic version of the Arithmetic are extant … we have several early Latin translations”13. The Latin translation of the Arabic title was Algoritmi de numero Indorum, and this is where the word “algorithm” originated from. In this treatise, Al-Khwarizmi talks about how the Hindu numeral system uses the digits from 0 to 9 for the place value system. Algorithm also explained “the basic operations of addition, subtraction, multiplication, and division and showed how to work with fractions and how to extract square roots”14, which become easier with the new place value system.
10 Ji, Shanyu. History of Mathematics. 11 Al-Khwarizmi, Muhammad Ibn Musa. The Algebra of Mohammed Ben Musa. Edited and Translated by Frederic Rosen. 12 The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." 13 Arndt, A. B. "Al-Khwarizmi." 14 Arndt, A. B. "Al-Khwarizmi." Manasiya 4
Abu Ja’far Muhammad ibn Musa Al-Khwarizmi contributed many ideas to mathematics through his treatises in algebra and in Hindu-Arabic numerals. Al-Khwarizmi didn’t just write about mathematics, he also wrote Kitāb ṣūrat al-arḍ, which translates to “The Image of the
Earth”15, and he wrote a “set of astronomical tables (Zīj), based on a variety of Hindu and Greek sources”16. But Al-Khwarizmi’s mathematical works “were the principal source of mathematical knowledge for centuries to come in the East and the West”17. The Europeans benefitted greatly from Al-Khwarizmi’s work, and this allowed them to advance Al-Khwarizmi’s works and discover and create new mathematical ideas. Al-Khwarizmi passed away in 850 C.E.18, but his work helped lay the foundation for the growth of future mathematics.
15 The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." 16 The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." 17 Al-Daffa, Ali Abdullah. The Muslim Contribution to Mathematics. 18 Ji, Shanyu. History of Mathematics. Manasiya 5
Works Cited
Al-Daffa, Ali Abdullah. The Muslim Contribution to Mathematics. N.p.: Croom Helm, 1977.
Print.
Al-Khwarizmi, Muhammad Ibn Musa. The Algebra of Mohammed Ben Musa. Edited and
Translated by Frederic Rosen. Trans. Frederic Rosen. N.p.: Wentworth, n.d. Print.
Arndt, A. B. "Al-Khwarizmi." The Mathematics Teacher. 8th ed. Vol. 76. N.p.: National Council
of Teachers of Mathematics, n.d. 668-70. Print.
Ji, Shanyu. History of Mathematics.
O'Conner, J. J., and E. F. Robertson. "Abu Ja'far Muhammad Ibn Musa Al-Khwarizmi."
MacTutor History of Mathematics Archive. School of Mathematics and Statistics,
University of St Andrews, Scotland, July 1999. Web. 15 Oct. 2018.
Overbay, Shawn, Jimmy Schorer, and Heather Conger. "Al-Khwarizmi." N.p., n.d. Web. 14 Oct.
2018.
Stewart, Doug. "Muhammad Ibn Musa Al-Khwarizmi." Famous Scientists. N.p., n.d. Web. 14
Oct. 2018.
The Editors of Encyclopaedia Britannica. "Al-Khwārizmī." Encyclopædia Britannica.
Encyclopædia Britannica, Inc., 17 Feb. 2017. Web. 15 Oct. 2018.