Algebra's Father

MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

Al-Khwarizmi, also known as, Abu Jafar Muhammad ibn Musa al-Khwarizmi was born in Baghdad, ca 780 AD, a city now in Iraq, around 780 and died around 850. Although there is little information on the life of Al-Khwarizmi, what we do know is rather profound. He is known today as the "father of algebra". Diophantus was also given this title, but Al-Khwarizmi was more often thought of as the father of algebra because he was the first to teach algebra in such an elementary way. Diophantus, knowing many of things Al-Khwarizmi knew, but instead of focusing on the algebraic techniques, he was more interested in number theory. He died around 850 AD.

Along with many other scholars, Al-Khwarizmi worked and studied at the House of Wisdom, which is similar to colleges in today’s time. Approximately 20 to 30 years before Al-Khwarizmi’s death, he authored what is probably the best-recognized book of his time. Al-Khwarizmi authored the book entitled, Al-Kitab Al-Mukhtasar fi hisab al-jabr wa’l mugabala, or translated, the book on calculations by completion and balancing. The original of his book has been lost, but many translations still remain. His book eventually helped him become the "father of algebra" because his book is where we pull the word algebra from, al-jabr. In Al-Khwarizmi’s book he starts with a discussion of algebra of the first and second degree, today known as linear and quadratic terms. It wasn’t until many years later when Cardano solved equations of the cubic, or third degree terms. The second part of his book focused on the aspect of business and the applications involved.

Al-Khwarizmi, who was probably the greatest mathematician of his time, wrote and gave examples of linear and quadratic equations, as mentioned above, into six standard forms:

	1. Squares equal to roots 10x²= 20x
	2. Squares equal to numbers 10x² = 25
	3. Roots equal to numbers 10x = 20
	4. Squares and roots equal to numbers x² + 10x = 39
	5. Squares and numbers equal to roots x² + 39 = 10x
	6. Roots and numbers equal to squares 10x + 39 = x²

Of course, these above forms are in our modern day notation. Al-Khwarizmi did no use notation, as we know today. All of his work was done rhetorically, or spoken. Notation didn’t actually appear until the sixteenth century with Francois Viete. In our modern day notation we would also have the use of negative numbers. Al-Khwarizmi possibly understood the idea of "wanting" or "owing something, but did not use the concept of negative numbers. In all of his examples he never uses the concept of negative. In fact in his examples he uses both algebraic methods as well as geometric methods to state his mathematical problem. This leads us to believe that he had read Euclid’s Elements, which explains the non-use of negative numbers. One of the examples that Al-Khwarizmi liked to use and probably doctored it up is from his book and is that of completing the square. Al-Khwarizmi used the equation x² + 10x = 39.

"The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x² (Fig 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Fig 2). Now Fig 2 has area x² + 10x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 x 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 x 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3."

Another of Al-Khwarizmi’s best-recognized works is that which is named after him, Liber Algorismi de numero Indorum, or the work of Al-Khwarizmi. This book gave us the mathematical concept of an algorithm. The meaning of algorithm today relates to a specific way for solving a certain problem.

Not only is Al-Khwarizmi the "father of algebra", he founded several other branches and basic concepts of astronomy and geography. His influence on mathematics, astronomy, and geography are well established and his approach was methodical and commonsensical. He developed in detail trigonometric tables containing the sine function, which later helped to form the tangent function. Al-Khwarizmi also developed the calculus of two errors, which led him to the concept of differentiation. He worked on integrating Greek and Hindu knowledge, which led him to adopt the use of the number zero. Al-Khwarizmi also wrote a book on the astronomical tables and revised Ptolemy’s views on geography and corrected them in detail. Many other geographers worked under the direction of Al-Khwarizmi leadership and they ended up producing the first map of the known world onto a globe.

It is such a shame that there is so little known about the life of Al-Khwarizmi. What he has given us, as mathematicians, is so profound that it is understandable that he would receive the title of "father of algebra". Without his contributions and ability to relay knowledge is such an elementary way, it would be interesting to see how "algebra" would be affected.

Bibliography Page

1. http://khwarzimic.org/frontline/khwarzimi.html

2. http://www.sjsu.edu/depts/Museum/alkhwa.html

3. http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html

4. Math 460, History of Mathematics, BSU, Dr. Jones. Class Discussion

5. Biography in Encyclopaedia Britannica. (WWW version)

6. http://users.erols.com/zenithco/khawariz.html

7. http://www.lib.virginia.edu/science/parshall/algebra.html

8. Calinger, Ronald. Moore Publishing Co., 1982.

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