Al-Khwarizmi: Arabic Numerals and Geometric Justification to Algorithms

by Jordan Vidmore

In today’s western schooling system students take for granted the rules of algebra when they are taught at a young age how to solve simple equations. When presented with the equation 3x-5=7, students immediately solve it by getting rid of the negatives and then dividing to get 4. It wasn’t always like this though. It took the works of a persian mathematician named Muḥammad ibn Musa Al-Khwarizmi to make this possible. Without Al-Khwarizmi’s works, math would not be the coherent process it is today.

Al-Khwarizmi did not invent the idea of algebra. “Finding solutions to equations is a pursuit that dates back to the ancient Egyptians and Babylonians.” (Tanton, 2005) Clay tables from the Babylonians (1700 BCE) show the existence of crude algebra in the form of a few quadratic equations worked out in full. However they lacked a general method to work out new ones. These tablets were used by finding a worked out equation on them similar to the equation they are currently faced with. Then they adjust the worked out equation’s values to fit the current situation. This method had major setbacks and was extremely limited as to which equations it could solve. (Tanton, 2005)

Early Greek mathematics improved algebra by using the method of false position. This method was taught in the Rhind Papyrus, a mathematical text of theirs, which contained the problem “A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?” (Chace 1979)The reader was asked to solve this problem by false position. One would guess a solution to the problem, and then adjust to get the answer. For example, if one guessed 5 as the solution, they would get 25. They would then subtract 15, the answer from the original problem, from 25 to get 10 and divide 10 by 5 to get 2. This 2 would then be subtracted from the 5 to get 3, which is the answer. This method always worked for linear equations but could not be developed into a viable formula to work on quadratic equations successfully. (Tanton, 2005)

Under the teachings of Pythagoras, followers gave geometric proofs to the distributive property and created the difference of two squares formula. This new formula made it possible to invent a quadratic equation and then find the solution to that equation. Even with this formula, though complex or large equations were far to difficult to solve because the Greeks lacked symbols for math and were forced to write out their equations. An equation such as 23x+x=82 in modern notation would be written as: “A number, its twenty-third is added to it. It becomes eighty-two. What is the quantity?” As the numbers get larger and the equation bigger, the complexity to solve it increases exponentially. It wasn’t until later that symbols started to appear, and were use by Diophantus of Alexandria; but these were only used as shorthand and not actually used to solve the equations. These symbols existed only as a way of writing the problem; this is, until a brilliant Persian mathematician discovered how to use them to solve equations.

Born in CE 780, Al-Khwarizmi – full name Muhammad ibn Musa Al-Khwarizmi – would grow up to be a great mathematician and be considered one of the “eight great brains of ancient world scientists.” (Graham 2009) Not much is know about his early life other than he grew up when Arab science and culture were beginning to flourish and transferring into the golden age of Islamic science. As an adult, Al-Khwarizmi spent most of his life in Baghdad where he had access to avast amounts of mathematical manuscripts as he studied in the House of Wisdom. One of his tasks here was to translate the works of ancient societies to make it available to Arabic scholars. With the knowledge of ancient mathematics, Al-Khwarizmi wrote two books titled al-Kitab al-mukhtasar fihisab al-jabr wa’l-muqabala and Al-jam’ w’al-tarfriq ib hisab al-hind. Know as the “father of modern algebra,” the word algorithm is actually a corruption of Al-Khwarizmi name. (Britannica 2011) He died in CE 850 but left behind these two texts which contain knowledge that has shaped modern algebra into what it is today.

Al-Khwarizmi’s text, Addition and Subtraction in India Arithmetic, introduced the Indian numerical system to the western world. In this book he explains the advantages of using the decimal-place system, such as in Indian numerals, over the current Egyptian technique based on finger counting. Al-Khwarizmi firmly believed that the Hindu method of arithmetic, based of fixed values, was more useful then the tradition form. (Healey 2006) This book actually didn’t contain any significant new knowledge that wasn’t already available in the Hindu language but his translation into Arabic, and Fibonacci’s subsequent translation into latin 300 years later, made the text available to many educated scholars and the numerals to be know as “Arabic numerals”. (Healey 2006) In this text, Al-Khwarizmi uses a symbol for zero which he calls “sifr”, and explains it as a placeholder but not as a number. The significance of which, for example, was to make 23 appear different than 203 when written.

This text created a basis for Al-Khwarizmi’s subsequent texts on algebra which would be based off of this system of numerals. These texts would not work with the old finger counting method and therefore caused Al-Khwarizmi to have to use the Arabic numerals based off of the Hindu ones. (Hutchinson’s 2011)

Al-Khwarizmi’s Calculation by Restoration and Compensation was designed as a teaching guide. “It aimed to offer an array of techniques and methods for solving very practical problems in matters of trade, inheritance, law, surveying, and architecture.” (Tanton 2005) This book begins with the basics and defines natural numbers and how to count them. It then moves on to the process of solving simple linear and quadratic equations. In this text, Al-Khwarizmi uses two simple methods, al-jabr (completion) and al-muqabala (balancing), which he says could reduce any linear or quadratic equation into one of six basic types. Al-jabr, or completion, is the process of removing any negative terms from an equation. An example of this would be changing 2x^2-6x=8 into 2x^2=6x+8. The word algebra comes from Al-Khwarizmi’s term al-jabr. The second, al-muqabala, or balancing, is the process of subtracting terms with the same power that appear on each side. 2x^2+3=x^2+8x would be simplified down to x^2+3=8x. The text goes on to explain the concept of completing the square and how this method could be used on all of the six basic equations. (Tanton 2005)

In part 2 of the text, Al-Khwarizmi explains the practical applications of al-jabr and al-muqabala. In this section he explains how the quadratic equations yield both a positive and a negative answer; but he also advises the reader to reject the negative answer as it has no real world value.

The final, and larges section of the book does not develop any new mathematical content, but instead focuses on the complicated Islamic rules for inheritance. All of the equations in this section are completely worked out and show how they can be used in different situations, such as inheritance land division. (Healey 2006)

When developing Calculation by Restoration and Compensation, Al-Khwarizmi attempted to be practical by including equations that would very likely be seen in relevant fields such as law, business, geography, engineering, and trade. (Healey 2006)

One of Al-Khwarizmi’s main concerns was to find the roots of quadratic equations. He accomplished this with the use of proofs based on geometric equations. Each of the six basic types of equations has its own geometrical proof. Consider the problem x^2+10x=39. Al-Khwarizmi writes:

“The manner of solving this type of equation is to take one half of the root [Coefficient]. Now the root in this problem is 10. Therefore, take 5 which multiplied by itself is 25, an amount which you add to 39 to give 64, having taken then the square root of this which is 8, subtract from it the half of the root 6, leaving 3”
From al-Kitab al-mukhtasar fihisab al-jabr wa’l-muqabal

Al-Khwarizmi also includes two geometric justifications for this equation.

“In our first, you start with a square of side x; adjoin to it two identical rectangles, each of length 5 and width x so as to form an L shape whose base and height are each x+5, and then superimpose on it a big square of side x+5. the area of the L shape is x^2+5x+5x, that is, x^2+10x which, by our original equation, must equal 39; the area of the big square, viz. (x+5)^2, is 25 more units than the area of the L shape and so much equal 39+25=64. Thus, (x+5)^2=64 and so, taking positive square roots, x+5=8, that is, x=3.”
Al-Khwarizmi through Maher 1998

The second:

“The small square of side x is centered in the middle of a big square, a gain of side x+5. Here, the area of the four corner squares is 4×2.5x.5=25 and the area of the cross shape is x^2+4(2.5x)=x^2+10x, so again the area of the big square is 64, whence x=3.”
Al-Khwarizmi through Maher 1998

These methods that Al-Khwarizmi used only gave positive solutions, which since the negative solutions didn’t have a practical use, he didn’t mind. Al-Khwarizmi’s work still had a large impact on the mathematical community.

Al-Khwarizmi’s work was most influential after Fibonacci translated it into Latin. Quickly thereafter his work became quite important to scholars and businessmen alike during the Middle Ages. (Healey 2006) The use of his Arabic numeral system was quickly adopted and became the standard. This had a huge impact on the education system by giving a ten-digit system to use rather then Roman numerals. There are two main impacts this had on the mathematical community in the university system. The first was that Arabic numerals are far easier to write, read, and work with than Roman numerals. A problem written as XXVx+XIV=CVII is much harder to work with than 25x+14=107. What makes the Roman numerals so difficult to work with is that they are not based off of ten, but rather a system of 5. The single digits change at 5; I,II,III,IV,V, the ten digits at 50; X,XX,XXX,XL,L, and so forth. The major problem with this, and its relation to math, is that formulas can not be made to work with all the values. The second main impact was that they made it possible for someone to solve the problem in symbol form.

Al-Khwarizmi’s work, Calculation by Restoration and Compensation, also had a large impact on the school system of the Middle Ages and beyond. Al-Khwarizmi may not have invented algebra, but he perfected it in this book. His perfection, and geometric proofs, of algebra is the foundation for many of todays upper level mathematics. All upper level math uses the idea of al-jabr and al-muqabala to simplify problems into a manageable and familiar equation.

As the university system spread, they adopted Al-Khwarizmi’s Arabic numerals and ideas of algebra as a foundation to their math. This led to the end of using the false position method and the practice of finger counting and being replaced by Al-Khwarizmi’s methods. If it wasn’t for Al-Khwarizmi, math would not be the coherent process it is today.

Bibliography

“Al-, Muhammad Ibn-Musa Khwarizmi (C. 780-C 850).” Hutchinson’s Biography Database (2011): 1. MAS Ultra – School Edition. Web. 18 Jan. 2012.

Brezina, Corona. Al-Khwarizmi : The Inventor Of Algebra. Rosen Central, 2006.

Chace, Arnold Buffum. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Transcriptions, Transliterations, and Literal Translations. Reston, VA: National Council of Teachers of Mathematics, 1979. Print.

Graham, Amy. Astonishing Ancient World Scientists : Eight Great Brains. Books, 2009.

Healey, Christina. “Al-Khwarizmi.” Al-Khwarizmi (2006): 1. MasterFILE Premier. Web. 18 Jan. 2012

“Khwarizmi, Al-” Britannica Biographies (2011): 1. MAS Ultra – School Edition. Web. 18 Jan. 2012.

Maher, Philip. From Al-Jabr to Algebra. Vol. 27. Print.

Tanton, James Stuart. Encyclopedia of Mathematics. New York: Facts on File, 2005. Print.