The Arabs and the Development of Algebra
4.1 Introductory Remarks
In the end of seventh century, under the inspiration Mohammad’s leadership, Muslims conquered lands from India to Spain-including parts of North Africa and southern Italy. This social situation led to the contact between two different cultures which ultimately led to the transmission of mathematical knowledge.
Around the year 800, the caliph Haroun Al Raschid ordered many of the works of Hippocrates, Aristotle, and Galen to be translated into Arabic. In the twelfth century, these Arab translations were translated into Latin.
4.2 The Development of Algebra
4.2.1 Al-Khwarizmi and the Basics of Algebra
Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (780 C.E.-850 C.E.) was the most illustrious and most famous of the ancient Arab mathematicians.
In 830 C.E. he wrote an algebra text called Kitab fi al-jabr wa’l-mugabala that introduced the now commonly used term “algebra” (from “al-jabr”). The word “jabr” referred to the balance maintained in an equation when the same quantity is added to both sides (curiously, the phrase “al-jabr” also came to mean “bonesetter”) and the word “mugabala” refers to cancelling like amounts from both sided of an equation.
Then he wrote book called Art of Hindu Reckoning that introduced the notational system. Now, we call it as Arabic numerals: 1, 2, 3, 4…
Al-Khwarizmi also introduced the concept, and the word, that has now come to be known as “algorithm”.
4.2.2 The Life of Al-Khwarizmi
Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (780 C.E.-850 C.E.) was likely born in Baghdad, now part of Iraq.
Harun al-Rashid became the fifth Caliph of the Abasid dynasty on 14 September 786, at that time Al-Khwarizmi was born. Harun died in 809. Thus his son, al-Mamun, became Caliph and ruled the empire. Al-Mamun continued the patronage of learning started by his father and founded an academy called the House Wisdom where Greek philosophical and scientific works were translated.
Al-Khwarizmi and his colleagues called the Banu Musa were scholars at the house of Wisdom in Baghdad. He dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w’al-muqobala was the most famous and significant of all of Al-Khwarizmi’s works.
For many centuries, the motivation for the study of algebra was the solution of equation. Al-Khwarizmi’s day, these were linier and quadratic equations. His equations were composed of units, roots and squares. For example, a unit was a number, a root was x, and a square was x2. However, he did his algebra with no symbols-only words.
Al-Khwarizmi first reduces an equation (linear or quadratic) to one of six standard forms:
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
6. Roots and numbers equal to squares
The reduction is carried out using the two operations of “al-jabr” and “al-muqobala”.
Here “al-jabr” means “completion” and is the process of removing negative terms from an equation. The term “al-muqobala” means “balancing” and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de numero indorum (rendered in English, the title is Al-Khwarizmi on the Hindu Art of Reckoning) gave rise to the word “algorithm”. Another important work by A-Khwarizmi was his work Sindhind zij on astronomy. The main topics are calendars; calculating true positions of the sun, moon and planets, tables of sinus and tangents; spherical astronomy; astrological tables; parallax and eclipse calculations; and visibility of the moon. In addition, he wrote a major on geography which gives latitudes and longitudes for 2402 localities as a basis for a world map.
4.2.3 The Ideas of Al-Khwarizmi
The ideas are perhaps best illustrated by example.
Solve this problem of Al-Khwarizmi:
A square and ten roots equal thirty-nine dirham. (The Arabs could not deal with negative numbers).
4.2.4 Omar Khayyam and the Resolution of the Cubic
Omar Khayyam (1050-1123) is famed, and still well-remembered, for his beautiful poem The Rubaiyat. Khayyam was an accomplished astronomer and mathematician. He is remembered particularly for his geometric method of solving the cubic equation.
4.3 The Geometry of the Arabs
4.3.1 The Generalized Pythagorean Theorem
Arab geometry took many forms. They used geometry to analyze the roots of polynomial equations. The Arabs took a great interest in the parallel postulate and the existence of non-Euclidean geometry. Thabit ibn- Qurra discovered a remarkable generalization of the Pythagorean Theorem that applies to not only in right triangle but also all triangles. We can find the formula using the concept of similarity of triangles.
Consider the two triangles ∆ ABC and ∆ A’B’C’
The angle at A equals the angle at A’ and the angle at C equals the angle at C’. Thus, in order to test two triangles for similarity, we need only establish that two of the corresponding pairs of angles are equal.
The generalized Pythagorean Theorem was discovered by the Arabs.
Theorem: Let ∆ ABC be a planar triangle, with its longest side. Choose the point B’ on the segment so that the angle B’AB (in dashes) is equal to angle C (i.e., the angle in the vertex C in the triangle). Choose the point C’ on the segment so that the angle C’AC (in dots) is equal to angle B (the angle at vertex B in the triangle). Then
4.3.2 Inscribing a Square in an Isosceles Triangle
In fact, Al-Khwarizmi examined a problem based on the isosceles triangle. Then he made the inscribed square in the isosceles triangle. He would have used the name “thing” to refer to the side length of the square and the analysis might have been done more than 1000 years ago.
The area of the square is of course (thing) x (thing). Notice, that in the figure, we denote the side of the square by “X”. How do you determine the value of X? (The solution to Al-Khwarizmi’s problem is X = 4, 8)
4.4 A Little Arab Number Theory
The Arabs were fascinated by a technique that has come down through the ages as “Casting out Nines”. The basic rule for casting out nines for a positive integer N is to add its digits together.
4873 → 4 + 8 + 7 + 3 = 22 → 2 + 2 = 4
Part of the rule of casting out nines is that the process respects addition and multiplication. If we let “c.o.n” stands for casting out nines, then we have:
c.o.n. [ k + m ] = c.o.n. (k) + c.o.n. (m)
c.o.n. [ k · m ] = c.o.n. (k) · c.o.n. (m)
Thus, casting out nines can be used to check arithmetic problems.
Using casting out nines to check whether:
1) 693 x 42 = 29206
2) 6 x 8 = 14
If casting out nines does not work then the original arithmetic problem is incorrect.
If casting out nines does work then it is likely that the original arithmetic problem was correct. But it is not guaranteed