**Arab Contributions**

Within a century of Muhammad's conquest of Mecca, Islamic armies conquered lands from northern Africa, southern Europe, through the Middle East and east up to India. Within a century of that the Caliphate split up into several parts. The eastern segment, under the Abbasid caliphs, became a center of growth, of luxury, and of peace. In 766 the caliph al-Mansur founded his capitol in Baghdad and the caliph Harun al-Rashid, established a library. The stage was set for his successor, Al-Ma'mum.

In the 9 century Al-Ma'mum established Baghdad as the new center of
wisdom and learning. He establihed a research institute, the *Bayt al-Hikma* (House of Wisdom), which would last more than 200 years.
Al-Ma'mum was responsible for a large scale *
translation* project of as many ancient works as could be found. Greek
manuscripts were obtained through treaties. By the end of the century, the major works of the Greeks had been translated. In addition, they learned the mathematics of the Babylonnians and the Hindus.

What follows is a brief introduction to a few of the more prominent Arab mathematicians, and a sample of their work

**Abu l'Hasan al-Uqlidisi **
c.950

In al-Uqlidisi's book *Kita b al-fusul fi-l-hisab al-Hindii* (*The book of chapters on Hindu Arithmetic*), two new contributions are significant: (1) an algorithm for multiplication on paper is given, and (2) decimal fractions are used for the first time. Both methods do not resemble modern ones, but the methods are easily understood using modern terminology.

**Abu Ja'far Muhammad**

ibn Musa Al-Khwarizmi

Born: about 790
in Baghdad (now in Iraq)

Died: about 850

sometimes called the
*``Father of Algebra''*.

Al-Khwarizmi most important work *Hisab al-jabr w'al-muqabala* written in 830 gives us the word algebra . This treatise classifies the solution
of quadratic equations and gives geometric methods for completing the square. No symbols are used and no negative or zero coefficients were allowed.

Al-Khwarizmi also wrote on
Hindu-Arabic numerals. The Arabic text is lost, but a Latin
translation, *Algoritmi de numero Indorum* in English Al-Khwarizmi
on the Hindu Art of Reckoning gave rise to the word algorithm
deriving from his name in the
title.

To him we owe the words
*AlgebraAlgorithm*

His book *Al-jabr wál Mugabala*, on algebra, was translated into
Latin and used for generations in Europe.

- It isstrictly rhetorical - even numerals
and more elementary than
*Arithmetica*by Diophantus - It is a practical work, by design, being concerned with straightforward solutions of deterministic problems, linear and especially quadratic.
- Chapters I-VI covers cases of all quadratics with a positive solution is a systematic and exhaustive way.
- It would have been easy for any student to master the solutions. Mostly he shows his methods using examples - as others have done.

- Al-Khwarizmi then establishes geometric proofs for the same solutions of these quadratics. However, the proofs are more in the Babylonnian style.
- He dealt with three types of quantities: the
**square**of a number, the**root**of the square (i.e. the unknown), and**absolute numbers.**He notes six different types of quadratics: - The reason: no negative numbers -- no non-positive solutions
- He then solves the equation using essentially a rhetorical form for the quadratic equation. Again note: he considers examples only. There are no ``general" solutons.

**Other Arab mathematicians**

**
Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja**

Born: about 850 in (possibly) Egypt

Died: about 930

Abu Kamil Shuja is sometimes known as al'Hasib and he worked on integer solutions of equations. He also gave the solution of a fourth degree equation and of a quadratic equations with irrational coefficients.

Abu Kamil's work was the basis of Fibonacci's books. He lived later than Al-Khwarizmi; his biggest advance was in the use of irrational coefficients (surds).

Abu'l-Hasan Thabit ibn Qurra

Born: 826 in Harran, Mesopotamia (now Turkey)

Died: 18 Feb 901 in Baghdad, (now in Iraq)

Thabit was a native of Harran and inherited a large family fortune which enabled him to go to Baghdad where he obtained his mathematical training. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad.

Thabit generalized Pythagoras's theorem to an arbitrary triangle (as did Pappus. He also considers parabolas, angle trisection and magic squares.

He was regarded as Arabic equivalent of Pappus, the commentator on higher mathematics.

He was also founder of the school that translated works by Euclid, Archimedes, Ptolemy, Eutocius but Diophantus and Pappus were unknown to the Arabs until the 10 century. Without his efforts many more of the ancient books would have been lost.

Perhaps most impressive is his contribution to **amicable** numbers, that is two numbers who are each the sum of the divisors of the other.

Theorem., then and are amicable.

**Theorem.** (Generalization of Pythagorean Theorem.) From the vertex
*A* of , construct *B*' and
*C*' so that
Then

**Proof.** Apply similarity ideas

Note: If , this is the Pythagorean Theorem.

This is the third generalization of the Pythagorean Theorem.

**Mohammad Abu'l-Wafa al'Buzjani**

Born: 10 June 940 in Buzjan (now in Iran)

Died: 15 July 998 in Baghdad (now in Iraq)

Abu'l-Wafa translated and wrote commentaries, since lost, on the works of Euclid, Diophantus and Al-Khwarizmi. For example, he translated *Arithmetica* by Diophantus.

He is best known for the first use of the *tangent* function and compiling tables of sines and tangents at 15' intervals. This work was done as part of an investigation into the orbit of
the Moon.

His trigonometric tables are accurate to **8 decimal places** (converted to decimal notation) while Ptolemy's were only accurate to 3 places!!

**Abu Bakr al-Karaji**

( al-Karkhi)

early 11 century)

Arabic disciple of Diophantus - without Diophantine analysis.

Gave numerical solution to equations of the form

(only positive roots were considered).

He proved

in such a way that it was extendable to every integer. The proof is interesting in the sense that it uses the two essential steps of **mathematical induction.** Nevertheless, this is the first known proof.

al-Karkji's mathematics, more that most other Arab mathematics, pointed to the direction of Renaissance. mathematics.

**
Omar Khayyam**

Born: May 1048 in Nishapur, Persia (now Iran)

Died: Dec 1122 in Nishapur, Persia (now Iran)

Omar Khayyam's full name was Abu al-Fath Omar ben Ibrahim al-Khayyam. A literal translation of his name means 'tent maker' and this may have been his fathers trade. Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems, the *Rubaiyat*.

Khayyam was a poet as well as a mathematician. He discovered a geometrical method to solve cubic equations by intersecting a parabola with a circle but, at least in part, these methods had been described by earlier authors such as Abu al-Jud.

Consider the circle and parabola

Substitute and simplify to get

which factored gives

So, the intersection *x* is the solution of the cubic:

Khayyam was an outstanding mathematician and astronomer. His work on algebra was known throughout Europe in the Middle Ages, and he also contributed to a calendar reform. Khayyam refers in his algebra book to another work of his which is now lost. In that lost work Khayyam discusses Pascal's triangle but the Chinese may have discussed triangle slightly before this date.

The algebra of Khayyam is geometrical, solving linear and quadratic equations by methods appearing in Euclid's *Elements*.

Khayyam also gave important results on ratios giving a new definition and extending Euclid's work to include the multiplication of ratios. He poses the question of whether a ratio can be regarded as a number but leaves the question unanswered.

Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the *Rubaiyat* existed in Persian literature before Khayyam, and few of its verses can be attributed to him with certainty.

Ghiyath al'Din Jamshid Mas'ud al'Kashi

Born: 1390 in Kashan, Iran

Died: 1450 in Samarkand (now Uzbek)

Al-Kashi worked at Samarkand, having partron Ulugh Beg.

He calculated to 16 decimal places and considered himself the inventor of decimal fractions. In fact, he gives as

which was the best until about 1700.

He wrote The Reckoners' Key which summarizes arithmetic and contains work on algebra and geometry.

In another work, al'Kashi applied the method now known as fixed-point iteration to solve a cubic equation having as a root.

Generally, for an equation of the form

we define the iteration

where is some initial ``guess".
If the iterations converge, then it is a solution of the equation.
Such a method is called a fixed point iteration. Another more famous fixed point iteration is *Newton's Method*

He also worked on solutions of systems of
equations
and developed methods for finding
the root of
a number - Horner's method today. [**Note.** This
method also appeared in Chinese mathematics in 1303 in the *
Ssu-yüan-yü-chien* (Precious Mirror of the Four Elements)]

**Horner's Method**

Example. Solve

First determine that a solution lies between *x*=19 and *x*=20. Now apply
the transformation

to obtain

We know there is a root between *y*=0 and *y*=1. Thus there are two ways to approximate the solution for *y*:

If then is even closer to zero, and this term may be taken as zero, giving the approximate solution

so that . We may also factor the equation as

Letting the *y* in the parentheses be 1, solve for the other to get
hence the approximation

so that .

Clearly the first is slightly too large, while the second is slightly too small. Which should be selected? al'Kashi selects the second, . Why?

After Al-Kashi, Arabic mathematics closes as does the whole Muslim world. But scholarship in Europe at this time was on the up-swing.

Thu Mar 6 09:44:30 CST 1997