Islamic Mathematics and Mathematicians
Introduction
The torch of ancient learning passed first to one of the invading groups
that helped bring down the Eastern Empire. 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. The empire was immense, rivaling that of Rome itself. Though
the Arabs initially focused on conquest, nonetheless to them ancient
science became precious treasure. The Qur’an, the sacred book of Islam,
praised medicine as an art close to God. Astronomy and astrology
were believed to be a pathway to discover God’s will. 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 9th 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
translating as many ancient works as could be found. Greek manuscripts
were obtained through treaties. By the end of the 9th century,
the major works of the Greeks had been translated. In addition, they
learned the mathematics of the Babylonnians and the Hindus. The
Arabs did not stop with assimilation. They innovated and criticized.
They absorbed Babylonian and Greek astronomy and constructed large
scale astronomical observatories and made measurements against which
predictions of Ptolemy could be checked. Numbers, particularly numbers
as used in algebra fascinated the Islamic mathematicians. Surely,
if one measures Islamic mathematics against the ancients, it would be
in algebra where their originality and depth is most clearly evident.
What follows is a brief introduction to a few of the more prominent
Arab mathematicians, and a sample of their work in more or less chronological
order. Observe the progression of their mathematics over five
century of labor. One should not underestimate the importance of the
Islamic world for the preservation of ancient learning.
We recount here a few of the other prominent Islamic mathematicians.
However, it is important to recognise that this is just the tip of the
iceberg. Islamic mathematics and mathematicians is a very active area
of mathematics, one that will reveal much much more that the mere
sketch we have today.
Here are some of the prominent Islamic Mathematicins:
Abu Ja’far Muhammad ibn Musa Al-Khwarizmi(c. 790 - 850)
Al-Khawarizmi
Also known as Al-Khwarizmi(which is spelled in several ways), he is
no doubt the best known of the Islamic mathematicians, and his work
is among the most influential of the islamic school. Indeed, his books
were studied long into the Renaissance. By reason of his work on algebra
Al-Khwarizmi is sometimes called the “Father of Algebra”. Al-
Khwarizmi’s 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, which in
English is 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
Algebra Algorithm
His book Al-jabr w´al Mugabala, on algebra, was translated into Latin
and used for generations in Europe.
- It is strictly rhetorical – even the numerals are expressed as words.
And it is 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
in 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:
ax2 = bx
ax2 = c
bx = c
ax2 + bx = c
ax2 + c = bx
ax2 = bx + c
- 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.
Shuja ibn Aslam ibn Muhammad ibn Shuja (c. 850 - 930)
Abu Kamil Shuja
Abu Kamil Shuja,an Egyptian sometimes known as al’Hasib, 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 al-Uqlidisi, (c. 950)
Al-Uqlidsi
In al-Uqlidisi’s book Kitab 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.
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