The contributions by arabic culture towards mathematics between
800-1200 AD
• Library of Alexandria, was considered most important library in in ancient
times. It flourished particularly under Ptolemy II 200-330 BC. In 642 AD
Alexandria was captured by the Muslim army of Amr ibn al ‘Aas. There
are some stories that the library was ordered to be distroyed (“if the books
are in accordance with the Qumran, we don’t need them, and if not destroy
them anyway....”)
• At any rate, arab scientist (‘eagerly’) absorbed greek text on mathematics,
and mostly due the translations of Euclid’s elements and Archimedes work
in arabic (in the 9-10 th century), these works have survived. (In the Byzantine empire also some texts were preserved, probably in the original greek
version).
• Bagdad became the new cosmopolitan center replacing Alexandria. The
mathematician and astronomer al-Khowarizimi wrote historimcally important textbooks on arithmetic and algebra. He combined Hindu influences
(the art of reckoning), based on fascination for numbers and computations.
• In particular the Hindu number system combining decimals, positions, and
0 was used and promoted through al-Khowarizimi’s work.
• Al-Khowarizimi’s book algebra deals with equations. The title Al-jabr wa’l
muqabalah refers to the technique of moving negative terms of an equation
to the other side, “al-jabr”, “muqabalah” refers to cancelation of equal terms
on both sides. This is the source of our word algebra.
• Al-Khowarizimi is the root of the word Al-Khowarizimi– algorisimi–algorithm.
• In terms of notation and symbols and compared to the work of greek mathe-
maticians after AD, Al-Khowarizimi treatments is a step back, compared to
Diophantus work on Arithmetic in the third century AD. Indeed, Diophantus used ∆γ for a square and K γ for a square of an unknown, so that ∆γ ∆
would be the forth power.1
1Edwards
book often ignores progress in number theory as not being relevant for calculus
1
2
• Al-Khowarizimi’s text is purely verbal: “A square and and a ten of its roots
are equal to thirty nine”. The solution would read like this. Take the half
the number and add its square to both sides. Take the square root, here
eight, and subtract from it half of the number of (in front of) the roots. In
our case
x2 + 10x = 39 ,
√
and the solution is x = 64 − 5.
(x + 5)2 = 64
• The arab mathematical science in this area reached its apex in the eleventh
century. Al-Haithan(965-1039, also known as Alhazen) wrote an influential
treatise on geometrical optics, and expetend some of Archimdes work on
rotated parabola and ellipsoids. This required better knowledge of sums of
cubes in a arithmetric progression.
• Alhazen found a new algorithm calculating sums of powers. In fact the
following pictures justifies
(n + 1)
n
!
i=1
ik =
n
!
i=1
ik+1 +
p
n !
!
(
ik ) .
p=1 i=1
3
• Application:
C |an | ≤ Cn.
!n
i=1
ik =
nk+1
k+1
+ O(n). Here an ∈ O(n) of for some constant