Mathematical Contributions of the Persians and Arabs

1
Mathematical Contributions of the Persians and
Arabs
Part II
2
Last time we saw how Arab mathematicians
collected and preserved scholarship that had been
exiled from Europe and Egypt. Al-Khwarizmi built
on this knowledge by making advances in algebra.
A Latin translation of his On the Indian Numbers
became one of the ways in which European
mathematicians learned the base ten number system
invented by medieval Indian mathematicians. In
The Condensed Book of Completion and Restoration
al-Khwarizmi demonstrated the process of solving
equations.
3
Abu Kamil wrote a commentary on al-Khwarizmis
algebra text that focused more closely on the
proofs of algebraic identities. What is an
algebraic identity?
Abu Kamil proved the following algebraic
identities If then
4
In the Book of Rare Things in the Art of
Calculation, Abu Kamil solved indeterminate
equations. He did this work before Arab
mathematicians knew the methods developed by
Diophantus. What is an indeterminate equation?
Al Karaji made important contributions to the
advancement of algebra by working with powers of
the unknown greater than three. His system of
naming higher powers appears in the table on the
top of page 254.
Al Karaji continued the arithmetization of
algebra begun by al-Khwarizmi by extending
arithmetic to powers of the unknown. In
particular he showed how an expression such as
a could be added,
subtracted, or multiplied by an expression such as
5
Al-Khayyamis On Completion and Restoration is
seen by some as the culmination of medieval
algebra in the Middle East. It is the most
rigorous of all the algebra texts introduced by
Chapter 9, and the proofs require fluency with
the geometry of Euclid and Apollonius.
Since al-Khayyami proved his results
geometrically, he became concerned with what it
meant to add, for example a cube to a square.

?
6
Last time we saw how al-Kwharizmi solved the
equation one square, and ten roots of the same,
amount to thirty-nine dirhems, i.e., x2 10x
39.
Al-Khayyami solves this same problem by first
converting the square and ten roots to geometric
objects of the same dimension, in this case
rectangles (see page 256).
one root of the square
ten roots of the square
one square

39
7
To solve cubic equations, al-Khayyami needed to
find two mean proportionals between two given
numbers. What does it mean to find two mean
proportionals between numbers a and b?
If a/x x/y y/b, then x and y are two mean
proportionals between a and b.
Al-Khayyami found mean proportionals by means of
intersecting parabolas (see page 256).
8
Our text demonstrates how al-Khayyami solved the
problem of cube and sides equal to a number,
i.e. x3 ax b (see page 257). To solve this
problem, he considered a p2 and b p2q. The
construction leads to x being the length of BE.
Al-Khayyamis solutions to cubic equations
include elegant applications of conic sections,
but they are hardly practical since the required
conic sections can be difficult to construct.
9
On the bottom of page 258, we read that
al-Khayyami disliked the purely formal nature of
ratios as presented in Euclids Elements. What
does this mean?
r
lt
r
r
r
r
10
r
lt
r
r
r
r
because
gt
lt
r
but
11
Al-Khayyami preferred to compare magnitudes using
the Euclidean algorithm. The example on page 258
considers magnitudes of size 28 and 10. The
Euclidean algorithm gives 28 2 10 8 10 1
8 2 8 4 2 with sequence 2, 1, 4 of partial
quotients. The ratio of any other pair of
magnitudes that share the same sequence of
partial quotient is equal to the ratio 28/10.
This meant that the ratios could be dealt with
numerically.
12
So far weve seen how the arithmetization of
algebra began with al-Khwarizmi and was extended
by al Karaji. The details left out by al Karaji
were filled in by al Samawal in his Shining Book
of Calculations. In particular he gave the first
examples of polynomial division. See page 260
for an example of the polynomials he could
divide. Al Samawal also introduced decimal (as
opposed to the more common sexigesimal) notation
centuries before its independent invention in
Europe.
13
Arab mathematicians also made contributions to
geometry, particularly in the accurate
construction of conic sections. One of the
homework problems asks you to explain why they
were so interested in conic sections.
The Arabs also tried to prove Euclids fifth
postulate.
14
On page 267 the author of our textbook claims
that Trigonometry as we study it today was
largely an invention of the Arab and Persian
mathematicians of the Islamic era. One of the
homework problems asks you to explain why he
makes this claim.