Mathematical Contributions of the Persians and Arabs, Part I

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Mathematical Contributions of the Persians and
Arabs, Part I
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Last time we ended with a discussion of the
contributions of Indian mathematicians working
between the 6th and 12th centuries. Indian
mathematicians invented the base ten number
system with nine distinct symbols to represent
the numbers one through nine that we use today.
They also introduced the use of a place-holder to
indicate the absence of a number in a particular
position of the base ten system, and demonstrated
how it could be treated as a number by
introducing rules for addition, subtraction,
multiplication, and division. The place-holder
became our zero. Why did these rules have to be
developed? Why were they not obvious?
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The place-holder was not a number in the same
sense that an orange, for example, is not a
number. What does it mean to multiply a number
by an orange, or to take an orange away from a
number? The question doesnt make sense.
In the beginning numbers counted things (or
portions of things in the case of fractions).
The place-holder didnt count anything. It was a
separate symbol used to mark an empty position in
the base ten system.
Medieval Indian mathematicians generalized the
concept of number to include zero, but they
didnt quite do it for negative numbers. They
did, however, make important first steps.
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What does it mean to generalize the number
concept to include negative numbers and zero?
We generalize any concept when we include it as a
special case in a broader context. At first
numbers measured quantity. They counted
individual objects (natural numbers) or portions
of individual objects (fractions).
We could generalize the number concept to include
zero by explaining that zero is the quantity
which remains when all objects are removed from a
container.
This concept would be hard to extend to negative
numbers.
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How could we generalize the number concept to
include negative numbers?
Last time we saw how negative numbers arise
naturally as solutions to equations such as the
one suggested by the following problem
I am 7 years old and my sister is 2. When will I
be exactly twice as old as my sister?
The equation is (x 7) 2(2 x) and the
solution is x 3.
If we replace the ages 7 and 2 by 18 and 11, the
equation becomes (x 18) 2(11 x), and the
solution is x 4.
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We could we generalize the number concept by
explaining that numbers are the solutions to
equations such as (x
7) 2(2 x) and (x 18) 2(11 x).
With this definition, a number doesnt have to
count objects or portions of objects, but it
might. Equations have solutions that can be
either whole or fractional numbers, and are
either positive, negative, or zero. This new
concept of number includes the old one.
An appropriate generalization of any concept
includes the original as a special case. The
rules we introduce for addition, subtraction,
multiplication, and division of numbers in the
generalized sense shouldnt change when applied
to numbers in the original sense.
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Since (a b)(c d) ac bc ad bd for
positive numbers, this rule should hold for
negative numbers and zero as well.
We know (5 2 )(6 1) 15, so we should have
5 6 (2)6 5(1) (2)(1) 15 as well.
It makes sense that ( 2)6 12 and 5(1) 5,
so we have 13 (2)(1) 15. This shows that
(2)(1) 2.
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Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 15
9
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1)
10
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1)
11
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30
12
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30
13
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30
14
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30
15
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12
16
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12
17
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12
18
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12
19
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5
20
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5
21
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5
22
Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5
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Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5
The purple squares were subtracted twice and need
to be added back once.
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Repeating this argument for various examples
convinces one that a negative times a negative
should be a positive.
This result can also be justified geometrically.

(5 2 )(6 1) 30 12 5 2
The purple squares were subtracted twice and need
to be added back once.
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We say that medieval Indian mathematicians didnt
quite generalize the number concept to include
negative numbers because they treated positive
and negative numbers as different objects, as
fortunes and debts. They did, however,
generalize the number concept to include zero,
and they arrived at the right rules for operating
with negative numbers. Indian mathematicians
also made advances in algebra and trigonometry,
as we saw last time. When Europe emerged from
the Dark Ages, scholars became interested in
mathematical advances from other parts of the
world. They learned Indian mathematics through
contact with their Arab and Persian neighbors to
the east.
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We know intellectual activity in the
Mediterranean declined in response to chaos
brought about by the rise of the Roman Empire.
Weve also seen how the influence of Christianity
diminished Greek scholarship. What little
scholarship remained in Alexandria was
extinguished by the rise of Islam. Mohammad
untied various nomadic Arab tribes in a jihad
that continued for a century after his death in
632. Lands from India to Spain, including North
Africa and southern Italy, came under Arab
control. In 755 the empire split into two
independent kingdoms the eastern part had its
capital in Baghdad, and the western part was
ruled from Cordoba.
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Once they had completed their conquests, the
Arabs settled down to build a civilization and a
culture. They quickly became interested in the
arts and sciences, and the capitals in both
halves of the empire attracted and supported
scholars. In Baghdad the caliph established a
House of Wisdom, similar to the Museum at
Alexandria, with an academy, a library, and an
astronomical observatory. There Arabs scholars
collected and translated as much Greek learning
as they could find.
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Arab scholars improved translations of Greek
manuscripts and wrote commentaries on them. In
many cases, it is only through these Arabic
translations and commentaries that European
scholars later gained access to ancient Greek
discoveries.
An Arabic translation of Euclids Elements.
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Chapter 9 begins by describing several methods
the Arabs used for multiplying, dividing, and
extracting roots from numbers using the base ten
system developed by the Indians. The grating
method described on page 233 is especially nice.
I find it much easier to use that the algorithm I
learned in school. One of the homework problems
asks you to explain why it works.
6
5
4
1
1
1
3
2
5
2
8
1
1
2
0
2
0
8
9
1
6
5
4
4
3
9
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The grating method and other techniques
introduced in this section are useful for doing
computations on paper using the Indian number
system. When an abacus is used for calculations,
the convenience of the Indian system is not
obvious. The advantage of having written
calculations is that they can be checked for
accuracy and stored for future reference.
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Arab mathematicians, like the Indians before
them, knew the triangle of numbers which would
later become known as Pascals triangle. I will
not focus on results related to this triangle,
but instead highlight better known contributions
from al-Khwarizmi and al-Khayyami. These
mathematicians are both Persians (not Arabs) and
the Latin spellings of their names vary.
Al-Khayyami in particular is better known by the
name Omar Khayyam. He is famous not only as a
mathematician, but also as a poet.
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The word algebra comes from the title of a book
by al-Khwarizmi, Al-jabr wal muqabala, which
means restoration and completion. In the
equation x2 3x 6 2x the 3x is restored
when we write x2 6 3x 2x and we complete
the right hand side when we write x2 6
5x. The word algorithm also comes from
al-Khwarizmi, in the form of a misunderstanding
of his name.
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In Restoration and Completion, al-Khwarizmi
distinguishes three types of simple equations and
three types of compound equations. (See the
bottom half of page 247.) How would we describe
these types using modern notation?
Lets work through his method of solution for the
equation x2 10x 39. Why does it work?
x
5
x2 10x 39 25 39 64 (x 5)2 64 (x 5)
8
x
x2
5x
5x
5
25
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One of the homework problems asks you to solve
equations in this way. The second full sentence
on the top of page 248 will help you get started.
With this method you can solve equations of the
form ax2 bx c.
The equation in Problem 9.2 has the form ax2 c
bx.
By distinguishing various types of equations,
al-Khwarizmi could avoid the use of negative
numbers. He was familiar with negative numbers
and knew the rules for operating with them, but
still chose to work around them, perhaps because
he justified his procedures geometrically. Since
algebra had not yet been developed axiomatically,
mathematicians often chose to justify algebraic
techniques using the more rigorous deductive
geometry.
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Al-Khwarizmi also made important steps in the
arithmetization of algebra, i.e., in explaining
how to do arithmetic with unknown quantities. On
the bottom of page 249 and the top of page 250 we
see his explanation for how to multiply two
numbers. He extends the process to unknown
things by analogy.
Note how the algebra in this chapter is done
without symbolism despite awareness of the work
of Diophantus.