Arabic mathematics 17

applied mathematics The study and use of the math-

ematical techniques to solve practical problems is called

applied mathematics. The field has various branches

including

STATISTICS

,

PROBABILITY

, mechanics, mathe-

matical physics, and the topics that derive from them,

but the distinction from

PURE MATHEMATICS

might not

be sharp. For instance, the general study of

VECTOR

s

and

VECTOR SPACE

s can be viewed as either an abstract

study or a practical one if one later has in mind to use

this theory to analyze force diagrams in mechanics.

Many research universities of today possess two

departments of mathematics, one considered pure and

the other applied. Students can obtain advanced degrees

in either field.

approximation A numerical answer to a problem

that is not exact but is sufficient for all practical pur-

poses is called an approximation. For example, noting

that 210 is approximately 1,000 allows us to quickly

estimate the value of 2100 = (210)10 as 1030. Students are

often encouraged to use the fraction 22/7 as an approx-

imate value for π.

Mathematicians use the notation “≈” to denote

approximately equal to. Thus, for example, π≈22/7.

Physicists and engineers often approximate func-

tions by their T

AYLOR SERIES

with the higher-order

terms dropped. For example, , at

least for small values of x. The theory of

INTEGRAL

CALCULUS

begins by approximating areas under curves

as sums of areas of rectangles.

See also

ERROR

;

FACTORIAL

;

NUMERICAL DIFFEREN

-

TIATION

;

NUMERICAL INTEGRATION

.

Arabic mathematics Mathematical historians of

today are grateful to the Arabic scholars of the past for

preserving, translating, and honoring the great Indian,

Greek, and Islamic mathematical works of the scholars

before them, and for their own significant contributions

to the development of mathematics. At the end of the

eighth century, with the great Library of Alexandria

destroyed, Caliph al-Ma’mun set up a House of Wis-

dom in Baghdad, Iraq, which became the next promi-

nent center of learning and research, as well as the

repository of important academic texts. Many scholars

were employed by the caliph to translate the mathemat-

ical works of the past and develop further the ideas they

contained. As the Islamic empire grew over the follow-

ing seven centuries, the culture of intellectual pursuit

also spread. Many scholars of 12th-century Europe, and

later, visited the Islamic libraries of Spain to read the

texts of the Arabic academics and to learn of the

advances that had occurred in the East during the dark

ages of the West. A significant amount of mathematical

material was transmitted to Europe via these means.

One of the first Greek texts to be translated at the

House of Wisdom was E

UCLID

’s famous treatise, T

HE

E

LEMENTS

. This work made a tremendous impact on

the Arab scholars of the period, and many of them,

when conducting their own research, formulated theo-

rems and proved results precisely in the style of Euclid.

Members of the House of Wisdom also translated the

works of A

RCHIMEDES OF

S

YRACUSE

, D

IOPHANTUS OF

A

LEXANDRIA

, M

ENELAUS OF

A

LEXANDRIA

, and others,

and so they were certainly familiar with all the great

Greek advances in the topics of

GEOMETRY

,

NUMBER

THEORY

, mechanics, and analysis. They also translated

the works of Indian scholars, –

A

RYABHATA

and

B

H

–

ASKARA

, for instance, and were familiar with the the-

ory of

TRIGONOMETRY

, methods in astronomy, and fur-

ther topics in geometry and number theory. Any Arab

scholar who visited the House of Wisdom had, essen-

tially, the entire bulk of human mathematical knowledge

available to him in his own language.

Arab mathematician M

UHAMMAD IBN

M–

US

–

A

AL

-K

HW

–

ARIZM

–

ı(ca. 800) wrote a number of original

texts that were enormously influential. His first piece

simply described the decimal place-value system he had

learned from Indian sources. Three hundred years later,

when translated into Latin, this work became the pri-

mary source for Europeans who wanted to learn the

new system for writing and manipulating numbers. But

more important was al-Khw–

arizm

–

ı ’s piece Hisab al-jabr

w’al-muq¯abala (Calculation by restoration and reduc-

tion), from which the topic of “algebra” (“al-jabr”)

arose. Al-Khw–

arizm

–

ı was fortunate to have all sources

of mathematical knowledge available to him. He began

to see that the then-disparate notions of “number” and

“geometric magnitude” could be unified as one whole

by developing the concept of algebraic objects. This rep-

resented a significant departure from Greek thinking, in

which mathematics is synonymous with geometry.

Al-Khw–

arizm

–

ı’s insight provided a means to study both

arithmetic and geometry under a single framework, and