as advancement of the field of partial differential equa-
tions. In mechanics, he is honored with a principle of
motion named after him, a generalization of Sir Isaac
Newton’s third law of motion.
See also
DIFFERENTIAL CALCULUS
.
algebra The branch of mathematics concerned with
the general properties of numbers, and generalizations
arising from those properties, is called algebra. Often
symbols are used to represent generic numbers,
thereby distinguishing the topic from the study of
ARITHMETIC
. For instance, the equation 2 ×(5 + 7) =
2 ×5 + 2 ×7 is a (true) arithmetical statement about a
specific set of numbers, whereas, the equation x×
(y+z) = x×y+ x×zis a general statement describing
a property satisfied by any three numbers. It is a state-
ment in algebra.
Much of elementary algebra consists of methods of
manipulating equations to either put them in a more
convenient form, or to determine (that is, solve for)
permissible values of the variables that appear. For
instance, rewriting x2+ 6x+ 9 = 25 as (x+ 3)2= 25
allows an easy solution for x: either x+ 3 = 5, yielding
x= 2, or x+ 3 = –5, yielding x= –8.
The word algebra comes from the Arabic term
al-jabr w’al-muqa¯bala (meaning “restoration and
reduction”) used by the great M
UHAMMAD IBN
M–
US
–
A
AL
-K
HW
–
ARIZM
–
I
(ca. 780–850) in his writings on the
topic.
algebra 9
History of Equations and Algebra
Finding solutions to equations is a pursuit that dates back to
the ancient Egyptians and Babylonians and can be traced
through the early Greeks’ mathematics. The R
HIND PAPYRUS
,
dating from around 1650
B
.
C
.
E
., for instance, contains a
problem reading:
A quantity; its fourth is added to it. It becomes
fifteen. What is the quantity?
Readers are advised to solve problems like these by a
method of “false position,” where one guesses (posits) a
solution, likely to be wrong, and adjusts the guess accord-
ing to the result obtained. In this example, to make the divi-
sion straightforward, one might guess that the quantity is
four. Taking 4 and adding to it its fourth gives, however, only
4 + 1 = 5, one-third of the desired answer of 15. Multiplying
the guess by a factor of three gives the solution to the prob-
lem, namely 4 ×3, which is 12.
Although the method of false position works only for
LINEAR EQUATION
s of the form ax = b, it can nonetheless be an
effective tool. In fact, several of the problems presented in
the Rhind papyrus are quite complicated and are solved rel-
atively swiftly via this technique.
Clay tablets dating back to 1700
B
.
C
.
E
. indicate that
Babylonian mathematicians were capable of solving certain
QUADRATIC
equations by the method of
COMPLETING THE
SQUARE
. They did not, however, have a general method of
solution and worked only with a set of specific examples
fully worked out. Any other problem that arose was
matched with a previously solved example, and its solution
was found by adjusting the numbers appropriately.
Much of the knowledge built up by the old civilizations
of Egypt and Babylonia was passed on to the Greeks. They
took matters in a different direction and began examining all
problems geometrically by interpreting numbers as lengths
of line segments and the products of two numbers as areas
of rectangular regions. Followers of P
YTHAGORAS
from the
period 540 to 250
B
.
C
.
E
., for instance, gave geometric proofs
of the
DISTRIBUTIVE PROPERTY
and the
DIFFERENCE OF TWO
SQUARES
formula, for example, in much the same geometric
way we use today to explain the method of
EXPANDING BRACK
-
ETS
. The Greeks had considerable trouble solving
CUBIC EQUA
-
TION
s, however, since their practice of treating problems
geometrically led to complicated three-dimensional con-
structions for coping with the product of three quantities.
At this point, no symbols were used in algebraic prob-
lems, and all questions and solutions were written out in
words (and illustrated in diagrams). However, in the third
century, D
IOPHANTUS OF
A
LEXANDRIA
introduced the idea of
abbreviating the statement of an equation by replacing fre-
quently used quantities and operations with symbols as a
kind of shorthand. This new focus on symbols had the sub-
tle effect of turning Greek thinking away from geometry.
Unfortunately, the idea of actually using the symbols to
solve equations was ignored until the 16th century.
The Babylonian and Greek schools of thought also influ-
enced the development of mathematics in ancient India. The
scholar B
RAHMAGUPTA
(ca. 598–665) gave solutions to
(continues)