Extending this idea to three-dimensions, points in
space can be specified by a triple of numbers (x,y,z)
representing the distances along three mutually perpen-
dicular number lines. The coordinate axes are usually
called the x-, y-, and z-axes. They intersect at a point
O, called the origin, which is zero on all three number
lines. The axes could be oriented to either form a left-
handed or a right-handed system.
Coordinate Geometry
The advent of a coordinate system allowed mathemati-
cians, for the first time, to bring the power of algebra
to the study of geometry. For example, straight lines
are represented as sets of points (x,y) that satisfy equa-
tions of the form y= mx + b. Multiplying the
SLOPE
m
of one line with the slope of another quickly ascertains
whether or not those two lines are perpendicular, for
example. (The product of the slopes of two perpendicu-
lar lines is –1.)
French mathematician N
ICOLE
O
RESME
(1323–82)
was the first to describe a way of graphing the relation-
ship between an independent variable and a dependent
one, and thus the first to make steps toward uniting
geometry and algebra. The explicit construction of
what we would call a coordinate system first appeared
with the work of French lawyer and amateur mathe-
matician P
IERRE DE
F
ERMAT
(1601–65). Starting with
some horizontal reference line to represent an indepen-
dent variable x, Fermat would graphically depict the
relationship of a second variable yto it as a line seg-
ment, held at a fixed angle to the reference line, whose
length would vary according to the variable yas it
slides along the x-axis. Fermat did not think in terms,
however, of identifying a second axis, nor did he
require the line segment representing yto be perpendic-
ular to the x-axis.
In his famous 1637 text La géométrie (Geometry),
René Descartes independently described similar meth-
ods for representing algebraic relationships graphically.
Because the work of Fermat was not published until
after his death, the discovery of coordinate geometry
was attributed to Descartes.
Because Fermat and Descartes interpreted the
unknown variable yin an algebraic relationship as a
physical length, both scholars only ever considered posi-
tive coordinates. English mathematician J
OHN
W
ALLIS
(1616–1703) was the first to introduce the possibility of
negative coordinates. The idea of setting a fixed second
axis, the y-axis, perpendicular to the x-axis was not
popular until the mid 1700s. It was an idea that seemed
to evolve gradually. S
IR
I
SAAC
N
EWTON
(1642–1727) is
considered the originator of
POLAR COORDINATES
.
See also
COORDINATES
;
GRAPH OF A FUNCTION
.
Cartesian product (cross product, external direct
product, product set, set direct product) Given two
sets Aand B, their Cartesian product, denoted A×B, is
the set of all ordered pairs (a,b),where a∈Aand b∈
B. For example, if A= {1,2,3} and B= {α,β}, then:
A×B= { (1,α), (2,α), (3,α), (1,β), (2,β), (3,β) }
This is different from the set B×A.
If sets Aand Bare both finite, with nand mele-
ments, respectively, then A×Bis a finite set with nm
elements. German mathematician G
EORG
C
ANTOR
(1845–1918) showed that if Aand Bare both infinite
COUNTABLE
sets, then their Cartesian product A×Bis
again countable.
The Cartesian product of three sets A, B, and C,
denoted A×B×C, is defined as the set of all ordered
triples (a,b,c),with a∈A, b ∈B, and c∈C. The
Cartesian product of any finite collection of sets is
defined similarly. Any
SEQUENCE
can be thought of as
an element of the Cartesian product of a countable
number of sets.
If two sets Aand Bhave a particular structure
(they might both be
GROUP
s or
VECTOR SPACE
s, for
instance), then it is usually possible to give the Carte-
sian product A×Bthe same structure. For example, if
Aand Bare groups with group operations *and •,
respectively, then A×Bhas the structure of a group
with group operation given by:
(a1, b1) · (a2, b2) = (a1*a2, b1• b2)
The Klein four-group is the Cartesian product of the two-
element group Z2= {0,1} with itself. (The group opera-
tion for Z2is addition in mod 2
MODULAR ARITHMETIC
.)
See also
SET THEORY
.
casting out nines The
DIVISIBILITY RULES
show that
the remainder of any number, when divided by 9, is the
sum of its digits. For example, 59,432,641 leaves a
casting out nines 63