coordinates 105

corresponding series with all terms positive does not.

It is possible to multiply convergent series by con-

stants, and to add and subtract two convergent series.

Precisely, if and both converge, and kis a

number, then:

These properties can be used to evaluate new infinite

sums. For example, in 1740 L

EONHARD

E

ULER

showed

that a particular value of the

ZETA FUNCTION

is given

by . It then follows

that the alternating form of this series has value:

See also

ARITHMETIC SERIES

;

GEOMETRIC SERIES

;

POWER SERIES

.

converse (reverse implication) The converse of a

CONDITIONAL

statement “p implies q” is the statement:

“q implies p.” It is the statement obtained by reversing

the roles of the antecedent and consequent. The con-

verse of a conditional statement might, or might not, be

true. For example, the converse of the true statement,

“If a triangle has three equal sides, then it has three

equal angles,” is valid—a triangle with three equal

angles does indeed have three equal sides—whereas the

converse of the statement, “If n is divisible by 6, then n

is divisible by 2,” is false—an even number need not be

divisible by 6.

See also

ARGUMENT

;

CONTRAPOSITIVE

.

convex See

CONCAVE

/

CONVEX

.

coordinates A set of numbers used to locate a point

on a number line, in a plane, or in space are called the

coordinates of that point. For example, the coordinates

of points on a number line could be given by their dis-

tances from a fixed point O(called the origin), with

points on one specified side of Obeing deemed a posi-

tive distance from O, and the points on the opposite

side of Oa negative distance from O.

One way of assigning coordinates to points in the

plane is to establish a fixed point Oin the plane (again

called the origin), and two lines of reference (called

axes) that pass through O. Each axis is divided into a

positive side and a negative side by O. Given a point P

in the plane, one draws lines through Pparallel to each

of the axes. The distances along which these new lines

intersect the axes specify the location of the point P.

When the axes are drawn at right angles, the sys-

tem is called a Cartesian coordinate system, or a rect-

angular coordinate system. The axes are usually called

the x- and y-axes, and the pair of numbers (x,y) speci-

fying the location of a point P(as xunits along one

axis, and yunits along the second) are called the

C

ARTESIAN COORDINATES

of P. In three-dimensional

space, the location of points can be specified via three

mutually perpendicular (or oblique) axes passing

through a common point O.

The idea of assigning sets of numbers to points to

specify locations is on old one. By the third century

B

.

C

.

E

., Greek scholars A

POLLONIUS OF

P

ERGA

and

A

RCHIMEDES OF

S

YRACUSE

had used longitude, lati-

tude, and altitude to define the position of a point on

the Earth’s surface. Roman and Greek surveyors

labeled maps with grid lines, so as to specify locations

via row and column numbers.

See also

CYLINDRICAL COORDINATES

;

DIMENSION

;

E

ARTH

;

POLAR COORDINATES

;

RIGHT

-

HANDED

/

LEFT

-

HANDED SYSTEM

;

SPHERICAL COORDINATES

.