88 compound interest

which is a different function. As another example, if M

is the function that assigns to each person of the world

his or her biological mother, and Fis the analogous

biological father function, then (M°F)(John) represents

John’s paternal grandmother, whereas (F°M)(John) is

John’s maternal grandfather.

The notation g°fis a little confusing, for it needs to

be read backwards. The function fis called the “core

function” and needs to be applied first, with the

“external function” gapplied second. The composition

of three functions f, g, and his written h°f°g(here his

the external function), and the repeated composition of

a function fwith itself is written f(n). Thus, for exam-

ple, f(4) denotes the composition f°f°f°f. A set of

repeated compositions is called a

DYNAMICAL SYSTEM

.

Mathematicians have shown that the composition

of two

CONTINUOUS FUNCTION

s is itself continuous.

Precisely, if fis continuous at x= a, and gis continuous

at x= f(a),then g°fis continuous x= a.

The composition of two differentiable functions is

differentiable. The

CHAIN RULE

shows that the

DERIVA

-

TIVE

of g°fis given by (g°f)′(x) = g′(f(x))·f′(x).

compound interest See

INTEREST

.

compound statement See

TRUTH TABLE

.

computer An electronic device for automatically

performing either arithmetic operations on

DATA

or

sequences of manipulations on sets of symbols (as

required for

ALGEBRA

and

SET THEORY

, for instance),

all according to a precise set of predetermined

instructions, is called a computer. The most widely

used and versatile computer used today is the digital

computer in which data are represented as sequences

of discrete electronic pulses. As each pulse could

either be “on” or “off,” it is natural to think of

sequences of 0s and 1s in working in computer theory

and, consequently, to work with the system of

BINARY

NUMBERS

to represent data.

A digital computer has a number of separate parts:

1. An input device, such as a keyboard, for entering a

set of instructions (program) and data.

2. A central processing unit (CPU) that codes informa-

tion into binary form and carries out the instruc-

tions. (This unit consists of a series of electronic

circuit boards on which are embedded a large num-

ber of “logic gates,” akin to the

CONJUNCTION

and

DISJUNCTION

configurations.)

3. Memory units, such as disks and magnetic tape.

4. An output device for displaying results, such as a

monitor or a printer.

The study of computer science typically lends itself

to the theoretical capabilities of computing machines

defined in terms of their programs, not the physical

properties of actual computers. The

HALTING PROBLEM

and the question of being

NP COMPLETE

, for instance,

are issues of concern to scientists in this field.

See also

ABACUS

; C

HARLES

B

ABBAGE

;

DIGIT

.

concave/convex A curve or surface that curves

inward, like the circumference of a circle viewed from

the interior, or the hollow of a bowl, for example, is

called concave. A curve or surface that curves outward,

such as the boundary of a circle viewed from outside

the circle, or the surface of a sphere, is called convex.

A geometric shape in the plane or a three-

dimensional solid is called convex if the boundary of

the shape is a convex curve or surface. For example,

triangles, squares, and any regular

POLYGON

are con-

vex figures. Cubes and spheres are convex solids. Any

shape that is not convex is called concave. A deltoid

QUADRILATERAL

, for example is a concave polygon.

A polygon is convex if each of its interior angles

has value less than 180°. Equivalently, a polygon is

convex if the figure lies entirely on one side of any line

that contains a side of the polygon. A

POLYHEDRON

is

convex if it lies entirely on one side of any plane that

contains one of its faces.

A convex figure can also be characterized by the

property that, for any two points inside the figure, the

line segment connecting them also lies completely

within the figure.