composition 87

and, in polar form, if z= reiθ, then –

z= re –iθ. Taking the

conjugate of a complex number has the geometric

effect of reflecting that number across the real axis.

History of Complex Numbers

The first European to make serious use of the square

root of negative quantities was G

IROLAMO

C

ARDANO

(1501–76) of Italy in the development of his solutions

to

CUBIC EQUATION

s. He noted that quantities that

arose in his work, such as an expression of the

form for instance, could be

manipulated algebraically to yield a real solution to

equations. (We have = 4.)

Nonetheless, he deemed such a manipulation only as a

convenient artifice with no significant practical mean-

ing. French philosopher R

ENÉ

D

ESCARTES

(1596–1650)

agreed and coined the term imaginary for roots of neg-

ative quantities.

During the 18th century, mathematicians continued

to work with imaginary roots, despite general skepti-

cism as to their meaning. Euler introduced the symbol i

for √

–

–1, and Argand and Wessel introduced their geo-

metric model for complex numbers, which was later

popularized by C

ARL

F

RIEDRICH

G

AUSS

(1777–1855).

His proof of the fundamental theorem of algebra con-

vinced mathematicians of the importance and validity

of the complex number system.

Irish mathematician S

IR

W

ILLIAM

R

OWAN

H

AMIL

-

TON

(1805–65) is credited as taking the final step to

demystify the meaning of the complex-number system.

He extended the notion of the complex numbers as

arising from 90°rotations by showing that any rota-

tion in three-dimensional space can naturally and easily

be represented in terms of complex numbers. He also

noted that the complex numbers are nothing more than

ordered pairs of numbers together with a means for

adding and multiplying them. (We have (a,b) + (c,d) =

(a+ c,b + d) and (a,b)·(c,d) = (ac – bd,ad + bc).) In

Hamilton’s work, the number ibecame nothing more

than the point (0,1).

See also

NEGATIVE NUMBERS

;

STEREOGRAPHIC

PROJECTION

.

composite Used in any context where it is possible to

speak of the multiplication of two quantities, the term

composite means “having proper factors.” For exam-

ple, the number 12, which equals 3 ×4, is a

COMPOSITE

NUMBER

, and y= x2+ 2x– 3 = (x–1)(x+ 3) is a com-

posite polynomial (not to be confused with the

COMPO

-

SITION

of two polynomials).

A quantity that is not composite is called irre-

ducible, or, in the context of number theory,

PRIME

.

composite number A whole number with more than

two positive factors is called a composite number. For

example, the number 12 has six positive factors, and so

is composite, but 7, with only two positive factors, is

not composite. The number 1, with only one positive

factor, also is not composite. Numbers larger than one

that are not composite are called

PRIME

.

The sequence 8, 9, 10 is the smallest set of three

consecutive composite numbers, and 24, 25, 26, 27, 28

is the smallest set of five consecutive composites. It is

always possible to find arbitrarily long strings of com-

posite numbers. For example, making use of the

FAC

-

TORIAL

function we see that the string

(n+ 1)! + 2, (n+ 1)! + 3,…,(n+ 1)! + (n+ 1)

represents nconsecutive integers, all of which are com-

posite. (This shows, for example, that there are arbi-

trarily large gaps in the list of prime numbers.)

See also

FACTOR

.

composition (function of a function) If the outputs

of one function fare valid inputs for a second function

g, then the composition of gwith f, denoted g°f, is the

function that takes an input xfor fand returns the out-

put of feeding f(x) into g:

(g°f)(x) = g(f(x))

For example, if feeding 3 into freturns 5, and feeding 5

into greturns 2, then (g°f)(3) = 2. If, alternatively,

f(x)= x2+ 1 and g(x) = 2 + , then

(g°f)(x) = g(f(x))

= g(x2+ 1)

= 2 +

Typically g°fis not the same as f°g. In our last example,

for instance,

1

x2+ 1

1

x