344 multiplication principle

COMMUTATIVE PROPERTY

: a×b= b×afor all count-

ing numbers aand b.

If a number ais multiplied by a number bto form

a product a×b, then the first number ais called a

multiplicand and the second number ba multiplier.

(Of course, the commutative property of multiplica-

tion obviates the need to distinguish the multiplicand

from the multiplier.)

The symbol ×for multiplication was used in

W

ILLIAM

O

UGHTRED

’

S

(1574–1660) 1631 work Clavis

mathematicae (The key to mathematics), but historians

suspect that the symbol was in use up to 100 years ear-

lier. Mathematicians today also use a raised dot to indi-

cate multiplication (4 · 3 = 12, for instance), or they

simply write symbols side by side if variables are being

used (x×y= xy or 2 ×w= 2w, for example).

There are a number of methods for computing the

product of two large numbers, such as E

LIZABETHAN

MULTIPLICATION

, E

GYPTIAN MULTIPLICATION

, and R

US

-

SIAN MULTIPLICATION

.

The process of multiplication can be extended to

NEGATIVE NUMBERS

(yielding the necessary consequence

that the product of two negative quantities is positive),

FRACTION

s,

REAL NUMBERS

,

COMPLEX NUMBERS

, and

MATRIX

es. Two

VECTOR

s can be multiplied by a

DOT

PRODUCT

or a

CROSS PRODUCT

. The product of two sets

is called a C

ARTESIAN PRODUCT

.

The number 1 is a multiplicative

IDENTITY ELE

-

MENT

in the theory of arithmetic. We have that a×1 =

a= 1 ×afor any number a.

The product of two real-valued functions fand g

is the function f· g, whose value at any input xis the

product of the outputs of fand gat that input value:

(f· g)(x) = f(x) · g(x).For example, if f(x) = x2+ 2x

and g(x) = 5x+ 7, then (f· g)(x) = (x2+ 2x)(5x+ 7) =

5x3+ 17x2+ 14x.

The product formulae in

TRIGONOMETRY

assert:

See also

ASSOCIATIVE

;

DISTRIBUTIVE PROPERTY

;

INFI

-

NITE PRODUCT

.

multiplication principle (fundamental principle of

counting) Suppose that a task can be broken up into

two steps. If the first step can be done in one of aways,

and the second in bdifferent ways (regardless of the

result of the first step), then the multiplication principle

says that the original task can be done in a×bways.

As an example, imagine that five roads connect

town A to town B, and seven roads connect town B to

town C. Then one has 5 ×7 = 35 alternatives for driv-

ing from A to C. When rolling a die and tossing a coin,

6 ×2 = 12 different outcomes are possible.

The multiplication principle extends to tasks that

are composed of more than two steps. For example,

with three different sets of shoes, four different trousers,

and three different shirts, one has 3 ×4 ×3 = 36 outfits

to wear. There are 10 ×10 ×10 ×26 ×26 ×26 =

17,256,000 different license plate numbers composed of

three single-digit numbers followed by three letters.

See also

FACTORIAL

;

PERMUTATION

.

mutually exclusive events (disjoint events) Two

EVENT

s

are mutually exclusive if they cannot both occur in a sin-

gle run of an experiment. For example, in tossing a die,

the events “rolling a 3” and “rolling an even number”

are mutually exclusive, whereas, the events “rolling a

multiple of 3” and “rolling an even number” are not.

If Aand Bare two mutually exclusive events for an

experiment, then the probability that either one event

or the other occurs is given by the addition law:

P(A∪B) = P(A) + P(B)

See also

PROBABILITY

.