raincoat” is likely to be dependent on the event “it is

raining.”

CONDITIONAL PROBABILITY

is used to analyze

dependent events.

Experiments that involve drawing objects from a

finite source, such as balls from a bag, or cards from a

deck, without replacing the objects withdrawn, often

yield dependent events. For example, the events “the

first card drawn from a deck of cards is red” and “the

second card drawn is red” are dependent—the proba-

bility that the second card selected is the desired color

could either be 26/51 or 25/51 , depending on whether

the first card drawn is black or red.

A careful study of

PROBABILITY

shows that two inde-

pendent events Aand Bsatisfy the probability relation:

P(Aand B) = P(A) ×P(B)

Mathematicians take this relation as the definition of

two events Aand Bbeing independent.

indeterminate equation An equation, or a

SYSTEM

OF EQUATIONS

, with an infinite number of solutions is

called indeterminate. For instance, the equation:

x+ 2y= 5

is indeterminate (for each real number tthere is the

solution x= 5 – 2tand y= t), as is the system of two

equations:

a+ b+ c= 1

2a+ 3b– c= 0

in variables a, b, and c. (All triples of the form (a,b,c) =

(t, , ) are solutions.)

The single equation x2+ y2+ z2+ w2= 0 is not

indeterminate: it has the unique (real) solution x= y= z

= w= 0.

See also

DEGREES OF FREEDOM

.

index (plural, indices) A number that indicates a

place or describes the characteristic or the nature of a

function is called an index. For example, a

SEQUENCE

of numbers might be denoted {an}, with the index n

denoting the place in the sequence. (Thus a7, for

instance, would represent the seventh number in the

list.) An

EXPONENT

is sometimes called an index. For

example, one might say that the function f(x) = x5has

index 5 to indicate that quantities are being raised to

the fifth power. The radical function g(x) = 3

√

–

x, for

example, has index 3, and the logarithmic function

h(x) = log10 xhas index 10.

In statistics and business, an index is a figure used

to show the variation in some quantity over a period of

time, usually standardized relative to some base value.

For example, a retail price index is used to measure

changes in the cost of household items. For the base

year, the index is usually set at 100. If, for instance, the

cost of dish detergent in the base year is $1.60, and the

price rises to $1.76 and to $2.00 in the two subsequent

years, then the price index for this item would be 100,

110, and 125 in each of those years, respectively.

It is possible to design an index that gives a general

measure of the value of more than one commodity

simultaneously. For example, the Dow Jones index of

the New York Stock Exchange is an indicator of the

worth of a representative selection of industrial, trans-

portation, and utility stocks.

Indian mathematics The entire course of Western

mathematics was profoundly affected by a single Indian

invention, that of the place-value decimal system. That

every possible number can be expressed via a set of just

10 symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, by making

careful use of the place of each symbol, seems such a

simple notion nowadays that it is hard to appreciate its

profound importance. This elegant notation system

provided the means for Indian scholars to perform

complicated arithmetical computations with relative

ease, which in turn led to significant developments in

numerical techniques, approximation methods, and the

theory of arithmetic. Only when other cultures adopted

the place-value decimal system from India could they

accomplish the same mathematical feats that this cul-

ture had already developed.

The earliest dated evidence of mathematical activ-

ity in the Indian subcontinent goes back to the Indus

civilization of 2500

B

.

C

.

E

. Bronze weights and graded

rods (rulers) from the period show that these people

were already working with a decimal system. The Indus

people worked with a basic unit of length 1.32 in. long

(today called the “Indus inch”), 10 of which make their

version of a “foot.” Excavations show that the weights

and graded rods were used extensively in construction.

2 – t

—

—

3

1 – 4t

——–

3

Indian mathematics 263