rational function 439

p

8

–

40

p(x)

––

q(x)

of each entrant and then comparing the result with the

maximum sum possible.

To illustrate: in the above example, apple received

ranks of 5 and 4. The rank difference squared is (5 – 4)2

= 12= 1. Similarly, the rank difference squared for blue-

berry is (2 – 1)2= 1, and for the remaining pies, in

alphabetical order, (3 – 2)2= 1, (1 – 3)2= 4, and (4 – 5)2

= 1. Summing all five rank differences squared yields the

total score: D2= 1 + 1 + 1 + 4 + 1 = 8.

With nentrants, it can be shown that the maximal

sum possible is . It occurs when the ranks

assigned by one judge are in reverse order to the ranks

assigned by the second. (To see this, observe how the

quantity D2changes when the ranks of two entrants

are switched. This will show that the quantity D2is

largest when, for any pair of entrants, their assigned

ranks are in reverse order.) For n= 5, the maximum

sum possible is 40.

Spearman’s rank correlation coefficient is given by:

Notice that a value ρ= 1 occurs when D2= 0, that is,

there is perfect agreement. A value ρ= –1 occurs when

D2has the maximum value, i.e., perfect disagreement.

In our example, we have ρ= 1 – 2 · = 0.6. This value

again indicates some disagreement and an inconsis-

tency in the two ranking schemes.

rate of change See

DIFFERENTIAL CALCULUS

.

ratio The relationship of one numerical quantity to

another expressed as a

QUOTIENT

so as to indicate their

relative sizes is called a ratio. We write a:b for the ratio

of a quantity ato a quantity b, or, alternatively, express

the ratio as a

FRACTION

a/b. The first term of a ratio

a:b is called the antecedent and the second term the

consequent. For example, the ratio of 10 to 4 is written

10:4 or 10/4, which can be simplified to 5/2. (Thus the

number 10 is 2 1/2 times as large as 4.) Here, 10 is the

antecedent, and 4 is the consequent.

The value of a ratio is unchanged if both terms in

the ratio are multiplied or divided by the same quan-

tity. For example, 10:4, 30:12, and 2.5:1, are equiva-

lent ratios. A ratio for which one of its terms is equal

to 1 is called a unitary ratio. Thus, 2.5:1, for instance,

is a unitary ratio.

If the ratio of two variables yand xis constant,

then the two variables are said to be

PROPORTIONAL

.

For example, if y/x has constant value k, then it must

be the case that y= kx.

In

PROBABILITY

theory, the

ODDS

of a game are

expressed as a ratio. For example, if in a game of

chance aof the possible outcomes are deemed favor-

able and the remaining bpossible outcomes unfavor-

able, then the odds in favor for the game is the ratio

a:b. Thus for a game with favorable odds of 3:2, for

instance, three of the five outcomes are deemed favor-

able, and there is 3/5, or 60 percent, chance of win-

ning the game.

A number that can be expressed as a ratio of two

integers is called a

RATIONAL NUMBER

. Not all numbers

are rational. The

FUNDAMENTAL THEOREM OF ARITH

-

METIC

establishes, for example, that √

–

2 is not rational.

The

RATIO TEST

is often used to determine the conver-

gence of an infinite

SERIES

.

In the theory of

VECTOR

s, the ratio theorem (also

known as the section formula) states the following:

If a point Pdivides a line segment AB in the

ratio a:b, then the position vector pof Pis

given in terms of the position vectors aand b

of Aand B, respectively, by the formula:

The ratio notation for numbers can be extended to

indicate the relative sizes of more than two quantities.

For instance, the ratio a:b:c states that the ratio of the

first quantity to the second is a:b, that the ratio of the

second quantity to the third is b:c, and that the ratio of

the first quantity to the third is a:c. For example, the

numbers 40, 80, and 200 are in the ratio 1:2:5.

See also

GOLDEN RATIO

.

rational function (rational expression) Any function

f(x) = given as a

RATIO

of two polynomials p(x) and

q(x) is called a rational function. Such a function is

defined at all values xfor which q(x) ≠0. For example,

is a rational function with a domain