p(x)

–

q(x)

p(x)

–

1

440 rationalizing the denominator

p(x)

–

q(x)

of all real values except x= 2 and x= –2. Any single

polynomial p(x) can be regarded as the rational function

.

If, for a rational function f(x) = , the denom-

inator q(x) is zero at x= a, then the function either has

a vertical

ASYMPTOTE

at x= aor a

REMOVABLE DISCON

-

TINUITY

at x= a. In the latter case, it must be that the

numerator p(x) is also zero at x= a. For example, the

rational function , although

not defined at x= 3, equals the continuous function x–

2 at all values different from 3, and so has a removable

discontinuity at that point.

If a rational function f(x) = tends to a limit c

as x→∞or as x→–∞, then the line y= cis a hori-

zontal asymptote for f. For example, the line y= 2 is a

horizontal asymptote of . (Write =

. This approaches the value

as x→±∞.)

Every rational function can be rewritten in terms of

PARTIAL FRACTIONS

.

See also

CONTINUOUS FUNCTION

.

rationalizing the denominator The process of elim-

inating any square-root terms in the denominator of a

rational expression, without changing the value of that

expression, is called rationalizing the denominator. For

example, the quantity can be rationalized by

multiplying both the numerator and denominator by

√

–

2 to obtain: .

In general, an expression of the form

is rationalized by multiplying through by the conjugate,

√

–

a+ √

–

b, to produce:

During the 1950s and 1960s, before the advent of

pocket calculators, it was deemed necessary to always

rationalize the denominator of a numerical expression.

It has the advantage of making computations relatively

straightforward. For instance, although one can read

from a book of tables that √

–

2 is approximately

1.414, computing the approximate value of via the

process of

LONG DIVISION

is tedious. However, rewriting

as allows us to see immediately that this

quantity is simply half of √

–

2, and so has approximate

value 0.707.

Today, the quantity is regarded as a valid

expression of a numerical quantity, and there is abso-

lutely no need to rationalize the denominator. Unfortu-

nately, many high school mathematics programs still

insist on rewriting such expressions, even though the

reasons for doing so are now obsolete.

rational numbers (rationals) Any number that can

be expressed as a

RATIO

, a/b, of two integers aand b,

with b≠0, is called a rational number. For example,

2/5 and –6/2 (which is equivalent to –3) are rational

numbers. Every rational number is a fraction, and as

such, the rules for adding, subtracting, multiplying, and

dividing fractions apply to the rationals. The set of all

rational numbers is denoted Q(for quotient), and the

set of all rationals constitutes a mathematical

FIELD

.

Any real number whose decimal representation

eventually repeats in a cycle or terminates (and so can

be regarded as containing a repeated cycle of zeros) is a

rational. For example, multiplying x= 0.34 by 100

yields 100x= 34, and so x= = . Multiplying

y= 2.3181818… by 10 and by 1,000 and then sub-

tracting yields:

1000y= 2318.181818…

10y= 23.181818…

990y= 2295

establishing that y= 2295/990 = 51/22. Conversely, the

process of

LONG DIVISION

shows that every rational

number has a decimal expansion that repeats. (For

instance, in dividing 3 by 7, each remainder that appears

17

––

50

34

––

100