c

–

b

a

–

b

real numbers 441

1

––

1 – x

in the process of long division can only be one of seven

numbers—0, 1, 2, 3, 4, 5, 6—and so, eventually, some

remainder must appear twice. As soon as this occurs, the

process of long division cycles.) This establishes:

The rational numbers are precisely those num-

bers whose decimal expansions repeat.

Thus any number with a nonrepeating decimal expan-

sion must be an

IRRATIONAL NUMBER

. In a similar

manner, one can show that the rationals are precisely

those numbers with terminating

CONTINUED FRACTION

expansions.

Between any two rational numbers and lies

another rational—their midpoint , for

instance. Repeating this reasoning establishes:

Infinitely many rational numbers lie between

any two given rationals.

We say that the set of rationals represents a “dense sub-

set” of the real

NUMBER LINE

. Moreover, between any

two real numbers xand yon the number line, with x<

y, there exists a rational qthat lies between them.

(Write xand yas decimal expansions and choose a ter-

minating decimal value that lies between them.)

These density arguments show that there is no

smallest positive rational number, that is, there is no

“first” rational number on the positive number line.

Nonetheless, German mathematician G

EORG

C

ANTOR

(1845–1918) showed with his famous

DIAGONAL ARGU

-

MENT

that it is possible to arrange all the positive (and

negative) rationals in a list: q1,q2,q3,… This shows:

The set of rational numbers is

COUNTABLE

.

With the aid of the

GEOMETRIC SERIES

1 + x+ x2

+x3+… = we can now show that the amount of

space occupied by the rationals on the number line is

zero. (This is surprising given that the rationals form a

dense subset of the line.)

Choose x= 1/10 and cover the point q1on

the number line with an interval of length x,

the point q2with an interval of length x2=

0.01, the point q3with an interval of length

x3= 0.001, and so forth. These intervals likely

overlap, but they occupy no more than:

amount of space in total. Thus the set of ratio-

nal points occupies at most one-ninth of the real

number line.

In the same way, by working with x=

1/100, we deduce that the rationals occupy at

most 1/99 of the space on the number line, or

at most 1/999 of the space if we work with x=

1/1000. Continuing in this manner, we are

forced to conclude that the total amount of

space occupied by the rational numbers is zero.

We say:

The rationals form a set with a measure of zero

on the real-number line.

Given that the number line is of infinite length, we must

conclude then that “most” numbers are not rational.

See also

ALGEBRAIC NUMBER

;

NUMBER

;

REAL NUM

-

BERS

;

TRANSCENDENTAL NUMBER

.

ratio test See

CONVERGENT SERIES

.

ray (half-line) The portion of a straight line starting

at one point and going on forever in one direction is

called a ray. Two rays in a plane starting at the same

point create a figure called an

ANGLE

.

real numbers It is extraordinarily difficult to define

precisely what is meant by a real number. Many standard

texts in mathematics define a real number to be any

RATIONAL NUMBER

or any

IRRATIONAL NUMBER

. Unfor-

tunately, since an irrational number is declared to be any

number that is not rational, it is not clear from which

pool of numbers the irrational numbers come. Alterna-

tively, one can define a real number to be any number

that can be expressed as a decimal expansion. (The ratio-

nal numbers are then those that have repeating expan-

sions and the irrationals are those that do not.) The

difficulty with this approach is that it is not clear how to

perform arithmetic operations on these entities. For

example, when adding two large integers, 769 + 845, say,

one begins with the rightmost digits, adds them, carries a