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 numbers whose decimal expansions repeat.
Thus any number with a nonrepeating decimal expansion 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 subset” of the real NUMBER LINE. Moreover, between any
two real numbers x and y on the number line, with x <
y, there exists a rational q that lies between them.
(Write x and y as decimal expansions and choose a terminating 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 GEORG CANTOR
(1845–1918) showed with his famous DIAGONAL ARGUMENT that it is possible to arrange all the positive (and
negative) rationals in a list: q 1 ,q 2 ,q 3 ,… This shows:
The set of rational numbers is COUNTABLE.
With the aid of the GEOMETRIC SERIES 1 + x + x
2
+ x
3 +… =
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 q 1 on
the number line with an interval of length x,
the point q 2 with an interval of length x
2 =
0.01, the point q 3 with an interval of length
x
3 = 0.001, and so forth. These intervals likely
overlap, but they occupy no more than:
amount of space in total. Thus the set of rational 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 NUMBERS; 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. Unfortunately, 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. Alternatively, one can define a real number to be any number
that can be expressed as a decimal expansion. (The rational numbers are then those that have repeating expansions 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
x x
x
x
x
+
+
+ = ⋅ −
=
2
3
1
1
1
9
L
1
––
1 – x
a
b
c
d ad bc
bd
+
=
+
2
2
c
–
b
a
–
b
real numbers 441
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 numbers whose decimal expansions repeat.
Thus any number with a nonrepeating decimal expansion 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 subset” of the real NUMBER LINE. Moreover, between any
two real numbers x and y on the number line, with x <
y, there exists a rational q that lies between them.
(Write x and y as decimal expansions and choose a terminating 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 GEORG CANTOR
(1845–1918) showed with his famous DIAGONAL ARGUMENT that it is possible to arrange all the positive (and
negative) rationals in a list: q 1 ,q 2 ,q 3 ,… This shows:
The set of rational numbers is COUNTABLE.
With the aid of the GEOMETRIC SERIES 1 + x + x
2
+ x
3 +… =
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 q 1 on
the number line with an interval of length x,
the point q 2 with an interval of length x
2 =
0.01, the point q 3 with an interval of length
x
3 = 0.001, and so forth. These intervals likely
overlap, but they occupy no more than:
amount of space in total. Thus the set of rational 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 NUMBERS; 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. Unfortunately, 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. Alternatively, one can define a real number to be any number
that can be expressed as a decimal expansion. (The rational numbers are then those that have repeating expansions 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
x x
x
x
x
+
+
+ = ⋅ −
=
2
3
1
1
1
9
L
1
––
1 – x
a
b
c
d ad bc
bd
+
=
+
2
2
c
–
b
a
–
b
real numbers 441