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Introduction
1
Representations of rational numbers
Closure properties of rationals
2
Images and metaphors for the rationals
Density of the rationals
3
Posted 6 February 2016
RATIONAL NUMBERS
Introduction
Definition: A rational number is a number that can be represented as a fraction m / n , where m
and n are integers and n ¹ 0 .
Examples
 The numbers 3/4 and –11/5 are rational.
 6 is rational because 6 = 6/1.
 0.33 is rational because 0.33 = 33 /100 . (See Approximations).
Theorem
Any integer is rational. Proof: The integer n is the same as the fraction n /1.
Terminology
Rational numbers may be referred to as “rationals”. The name comes from the fact that they
represent ratios and is not related to the meaning “able to reason” or “sane”.
Representations of rational numbers
Rational numbers have two familiar representations, as fractions and as decimals. Decimals are
discussed in the section on real numbers.
The definition of rational number says that it must be a number that can be represented as a
fraction of integers.
“Can be represented” does not mean “is represented”!
The number 0.25 is a rational number because it can be represented as 1/ 4 , but the expression
“0.25” is not itself a fraction representation.
An expression a / b does not automatically denote a rational number.
You must check that a and b are integers
p
you cannot conclude that it denotes a rational number,
e
p
because p and e are not integers. (I do not know if is rational.)
e
For example if you see the expression
1
3
Properties of the fraction representation
Non-uniqueness
The representation of a rational number as a fraction is not unique. For example,
3 6
- 9
= =
4 8 - 12
Two representations m / n and r / s give the same rational number if and only if ms = nr .
Example
3 / 4 = 6 / 8 because 3 ×8 = 6 ×4 .
Definition: lowest terms
Let m / n be the representation of a rational number with m ¹ 0 and n > 0 . The representation is
in lowest terms (or is reduced) if there is no integer d > 1 for which d divides m and d divides n. (See
parenthetic assertion.)
Usage
The symbols 3/4 and 6/8 are two representations of the same number. One of the
representations, 3/4, is in lowest terms and the other is not. So when someone says “3/4 is in lowest
terms”, the symbol “3/4” refers to the representation, not the rational number. See context sensitive
interpretation.
Examples
 3/4 is in lowest terms but 6/8 is not, because 6 and 8 have 2 as a common divisor.
 74/111 is not in lowest terms because 74 and 111 have 37 as a common divisor. In fact, 74/111 =
2/3.
To calculate the lowest terms representation you divide the numerator and denominator by the
largest integer that divides both of them. For example, the largest number dividing both 74 and 111 is
37. 74/37 = 2 and 111/37 = 3. So 74/111 = 2/3 and “2/3” is in lowest terms.
Theorem
Every nonzero rational number has a representation in lowest terms.
A proof of this will appear in the number theory section if I ever get around to writing it.
Closure properties of rationals
Rational numbers are closed under addition, subtraction, multiplication, as well as division
by a nonzero rational.
These operations are carried out according to the familiar rules for operating with fractions. Thus for
rational numbers a / b and c / d , we have
a c ac
´ =
b d bd
,
a c ad + bc
a c ad
and ¸
+ =
=
b d
bd
b d bc
[AMD]
The expressions ac / bd , (ad + bc ) / bd and ad / bc denote rational numbers because integers are
closed under addition and multiplication, so that ac, bd, ad, bc and ad+bc are integers.
However, if “ a / b ” and “ c / d ” are in lowest terms, “ ac / bd ” and “ (ad + bc ) / bd ” may
nevertheless not be in lowest terms. For example, the formulas give
1 1
8
2 3
6
and ´
+ =
=
4 4 16
3 4 12
In other words, “being in lowest terms” is not closed under addition and multiplication.
Images and metaphors for the rationals
Ratios
Suppose we have a line segment L that is 5/8 of a unit long. Then we can take a line segment M
that is 1 unit long and divide it into exactly 8 segments of 1/8 unit each, and we can divide L into
exactly 5 segments of the same length. That is because the ratio of L to M is 5:8.
Remarkably, we cannot take a
line of length 1 and a line of length
2 and divide them into whole
numbers of segments of the same
length, no matter how small we
make the segments. This means
that 2 is not rational. This is
discussed in the section on
irrational numbers.
This is the sense in which rational numbers represent ratios of
integers.
Density of the rationals
The integers can be thought of as beads or points in a row going to infinity in both directions. The
rationals go to infinity in both directions, too, but:
The rationals must not be thought of as a row of points.
That is because between any two rational numbers there is another one.
In general, if r and s are any distinct rational numbers, then
between them. This number
r+s
is a rational number
2
r+s
is the average (or mean) of r and s, so it makes sense that it is
2
between them. The preceding sentence is an example of using the rich view to see why something is
true. Here is a rigorous proof:
Theorem. Let r and s be distinct rational numbers. Assume WLOG that r < s. Then r <
Proof. Let r =
a
c
and s = , where a, b, c and d are integers. Then by AMD,
b
d
r + s 1æ
a c ö 1 çæad + bc ö
ad + bc
÷
= çç + ÷
= ç
=
÷
÷
÷
÷
ç
ç
è
è
ø
ø
2
2 b d
2
bd
2bd
r+s
< s.
2
which is rational because ad + bc and 2bd are integers.
I recommend that you check using AMD that if r =
Warning:
5
1
r + s 11
and s =
, then
.
=
2
24
12
2
r+s
is not the only rational number between r and s. In fact, between any two distinct
2
rational numbers there are infinitely many other rational numbers. This means that if you are given
a rational number r, there is no “next largest” rational number (or next smallest, either).
These properties are discussed for all real numbers here.