Posted 6 November 2008
RATIONAL NUMBERS
Introduction
Definition: A rational
number is a number that can be represented as a fraction ,
where m and n are integers and
.
Examples
¨
The numbers 3/4 and 11/5
are rational.
¨ 6 is rational because 6 = 6/1.
¨
0.33 is rational because .
(See Approximations).
Theorem
Any integer is rational. Proof:
The integer n is the same as the fraction .
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 ,
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
For
example if you see the expression you cannot conclude that it denotes a rational
number, because
and e
are not integers. (I do not know if
is rational.)
Properties of the fraction representation
Non-uniqueness
The representation of a rational number as a fraction is not unique. For example,
Two
representations and
give the same rational number if
and only if
.
Example
because
.
Definition: lowest terms
Let
be the representation of a rational number with
and
.
The representation is in lowest terms (or is reduced) if there is no integer
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
¨
74/111 is
not in lowest terms because 74 and
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 and
,
we have
The expressions ,
and
denote rational numbers because integers are
closed under addition and multiplication, so that ac, bd, ad, bc and ad+bc are integers.
However, if “ ”
and “
” are in lowest terms, “
” and “
” may nevertheless not be in lowest terms. For example, the formulas give
and
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.
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 is a rational number between them. This number
is the average (or mean)
of r and s, so it makes sense that it is 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 .
Proof. Let and
,
where a, b, c and d are integers. Then by
AMD,
which
is rational because and
are integers.
I
recommend that you check using AMD
that if and
,
then
.
Warning: is
not the only rational number between r and s. In fact, between any two distinct
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.