Posted 15 April 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 and
are rational.
¨
6 is rational because .
¨
0.33 is rational because .
(See Approximations).
Fact: 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.