Definition of a Rational Number
The set of rational numbers is the set 
a and
b are integers,
Equality of Rational Numbers
Let 
and 
be any rational numbers. Then 
if and only if
ad = bc.
Simplest Form of a Rational Number
A rational number is in simplest form when the numerator and denominator are
both
integers that have no common factors other than 1 and the denominator is greater
than
zero.
Addition of Rational Numbers
Let and be
any rational numbers. Then 
Properties of Addition of Rational Numbers
Closure Property
For rational numbers and
is a unique rational number.
Commutative Property
For rational number and
Associative Property
For rational numbers
and 
Identity Property
A unique rational number; 0, exists such that 
for every rational
number ; 0 is the additive identity element.
Additive Inverse Property
For every rational number , a unique rational
number - exists such that
Subtraction of Rational Numbers: Adding the opposite
Let and
be any rational
numbers. Then
Multiplication of Rational Numbers
Let and
be any rational
numbers. Then 
Basic Properties for Multiplication of Rational Numbers
The basic properties of multiplication for integers hold for rational numbers:
(Closure property, Identity property, Zero property , Commutative property,
Associative
property, and Distributive property )
BUT …. A new basic property of multiplication appears for rational numbers, a
property that didn’t exist for integers.
Multiplicative inverse or reciprocal
For every nonzero rational number , a unique
rational number, 
, exists such that
Division of Rational Numbers
Let and
be any rational
numbers where is
nonzero. Then 
Using Models to Compare Rational Numbers
Fraction wall
Number Line
Common Positive Denominator
Approach
For rational numbers and
, where b > 0,
if and only if a > c.
When the denominators of two rational numbers are the same, the one with the
greater
numerator represents the larger rational number.
Number Line Approach
if and only if
is to the left of
on the rational
number line.
Addition Approach
if and only if there is a positive rational
number 
such that
Cross-Multiplication of Rational-Number Inequality
Let and
be any rational
numbers, where b > 0 and d > 0. Then if and
only if ad < bc.
