**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.