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The Rational Numbers

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.

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