# 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.

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.

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.

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

 Prev Next

Start solving your Algebra Problems in next 5 minutes!

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of September 19th you will pay only \$39.99 instead of our regular price of \$74.99 -- this is \$35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1\$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

• simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
• factoring and expanding expressions
• finding LCM and GCF
• (simplifying, rationalizing complex denominators...)
• solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
• solving a system of two and three linear equations (including Cramer's rule)
• graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
• graphing general functions
• operations with functions (composition, inverse, range, domain...)
• simplifying logarithms
• basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
• arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!