Properties of Real Numbers

Closure Property

If a and b are real numbers, then If a and b are real numbers, then

Closure Property

If a and b are real numbers, then

We should think of any combination of addition as a single, real number

If a and b are real numbers, then

We should think of any combination of multiplication as a single, real number

Commutative Property

If a and b are real numbers, then If a and b are real numbers, then

Commutative Property

If a and b are real numbers, then

We can change the order of addition without changing the result

If a and b are real numbers, then

We can change the order of multiplication without changing the result

Associative Property

If a, b, and c are real numbers, then

If a, b, and c are real numbers, then

Associative Property

If a, b, and c are real numbers, then

Under addition, we can place parentheses wherever we please, or choose not to use parentheses

If a, b, and c are real numbers, then

Under multiplication, we can place parentheses wherever we please or choose not to use parentheses

Identity Property

There exists a unique number called zero (0) such that, for any number a

There exists a unique number called one (1) such that, for any number a

Identity Property

There exists a unique number called zero (0) such that, for any number a

If we ever end up with zero plus a number, we can drop the zero

There exists a unique number called one (1) such that, for any number a

If we ever end up with one times a number, we can drop the 1

Inverse Property

For every non-zero real number a, there exists the number such that

For every non-zero real number a, there exists the number such that

Inverse Property

For every non-zero real number a, there exists the number such that

The “canceling” property for addition

For every non-zero real number a, there exists the number such that

The “canceling” property for multiplication

Distributive Property

If a, b, and c are real numbers, then

and

Distributive Property

If a, b, and c are real numbers, then

and

The top equation is multiplication. The bottom is factoring.

Examples

Use the:

a) Commutative Property for Addition

b) Commutative Property for Multiplication

c) Associative Property for Addition

d) Associative Property for Multiplication

Examples

What is the multiplicative inverse for

What is the additive inverse for

Definitions of Subtraction and Division

DEFINITION:

For real numbers a and b, we define subtraction to be

For real numbers a and b, with , we define division to be

Examples

Show that each equation is a true statement. Justify each step using the number properties.

a)

b)

c)

d)

e)

Examples

• Justification

• Definition of division

• Distributive Property

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Multiplication

• Commutative Property for Addition

Examples

• Justification

• Definition of subtraction

• Commutative Property for Addition

• Associative Property for Addition

• Inverse Property of Addition

• Identity Property of Addition

Examples

• Justification

• Definition of Division

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Inverse Property for Multiplication

• Identity Property for Multiplication

Examples

• Justification

• Definition of Division

• Distributive Property

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Inverse Property for Multiplication

• Identity Property for Multiplication

• Addition

Examples

• Justification

• Commutative Property of Addition

• Associative Property of Addition

• Distributive Property (Factoring)

• Addition

• Commutative Property for Addition