Other Properties of Real Numbers

Identity Properties

Identity properties tell us how we can

add or multiply and get an answer that

is identical to the number

we started with.

Identity Property of Addition

The identity property of addition tells us that we can add zero to any number and get that identical

number as the answer.

a + 0 = a

That makes the identity element for addition:

Identity Property of Multiplication

The identity property of multiplication tells us that we can multiply any number by 1 and get that

identical number as the answer.

That makes the identity element for multiplication:

1a a

Inverse Properties

The inverse properties tell us how we can create the identity elements out of other numbers.

Sort of like building a sand castle!

Additive Inverse Property

The identity element for addition is

How can we add and get an answer equal to zero?

Ah ha! - 5 + 5 = 0

In fact: - a + a = 0

Additive Inverse Property

-a + a = 0

Numbers that have the same magnitude but different signs are called opposites.

Another term for opposite is additive inverse.

Additive Inverse Property

The additive inverse property tells us that when we add any number and its opposite the answer

will be zero.

- a + a = 0

Double Negative Property

Closely related to the concept of opposite is the double negative property.

- ( - a ) = a

We can think if that as two negative signs in a row convert to one positive sign.

(signs are the same, replace with +)

Double Negative Property

Closely related to the concept of opposite is the double negative property.

- ( - a ) = a

Or we can think if it as ‘taking the opposite of an opposite gets you back where you started’!

Multiplicative Inverse Property

The identity element for multiplication is

How can we multiply and get an answer equal to one?

Ah ha!

In fact:

12 1

21

1aa

Reciprocals

Reciprocals are the gymnasts of

algebra.

They just love to do

headstands!

Watch them flip!

2 3 flip

3 21 5

flip or 55 1

16 flip

6

Multiplicative Inverse Property

The multiplicative inverse property tells us that when we multiply any number by its reciprocal

the answer will be one.

Another term for reciprocal is multiplicative inverse.

Multiplication Property of Zero

Zero has a unique property that we will use a lot later in the semester.

Any number multiplied by zero equals zero.

0 0a

Reduction Property

When we reduce fractions, we are using the reduction property.

In other words, as long as b and c are not equal to zero, we can cancel and reduce.

ac ac a

bc bc b

More Properties of Real Numbers

Identity properties

a + 0 = a and

Inverse properties – opposites and reciprocals

Double negative property

Multiplication property of zero

Reduction property

1a a