# The Properties of Real Numbers

## Properties of Identity, Inverses, and Zero


### Recognize the Identity Properties of Addition and Multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call  the additive identity.
For example,
What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call  the multiplicative identity.
For example,

### Use the Inverse Properties of Addition and Multiplication

Notice that in each case, the missing number was the opposite of the number.
We call  the additive inverse of  The opposite of a number is its additive inverse. A number and its opposite add to  which is the additive identity.
What number multiplied by  gives the multiplicative identity,  In other words, two-thirds times what results in 
What number multiplied by  gives the multiplicative identity,  In other words two times what results in 
Notice that in each case, the missing number was the reciprocal of the number.
We call  the multiplicative inverse of  The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to  which is the multiplicative identity.
We’ll formally state the Inverse Properties here:

### Use the Properties of Zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

### Multiplication by Zero

What happens when you multiply a number by  Multiplying by  makes the product equal zero. The product of any real number and  is 

### Dividing with Zero

What about dividing with  Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are  cookies to share, so each person gets  cookies.
Remember that we can always check division with the related multiplication fact. So, we know that
Zero divided by any real number except zero is zero.
Now let’s think about dividing a number by zero. What is the result of dividing  by  Think about the related multiplication fact. Is there a number that multiplied by  gives 
Since any real number multiplied by  equals  there is no real number that can be multiplied by  to obtain  We can conclude that there is no answer to  and so we say that division by zero is undefined.
Division by zero is undefined.
We summarize the properties of zero.

### Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.
All the properties of real numbers we have used in this chapter are summarized in .

### Key Concepts

1. Identity Properties
2. Inverse Properties
3. Properties of Zero

### Practice Makes Perfect

Recognize the Identity Properties of Addition and Multiplication 
In the following exercises, identify whether each example is using the identity property of addition or multiplication.
Use the Inverse Properties of Addition and Multiplication 
In the following exercises, find the multiplicative inverse.
Use the Properties of Zero 
In the following exercises, simplify using the properties of zero.
Simplify Expressions using the Properties of Identities, Inverses, and Zero 
In the following exercises, simplify using the properties of identities, inverses, and zero.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
