# Foundations

## Properties of Real Numbers


### Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add  or  the results are the same—they both equal 17. So,  The order in which we add does not matter!
Similarly, when multiplying two numbers, the order does not affect the result. If we multiply  or  the results are the same—they both equal 72. So,  The order in which we multiply does not matter!
These examples illustrate the Commutative Property.
The Commutative Property has to do with order. We subtract  and , and see that  Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.
Division is not commutative either. Since  changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!
  Addition and multiplication are commutative.
  Subtraction and division are not commutative.

When adding three numbers, changing the grouping of the numbers gives the same result. For example,  since each side of the equation equals 17.
This is true for multiplication, too. For example,  since each side of the equation equals 5.
These examples illustrate the Associative Property.
The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.
When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.

### Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity. The Identity Property of Addition that states that for any real number  and 
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity. The Identity Property of Multiplication that states that for any real number  and 
We summarize the Identity Properties here.
What number added to 5 gives the additive identity, 0? We know
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 zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number 
What number multiplied by  gives the multiplicative identity, 1? In other words,  times what results in 1? We know
The missing number was the reciprocal of the number!
We call  the multiplicative inverse of a. The reciprocal of a number is its multiplicative inverse. This leads to the Inverse Property of Multiplication that states that for any real number 
We’ll formally state the inverse properties here.
The Identity Property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.
What about division involving zero? What is  Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So, 
We can check division with the related multiplication fact. So we know  because 
Now think about dividing by zero. What is the result of dividing 4 by  Think about the related multiplication fact:
Is there a number that multiplied by 0 gives  Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4. We conclude that there is no answer to  and so we say that division by 0 is undefined.
We summarize the properties of zero here.
We will now practice using the properties of identities, inverses, and zero to simplify expressions.
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.
The next example makes us aware of the distinction between dividing 0 by some number or some number being divided by 0.

### Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?
You can think about the dollars separately from the quarters. They need 3 times $9 so $27 and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.
Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in  would look like this:
Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.
When we distribute a negative number, we need to be extra careful to get the signs correct!
In the next example, we will show how to use the Distributive Property to find the opposite of an expression.
There will be times when we’ll need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the Distributive Property, which removes the parentheses. The next two examples will illustrate this.
All the properties of real numbers we have used in this chapter are summarized here.

### Key Concepts


### Section Exercises


### Practice Makes Perfect

Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter Review Exercises


### Use the Language of Algebra

Identify Multiples and Factors
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.

### Integers

Simplify Expressions with Absolute Value
In the following exercise, fill in  or  for each of the following pairs of numbers.
In the following exercises, simplify.
Add and Subtract Integers
In the following exercises, simplify each expression.
Multiply and Divide Integers
In the following exercise, multiply or divide.
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
For the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercise, solve.

### Fractions

Simplify Fractions
In the following exercises, simplify.
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
Add and Subtract Fractions
In the following exercises, perform the indicated operation.
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.

### Decimals

Round Decimals
Add and Subtract Decimals
In the following exercises, perform the indicated operation.
Multiply and Divide Decimals
In the following exercises, perform the indicated operation.
Convert Decimals, Fractions and Percents
In the following exercises, convert each decimal to a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each decimal to a percent.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.

### Properties of Real Numbers

Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Practice Test

In the following exercises, simplify each expression.