Operations with Rational Numbers and an Introduction to Real Numbers

Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See (Figure 1). The fraction represents two of three equal parts. In the fraction , the 2 is called the numerator and the 3 is called the denominator.

Figure 1.

The circle on the left has been divided into 3 equal parts. Each part is of the 3 equal parts. In the circle on the right, of the circle is shaded (2 of the 3 equal parts).

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate pieces, or, in other words, one whole pie.

So . This leads us to the property of one that tells us that any number, except zero, divided by itself is 1

If a pie was cut in pieces and we ate all 6, we ate pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions and have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. (Figure 2) shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that is equivalent to . In other words, they are equivalent fractions.

Figure 2.

Since the same amount is of each pizza is shaded, we see that is equivalent to . They are equivalent fractions.

How can we use mathematics to change into How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into pieces instead of just 2. Mathematically, what we’ve described could be written like this as . See (Figure 3).

Figure 3.

Cutting each half of the pizza into pieces, gives us pizza cut into 8 pieces: .

This model leads to the following property:

If we had cut the pizza differently, we could get

So, we say are equivalent fractions.

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

• is simplified because there are no common factors of 2 and 3.
• is not simplified because is a common factor of 10 and 15.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In Example 1, we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent Fractions Property

If are numbers where ,

EXAMPLE 2

Simplify: .

Solution

Rewrite the numerator and denominator showing the common factors.
Simplify using the equivalent fractions property.

Notice that the fraction is simplified because there are no more common factors.

TRY IT 2.1

Simplify: .

Show answer

TRY IT 2.2

Simplify: .

Show answer

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.

We now summarize the steps you should follow to simplify fractions.

EXAMPLE 4

Simplify: .

Solution

Rewrite showing the common factors, then divide out the common factors.
Simplify.

TRY IT 4.1

Simplify: .

Show answer

TRY IT 4.2

Simplify: .

Show answer

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with .

Now we’ll take of .

Notice that now, the whole is divided into 8 equal parts. So .

To multiply fractions, we multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 5, we will multiply negative and a positive, so the product will be negative.

EXAMPLE 5

Multiply: .

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

Determine the sign of the product; multiply.
Are there any common factors in the numerator and the demoninator? No.

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as . So, for example, .

EXAMPLE 6

Multiply: .

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

Write as a fraction.
Multiply.
Rewrite 20 to show the common factor 5 and divide it out.
Simplify.

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of is .

Notice that . A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of is , since .

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say to be sure we don’t divide by zero!

EXAMPLE 7

Divide: .

Solution

To divide, multiply the first fraction by the reciprocal of the second.
Multiply.

EXAMPLE 8

Find the quotient: .

Solution

To divide, multiply the first fraction by the reciprocal of the second.
Determine the sign of the product, and then multiply..
Rewrite showing common factors.
Remove common factors.
Simplify.

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

• “To multiply fractions, multiply the numerators and multiply the denominators.”
• “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Some examples of complex fractions are:

To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction means .

EXAMPLE 9

Simplify: .

Solution

Rewrite as division.
Multiply the first fraction by the reciprocal of the second.
Multiply.
Look for common factors.
Divide out common factors and simplify.

EXAMPLE 10

Simplify: .

Solution

Rewrite as division.
Multiply the first fraction by the reciprocal of the second.
Multiply.
Look for common factors.
Divide out common factors and simplify.

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression , we first simplify the numerator and the denominator separately. Then we divide.

EXAMPLE 11

Simplify: .

Solution

Use the order of operations to simpliy the numerator and the denominator.
Simplify the numerator and the denominator.
Simplify. A negative divided by a positive is negative.

TRY IT 11.1

Simplify: .

Show answer

TRY IT 11.2

Simplify: .

Show answer

Placement of Negative Sign in a Fraction

For any positive numbers a and b,

EXAMPLE 12

Simplify: .

Solution

Multiply.
Simplify.
Divide.

TRY IT 12.1

Simplify: .

Show answer

4

TRY IT 12.2

Simplify: .

Show answer

2

Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of and is the result we get from dividing by , or .

EXAMPLE 13

Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.

Solution

We are looking for the quotient of the difference of m and n, and p. This means we want to divide the difference of .

Key Concepts

• Equivalent Fractions Property: If are numbers where , then
  and .
• Fraction Division: If are numbers where , then . To divide fractions, multiply the first fraction by the reciprocal of the second.
• Fraction Multiplication: If are numbers where , then . To multiply fractions, multiply the numerators and multiply the denominators.
• Placement of Negative Sign in a Fraction: For any positive numbers , .
• Property of One: Any number, except zero, divided by itself is one.
• Simplify a Fraction
  1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
  2. Simplify using the equivalent fractions property by dividing out common factors.
  3. Multiply any remaining factors.
• Simplify an Expression with a Fraction Bar
  1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Simplify the fraction.

Glossary

Practice Makes Perfect

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1.	2.
3.	4.

Simplify Fractions

In the following exercises, simplify.

5.	6.
7.	8.
9.	10.
11.	12.
13.	14.

Multiply Fractions

In the following exercises, multiply.

15.	16.
17.	18.
19.	20.
21.	22.
23.	24.
25.	26.
27.	28.
29.	30.

Divide Fractions

In the following exercises, divide.

31.	32.
33.	34.
35.	36.
37.	38.
39.	40.
41.	42.
43.	44.

In the following exercises, simplify.

45.	46.
47.	48.
49.	50.

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

51.	52.
53.	54.
55.	56.
57.	58.
59.	60.
61.	62.
63.	64.
65.	66.
67.	68.
69.	70.

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

71. the quotient of r and the sum of s and 10	72. the quotient of A and the difference of 3 and B
73. the quotient of the difference of	74. the quotient of the sum of

Everyday Math

75. Baking. A recipe for chocolate chip cookies calls for cup brown sugar. Imelda wants to double the recipe. a) How much brown sugar will Imelda need? Show your calculation. b) Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.	76. Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs cup of condensed milk. a) How much condensed milk will Nina need? Show your calculation. b) Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for pans of fudge.
77. Portions Don purchased a bulk package of candy that weighs pounds. He wants to sell the candy in little bags that hold pound. How many little bags of candy can he fill from the bulk package?	78. Portions Kristen has yards of ribbon that she wants to cut into equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?

Writing Exercises

79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.	80. Give an example from everyday life that demonstrates how .
81. Explain how you find the reciprocal of a fraction.	82. Explain how you find the reciprocal of a negative number.

Answers

1. answers may vary	3. answers may vary	5.
7.	9.	11.
13.	15.	17.
19.	21.	23.
25.	27. 9n	29.
31.	33.	35.
37.	39.	41.
43.	45.	47.
49.	51.	53.
55. 0	57.	59.
61.	63.	65.
67.	69.	71.
73.	75. a) cups b) answers will vary	77. 20 bags
79. Answers may vary.	81. Answers may vary.

Attributions

This chapter has been adapted from “Visualize Fractions” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

EXAMPLE 1

Find the sum: .

Solution

Add the numerators and place the sum over the common denominator.

TRY IT 1.1

Find the sum: .

Show answer

TRY IT 1.2

Find the sum: .

Show answer

EXAMPLE 2

Find the difference: .

Solution

Subtract the numerators and place the difference over the common denominator.
Simplify.
Simplify. Remember, .

TRY IT 2.1

Find the difference: .

Show answer

TRY IT 2.2

Find the difference: .

Show answer

EXAMPLE 3

Simplify: .

Solution

Subtract the numerators and place the difference over the common denominator.
Rewrite with the sign in front of the fraction.

TRY IT 3.1

Find the difference: .

Show answer

TRY IT 3.2

Find the difference: .

Show answer

Now we will do an example that has both addition and subtraction.

EXAMPLE 4

Simplify: .

Solution

Add and subtract fractions—do they have a common denominator? Yes.
Add and subtract the numerators and place the difference over the common denominator.
Simplify left to right.
Simplify.

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

In (Example 5), the LCD, 36, has two factors of 2 and two factors of .

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2

We will apply this method as we subtract the fractions in (Example 6).

EXAMPLE 6

Subtract: .

Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD.
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD.
Simplify.
Subtract.
Check to see if the answer can be simplified.
Both 39 and 120 have a factor of 3.
Simplify.

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

EXAMPLE 7

Add: .

Solution

The fractions have different denominators.

Find the LCD.
Rewrite as equivalent fractions with the LCD.
Simplify.
Add.

 

We now have all four operations for fractions. The table below summarizes fraction operations.

Summary of Fraction Operations

Fraction Operation	Sample Equation	What to Do
Fraction multiplication		Multiply the numerators and multiply the denominators
Fraction division		Multiply the first fraction by the reciprocal of the second.
Fraction addition		Add the numerators and place the sum over the common denominator.
Fraction subtraction		Subtract the numerators and place the difference over the common denominator.

To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.

EXAMPLE 8

Simplify: a) b) .

Solution

First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

a) What is the operation? The operation is subtraction.
Do the fractions have a common denominator? No.
Rewrite each fraction as an equivalent fraction with the LCD.
Subtract the numerators and place the difference over the common denominators.
Simplify, if possible.	There are no common factors. The fraction is simplified.

b) What is the operation? Multiplication.
To multiply fractions, multiply the numerators and multiply the denominators.
Rewrite, showing common factors. Remove common factors.
Simplify.

Notice we needed an LCD to add , but not to multiply .

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction by dividing by .

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

EXAMPLE 10

Simplify: .

Solution

It may help to put parentheses around the numerator and the denominator.

Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).
Simplify.
Divide the numerator by the denominator.
Simplify.
Divide out common factors.
Simplify.

TRY IT 10.1

Simplify: .

Show answer

2

TRY IT 10.2

Simplify: .

Show answer

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

EXAMPLE 12

Evaluate when .

Solution

Rewrite as equivalent fractions with the LCD, 6.
Subtract.
Simplify.

TRY IT 12.1

Evaluate when .

Show answer

TRY IT 12.2

Evaluate when .

Show answer

EXAMPLE 13

Evaluate when and .

Solution

Substitute the values into the expression.

Simplify exponents first.
Multiply. Divide out the common factors. Notice we write 16 as to make it easy to remove common factors.
Simplify.

TRY IT 13.1

Evaluate when and .

Show answer

TRY IT 13.2

Evaluate when and .

Show answer

The next example will have only variables, no constants.

EXAMPLE 14

Evaluate when .

Solution

To evaluate when , we substitute the values into the expression.

Add in the numerator first.
Simplify.

Key Concepts

• Fraction Addition and Subtraction: If are numbers where , then
  and .
  To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
• Strategy for Adding or Subtracting Fractions
  1. Do they have a common denominator?
     Yes—go to step 2.
     No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Add or subtract the fractions.
  3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
• Simplify Complex Fractions
  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator. Simplify if possible.

Glossary

Practice Makes Perfect

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

1.	2.
3.	4.
5.	6.
7.	8.
9.	10.

11.	12.
13.	14.
15.	16.
17.	18.
19.	20.
21.	22.
23.	24.

Mixed Practice

In the following exercises, simplify.

25.	26.
27.	28.
29.	30.
31.	32.

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

33.	34.
35.	36.
37.	38.
39.	40.
41.	42.
43.	44.
45.	46.
47.	48.
49.	50.
51.	52.
53.	54.
55.	56.

Mixed Practice

In the following exercises, simplify.

57. a) b)	58. a) b)
59. a) b)	60. a) b)
61.	62.
63.	64.
65.	66.
67.	68.
69.	70.

In the following exercises, simplify.

71.	72.
73.	74.
75.	76.
77.	78.
79.	80.
81.	82.
83.	84.
85.	86.
87.	88.
89.	90.
91.	92.

In the following exercises, evaluate.

93. when a) b)	94. when a) b)
95. when a) b)	96. when a) b)
97. when a) b)	98. when a) b)
99. when and	100. when and
101. when	102. when

Everyday Math

103. Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs yard of print fabric and yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

104. Baking Samuel is baking chocolate chip cookies and oatmeal cookies. He needs cup of sugar for the chocolate chip cookies and of sugar for the oatmeal cookies. How much sugar does he need altogether?

Writing Exercises

Answers

1.	3.	5.
7.	9.	11.
13.	15.	17.
19.	21.	23.
25.	27.	29.
31.	33.	35.
37.	39.	41.
43.	45.	47.
49.	51.	53.
55.	57. a)b) 4	59. a)b)
61.	63.	65.
67.	69.	71. 54
73.	75.	77.
79.	81.	83.
85.	87.	89. 1
91.	93. a) b)	95. a) b)
97. a) b)	99.	101.
103. yard	105. Answers may vary

Attributions

This chapter has been adapted from “Add and Subtract Fractions” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Name and Write Decimals

Decimals are another way of writing fractions whose denominators are powers of 10.

Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values. Figure 1 shows the names of the place values to the left and right of the decimal point.

Place value of decimal numbers are shown to the left and right of the decimal point.

Figure 1

EXAMPLE 1

Name the decimal 4.3

Solution

TRY IT 1.1

Name the decimal: .

Show answer

six and seven tenths

TRY IT 1.2

Name the decimal: .

Show answer

five and eight tenths

We summarize the steps needed to name a decimal below.

HOW TO: Name a Decimal

1. Name the number to the left of the decimal point.
2. Write “and” for the decimal point.
3. Name the “number” part to the right of the decimal point as if it were a whole number.
4. Name the decimal place of the last digit.

EXAMPLE 2

Name the decimal: .

Solution

Name the number to the left of the decimal point.	negative fifteen __________________________________
Write “and” for the decimal point.	negative fifteen and ______________________________
Name the number to the right of the decimal point.	negative fifteen and five hundred seventy-one __________
The 1 is in the thousandths place.	negative fifteen and five hundred seventy-one thousandths

TRY IT 2.1

Name the decimal: .

Show answer

negative thirteen and four hundred sixty-one thousandths

TRY IT 2.2

Name the decimal: .

Show answer

negative two and fifty-three thousandths

When we write a check we write both the numerals and the name of the number. Let’s see how to write the decimal from the name.

We summarize the steps to writing a decimal.

Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

We summarize the steps for rounding a decimal here.

EXAMPLE 5

Round 18.379 to the nearest a) tenth b) whole number.

Solution

Round 18.379

a) to the nearest tenth

Locate the tenths place with an arrow.
Underline the digit to the right of the given place value.
Because 7 is greater than or equal to 5, add 1 to the 3.
Rewrite the number, deleting all digits to the right of the rounding digit.
Notice that the deleted digits were NOT replaced with zeros.	So, 18.379 rounded to the nearest tenth is 18.4.

 

b) to the nearest whole number

Locate the ones place with an arrow.
Underline the digit to the right of the given place value.
Since 3 is not greater than or equal to 5, do not add 1 to the 8.
Rewrite the number, deleting all digits to the right of the rounding digit.
	So, 18.379 rounded to the nearest whole number is 18.

TRY IT 5.1

Round to the nearest a) hundredth b) tenth c) whole number.

Show answer

a) 6.58 b) 6.6 c) 7

TRY IT 5.2

Round to the nearest a) thousandth b) hundredth c) tenth.

Show answer

a) 15.218 b) 15.22 c) 15.2

Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

EXAMPLE 6

Add: .

Solution

Write the numbers so the decimal points line up vertically.
Put 0 as a placeholder after the 5 in 23.5. Remember, .
Add the numbers as if they were whole numbers. Then place the decimal point in the sum.

TRY IT 6.1

Add: .

Show answer

16.49

TRY IT 6.2

Add: .

Show answer

23.593

EXAMPLE 7

Subtract: .

Solution

Write the numbers so the decimal points line up vertically. Remember, 20 is a whole number, so place the decimal point after the 0.
Put in zeros to the right as placeholders.
Subtract and place the decimal point in the answer.

TRY IT 7.1

Subtract: .

Show answer

0.42

TRY IT 7.2

Subtract: .

Show answer

12.58

Multiply and Divide Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first convert them to fractions and then multiply.

So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern!

Convert to fractions.
Multiply.
Convert to decimals.

Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.

We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.

The rules for multiplying positive and negative numbers apply to decimals, too, of course!

When multiplying two numbers,

• if their signs are the same the product is positive.
• if their signs are different the product is negative.

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.

EXAMPLE 8

Multiply: .

Solution

	(−3.9)(4.075)
The signs are different. The product will be negative.
Write in vertical format, lining up the numbers on the right.
Multiply.
Add the number of decimal places in the factors (1 + 3). Place the decimal point 4 places from the right.
The signs are different, so the product is negative.	(−3.9)(4.075) = −15.8925

TRY IT 8.1

Multiply: .

Show answer

TRY IT 8.2

Multiply: .

Show answer

In many of your other classes, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.

EXAMPLE 9

Multiply 5.63 a) by 10 b) by 100 c) by 1,000.

Solution

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

a)

	5.63(10)
There is 1 zero in 10, so move the decimal point 1 place to the right.

b)

	5.63(100)
There are 2 zeros in 100, so move the decimal point 2 places to the right.

c)

	5.63(1,000)
There are 3 zeros in 1,000, so move the decimal point 3 places to the right.
A zero must be added at the end.

TRY IT 9.1

Multiply 2.58 a) by 10 b) by 100 c) by 1,000.

Show answer

a) 25.8 b) 258 c) 2,580

TRY IT 9.2

Multiply 14.2 a) by 10 b) by 100 c) by 1,000.

Show answer

a) 142 b) 1,420 c) 14,200

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To divide decimals, determine what power of 10 to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of . Because of the equivalent fractions property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example:

We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.

We review the notation and vocabulary for division:

We’ll write the steps to take when dividing decimals, for easy reference.

EXAMPLE 10

Divide: .

Solution

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property.

The signs are the same.	The quotient is positive.
Make the divisor a whole number by “moving” the decimal point all the way to the right.
“Move” the decimal point in the dividend the same number of places.
Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Write the quotient with the appropriate sign.

TRY IT 10.1

Divide: .

Show answer

687.3

TRY IT 10.2

Divide: .

Show answer

34.25

A common application of dividing whole numbers into decimals is when we want to find the price of one item that is sold as part of a multi-pack. For example, suppose a case of 24 water bottles costs $3.99. To find the price of one water bottle, we would divide $3.99 by 24. We show this division in Example 11. In calculations with money, we will round the answer to the nearest cent (hundredth).

EXAMPLE 11

Divide: .

Solution

Place the decimal point in the quotient above the decimal point in the dividend.
Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth.) To do this, we must carry the division to the thousandths place.
Round to the nearest cent.

Convert Decimals and Fractions

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03

Notice, when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator. Fractions like this are called proper fractions.

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

EXAMPLE 12

Write 0.374 as a fraction.

Solution

Determine the place value of the final digit.
Write the fraction for 0.374: The numerator is 374. The denominator is 1,000.
Simplify the fraction.
Divide out the common factors.	so,

Did you notice that the number of zeros in the denominator of is the same as the number of decimal places in 0.374?

TRY IT 12.1

Write 0.234 as a fraction.

Show answer

TRY IT 12.2

Write 0.024 as a fraction.

Show answer

We’ve learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar means division. So can be written or . This leads to the following method for converting a fraction to a decimal.

HOW TO: Covert a Fraction to a Decimal

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

Sometimes we may have to simplify expressions with fractions and decimals together.

EXAMPLE 15

Simplify: .

Solution

First we must change one number so both numbers are in the same form. We can change the fraction to a decimal, or change the decimal to a fraction. Usually it is easier to change the fraction to a decimal.

Change to a decimal.
Add.
		So,

Key Concepts

• Name a Decimal
  1. Name the number to the left of the decimal point.
  2. Write ”and” for the decimal point.
  3. Name the “number” part to the right of the decimal point as if it were a whole number.
  4. Name the decimal place of the last digit.
• Write a Decimal
  1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right.
  2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
  3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
  4. Fill in zeros for place holders as needed.
• Round a Decimal
  1. Locate the given place value and mark it with an arrow.
  2. Underline the digit to the right of the place value.
  3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value.
  4. Rewrite the number, deleting all digits to the right of the rounding digit.
• Add or Subtract Decimals
  1. Write the numbers so the decimal points line up vertically.
  2. Use zeros as place holders, as needed.
  3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
• Multiply Decimals
  1. Determine the sign of the product.
  2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
  3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors.
  4. Write the product with the appropriate sign.
• Multiply a Decimal by a Power of Ten
  1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Add zeros at the end of the number as needed.
• Divide Decimals
  1. Determine the sign of the quotient.
  2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places – adding zeros as needed.
  3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Write the quotient with the appropriate sign.
• Convert a Decimal to a Proper Fraction
  1. Determine the place value of the final digit.
  2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.
• Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

Practice Makes Perfect

Name and Write Decimals

In the following exercises, write as a decimal.

In the following exercises, name each decimal.

9. 5.5	10. 14.02
11. 8.71	12. 2.64
13. 0.002	14. 0.479
15.	16.

Round Decimals

In the following exercises, round each number to the nearest tenth.

In the following exercises, round each number to the nearest hundredth.

27. 5.781	28. 1.6381
29. 63.479	30.

Add and Subtract Decimals

In the following exercises, add or subtract.

31.	32.
33.	34.
35.	36.
37.	38.
39.	40.
41.	42.
43.	44.
45.	46.
47.	48.

In the following exercises, multiply.

49.	50.
51.	52.
53.	54.
55.	56.
57.	58.
59.	60.
61.	62.

63.	64.
65.	66.
67.	68.
69.	70.
71.	72.
73.	74.
75.	76.

In the following exercises, write each decimal as a fraction.

87.	88.
89.	90.
91.	92.
93.	94.
95.	96.
97.	98.

Everyday Math

Glossary

Answers

1. 29.81	3. 0.7	5. 0.029
7.	9. five and five tenths	11. eight and seventy-one hundredths
13. two thousandths	15. negative seventeen and nine tenths	17. 0.7
19. 2.8	21. 0.85	23. 0.30
25. 4.10	27. a) 5.78 b) 5.8 c) 6	29. a) 63.48 b) 63.5 c) 63
31.	33.	35.
37.	39.	41. 15.73
43. 102.212	45. 51.31	47.
49. 0.144	51. 42.008	53.
55. 337.8914	57. 1.305	59. 92.4
61. 55,200	63. 0.19	65. $2.44
67. 3	69.	71. 35
73. 2.08	75. 20	77.
79.	81.	83.
85.	87. 0.85	89. 2.75
91.	93.	95.
97. 3.025	99. a) $58,966 b) $59,000 c) $60,000	101. a) $142.19; b) $142
103. a) $243.57 b) $79.35	105. Answers may vary.	107. Answers may vary.

Attributions

This chapter has been adapted from “Decimals” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers?

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

All signed fractions, such as are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer could be written as the decimal . So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means , we can write it as an improper fraction, . So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

EXAMPLE 1

Write as the ratio of two integers: a) b) 7.31

Solution

a) Write it as a fraction with denominator 1.
b) Write it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths. Convert to an improper fraction.

So we see that and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number, since for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we considered above.

Ratio of integers		–		–
The decimal form				.
These decimals either stop or repeat.

What do these examples tell us?

Every rational number can be written both as a ratio of integers, ,where p and q are integers and ,and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

	Fractions				Integers
Number
Ratio of Integers
Decimal Form

The number (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

.

We can even create a decimal pattern that does not stop or repeat, such as

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational. More on irrational numbers later on is this course.

EXAMPLE 2

Given the numbers . list the a) rational numbers b) irrational numbers.

Solution

a) Look for decimals that repeat or stop.	The 3 repeats in . The decimal 0.47 stops after the 7. So and 0.47 are rational.
b) Look for decimals that neither stop nor repeat.	has no repeating block of digits and it does not stop. So is irrational.

TRY IT 2.1

For the given numbers list the a) rational numbers b) irrational numbers: .

Show answer

a) b)

TRY IT 2.2

For the given numbers list the a) rational numbers b) irrational numbers:

Show answer

a) b)

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

All the numbers we use in algebra are real numbers.  Figure 1 illustrates how the number sets we’ve discussed in this section fit together.

Figure 1 This chart shows the number sets that make up the set of real numbers. Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be?

Locate Fractions on the Number Line

The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.

Let’s start with fractions and locate on the number line.

We’ll start with the whole numbers and . because they are the easiest to plot. See Figure 2.

The proper fractions listed are . We know the proper fraction has value less than one and so would be located between The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts . We plot . See Figure 2.

Similarly, is between 0 and . After dividing the unit into 5 equal parts we plot . See Figure 2.

Finally, look at the improper fractions . These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure 2.

Figure 2 shows the number line with all the points plotted.

Figure 2

EXAMPLE 4

Locate and label the following on a number line: .

Solution

Locate and plot the integers, .

Locate the proper fraction first. The fraction is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot . Similarly plot .

Now locate the improper fractions . It is easier to plot them if we convert them to mixed numbers and then plot them as described above: .

TRY IT 4.1

Locate and label the following on a number line: .

Show answer

TRY IT 4.2

Locate and label the following on a number line: .

Show answer

In Example 5, we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a < b “a is less than b” when a is to the left of b on the number line
• a > b “a is greater than b” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

EXAMPLE 5

Order each of the following pairs of numbers, using < or >. It may be helpful to refer Figure 3.

a) –____ b) -3____ c) ____ – d) ____ –

Figure 3

Solution

a) – is to the right of on the number line.	–___-1 – > -1
b) – is to the right of on the number line.
c) is to the right of on the number line.
d) is to the right of on the number line.

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

EXAMPLE 6

Locate 0.4 on the number line.

Solution

A proper fraction has value less than one. The decimal number 0.4 is equivalent to , a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See Figure 4.

Figure 4

TRY IT 6.1

Locate on the number line: 0.6

Show answer

TRY IT 6.2

Locate on the number line: 0.9

Show answer

EXAMPLE 7

Locate on the number line.

Solution

The decimal is equivalent to , so it is located between 0 and . On a number line, mark off and label the hundredths in the interval between 0 and . See Figure 5.

Figure 5

TRY IT 7.1

Locate on the number line: .

Show answer

TRY IT 7.2

Locate on the number line: .

Show answer

Which is larger, 0.04 or 0.40? If you think of this as money, you know that ?0.40 (forty cents) is greater than ?0.04 (four cents). So,

>

Again, we can use the number line to order numbers.

• a < b “a is less than b” when a is to the left of b on the number line
• a > b “a is greater than b” when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line? See Figure 6.

Figure 6

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

	0.31	0.308
Convert to fractions.
We need a common denominator to compare them.

Because 310 > 308, we know that > . Therefore, 0.31 > 0.308

Notice what we did in converting 0.31 to a fraction—we started with the fraction and ended with the equivalent fraction . Converting back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

We say 0.31 and 0.310 are equivalent decimals.

The steps we take to order decimals are summarized here.

HOW TO: Order Decimals.

1. Write the numbers one under the other, lining up the decimal points.
2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

EXAMPLE 8

Order using < or >.

Solution

Write the numbers one under the other, lining up the decimal points.
Add a zero to 0.6 to make it a decimal with 2 decimal places. Now they are both hundredths.
64 is greater than 60.	>
64 hundredths is greater than 60 hundredths.	>
	>

TRY IT 8.1

Order each of the following pairs of numbers, using > .

Show answer

>

TRY IT 8.2

Order each of the following pairs of numbers, using > .

Show answer

>

EXAMPLE 9

Order using < or >.

Solution

Write the numbers one under the other, lining up the decimals.
They do not have the same number of digits. Write one zero at the end of 0.83.
Since > , 830 thousandths is greater than 803 thousandths.	>
	>

TRY IT 9.1

Order the following pair of numbers, using > .

Show answer

>

TRY IT 9.2

Order the following pair of numbers, using < > .

Show answer

<

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because lies to the right of on the number line, we know that > . Similarly, smaller numbers lie to the left on the number line. For example, because lies to the left of on the number line, we know that . See Figure 7.

Figure 7

If we zoomed in on the interval between 0 and , as shown in Example 10, we would see in the same way that > .

EXAMPLE 10

Use < or > to order .

Solution

Write the numbers one under the other, lining up the decimal points. They have the same number of digits.
Since > , −1 tenth is greater than −8 tenths.	>

Key Concepts

• Order Decimals
  1. Write the numbers one under the other, lining up the decimal points.
  2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
  3. Compare the numbers as if they were whole numbers.
  4. Order the numbers using the appropriate inequality sign.

Glossary

Practice Makes Perfect

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

1. a) 5 b) 3.19	2. a) 8 b) 1.61
3. a) b) 9.279	4. a) b) 4.399

In the following exercises, list the a) rational numbers, b) irrational numbers

5.	6.
7.	8.

In the following exercises, list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers for each set of numbers.

9.	10.
11.	12.

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

13.	14.
15.	16.
17.	18.
19.	20.

In the following exercises, order each of the pairs of numbers, using < or >.

21.	22.
23.	24.
25.	26.
27.	28.

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

29. 0.8	30.
31.	32. 3.1

In the following exercises, order each pair of numbers, using < or >.

33.	34.
35.	36.
37.	38.
39.	40.

Everyday Math

41. Field trip. All the 5th graders at Lord Selkirk Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people. a) How many buses will be needed? b) Why must the answer be a whole number? c) Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

42. Child care. Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care centre to serve 40 children. a) How many teachers will be needed? b) Why must the answer be a whole number? c) Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Writing Exercises

43. In your own words, explain the difference between a rational number and an irrational number.

44. Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

Answers

1. a) b)	3. a) b)	5. a) b)
7. a) b)	9. a) b) c) d) e)	11. a) none b) c) -7, d) none e)
13.	15.	17.
19.	21. <	23. >
25. >	27. <	29.
31.	33. <	35. >
37. <	39. <	41. a) 4 buses b) answers may vary c) answers may vary
43. Answers may vary.

Attributions

This chapter has been adapted from “The Real Numbers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Use the Commutative and Associative Properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying

Again, the results are the same!

The order in which we multiply does not matter!

These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.

The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

What about subtraction? Does order matter when we subtract numbers? Does give the same result as

The results are not the same.

Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

The results are not the same.

Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!

• Addition and multiplication are commutative.
• Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?

Some people would think and then . Others might start with and then .

Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.

Add . Add.
Add . Add.

When adding three numbers, changing the grouping of the numbers gives the same result.

This is true for multiplication, too.

Multiply. Multiply.
Multiply. . Multiply.

When multiplying three numbers, changing the grouping of the numbers gives the same result.

You probably know this, but the terminology may be new to you. These examples illustrate the associative property.

Let’s think again about multiplying . We got the same result both ways, but which way was easier? Multiplying and first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!

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.

EXAMPLE 1

Simplify: .

Solution

Use the commutative property of addition to re-order so that like terms are together.
Add like terms.

When we have to simplify algebraic expressions, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.

EXAMPLE 2

Simplify: .

Solution

Notice that the last 2 terms have a common denominator, so change the grouping.
Add in parentheses first.
Simplify the fraction.
Add.
Convert to an improper fraction.

EXAMPLE 3

Use the associative property to simplify .

Solution

Change the grouping.
Multiply in the parentheses.

Notice that we can multiply but we could not multiply 3x without having a value for x.

TRY 3.1

Use the associative property to simplify 8(4x).

Show answer

32x

Use the Identity and Inverse Properties of Addition and Multiplication

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.

For example,

These examples illustrate 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.

For example,

These examples illustrate the Identity Property of Multiplication that states that for any real number , and .

We summarize the Identity Properties below.

Notice that in each case, the missing number was the opposite of the number!

We call . the additive inverse of a. 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 . Remember, a number and its opposite add to zero.

What number multiplied by gives the multiplicative identity, 1? In other words, times what results in 1?

What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?

Notice that in each case, the missing number was the reciprocal of the number!

We call the multiplicative inverse of a. The reciprocal of number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number .

We’ll formally state the inverse properties here:

Inverse Property

of addition	For any real number , is the additive inverse of . A number and its opposite add to zero.
of multiplication	For any real number , is the multiplicative inverse of . A number and its reciprocal multiply to one.

EXAMPLE 4

Find the additive inverse of a) b) c) d) .

Solution

To find the additive inverse, we find the opposite.

1. The additive inverse of is the opposite of . The additive inverse of is .
2. The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is .
3. The additive inverse of is the opposite of . We write the opposite of as , and then simplify it to 8. Therefore, the additive inverse of is 8.
4. The additive inverse of is the opposite of . We write this as , and then simplify to . Thus, the additive inverse of is .

TRY IT 4.1

Find the additive inverse of: a) b) c) d) .

Show answer

a) b) c) d)

Exercises

Find the additive inverse of: a) b) c) d) .

Show answer

a) b) c) d)

Use the Properties of Zero

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.

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 0? Think about the related multiplication fact: means . Is there a number that multiplied by 0 gives 4? 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 below.

Properties of Zero

Multiplication by Zero: For any real number a,

The product of any number and 0 is 0.

Division of Zero, Division by Zero: For any real number

	Zero divided by any real number except itself is zero.
	Division by zero is undefined.

EXAMPLE 6

Simplify: a) b) c) .

Solution

a) The product of any real number and 0 is 0.
b) The product of any real number and 0 is 0.
c) Division by 0 is undefined.

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

EXAMPLE 7

Simplify: a) , where b) , where .

Solution

a) Zero divided by any real number except itself is 0.
b) Division by 0 is undefined.

EXAMPLE 8

Simplify: .

Solution

Notice that the first and third terms are opposites; use the commutative property of addition to re-order the terms.
Add left to right.
Add.

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1

EXAMPLE 9

Simplify: .

Solution

Notice that the first and third terms are reciprocals, so use the commutative property of multiplication to re-order the factors.
Multiply left to right.
Multiply.

EXAMPLE 10

Simplify: .

Solution

There is nothing to do in the parentheses, so multiply the two fractions first—notice, they are reciprocals.
Simplify by recognizing the multiplicative identity.

TRY IT 10.1

Simplify: .

Show answer

TRY IT 10.2

Simplify: .

Show answer

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.

Back to our friends at the movies, we could find the total amount of money they need like this:

In algebra, we use the distributive property to remove parentheses as we simplify expressions.

For example, if we are asked to simplify the expression , the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in (Example 11).

EXAMPLE 11

Simplify: .

Solution

Distribute.
Multiply.

Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in (Example 11) would look like this:

EXAMPLE 12

Simplify: .

Solution

Distribute.
Multiply.

Using the distributive property as shown in (Example 13) will be very useful when we solve money applications in later chapters.

EXAMPLE 13

Simplify: .

Solution

Distribute.
Multiply.

When we distribute a negative number, we need to be extra careful to get the signs correct!

EXAMPLE 14

Simplify: .

Solution

Distribute.
Multiply.

EXAMPLE 15

Simplify: .

Solution

Distribute.
Multiply.
Simplify.

Notice that you could also write the result as . Do you know why?

TRY IT 15.1

Simplify: .

Show answer

TRY IT 15.2

Simplify: .

Show answer

(Example 16) will show how to use the distributive property to find the opposite of an expression.

EXAMPLE 16

Simplify: .

Solution

Multiplying by −1 results in the opposite.
Distribute.
Simplify.

EXAMPLE 17

Simplify: .

Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

Solution

Distribute.
Multiply.
Combine like terms.

EXAMPLE 18

Simplify: .

Solution

Distribute.
Combine like terms.

All the properties of real numbers we have used in this chapter are summarized in the table below.

Commutative Property	of addition If are real numbers, then
	of multiplication If are real numbers, then
Associative Property	of addition If are real numbers, then
	of multiplication If are real numbers, then
Distributive Property	If are real numbers, then
Identity Property	of addition For any real number 0 is the additive identity
	of multiplication For any real number is the multiplicative identity
Inverse Property	of addition For any real number , is the additive inverse of
	of multiplication For any real number is the multiplicative inverse of .
Properties of Zero	For any real number a, For any real number For any real number	is undefined

Key Concepts

• Commutative Property of
  • Addition: If are real numbers, then .
  • Multiplication: If are real numbers, then . When adding or multiplying, changing the order gives the same result.
• Associative Property of
  • Addition: If are real numbers, then .
  • Multiplication: If are real numbers, then .
    When adding or multiplying, changing the grouping gives the same result.
• Distributive Property: If are real numbers, then
  • 
  • 
  • 
  • 
• Identity Property
  • of Addition: For any real number
    0 is the additive identity
  • of Multiplication: For any real number
    is the multiplicative identity
• Inverse Property
  • of Addition: For any real number . A number and its opposite add to zero. is the additive inverse of .
  • of Multiplication: For any real number . A number and its reciprocal multiply to one. is the multiplicative inverse of .
• Properties of Zero
  • For any real number ,
    – The product of any real number and 0 is 0.
  • for – Zero divided by any real number except zero is zero.
  • is undefined – Division by zero is undefined.

Glossary

Practice Makes Perfect

Use the Commutative and Associative Properties

In the following exercises, use the associative property to simplify.

1. 3(4x)	2. 4(7m)
3.	4.

5.	6.
7.	8.
9.	10.
11.	12.
13. 17(0.25)(4)	14. 36(0.2)(5)
15. [2.48(12)](0.5)	16. [9.731(4)](0.75)
17. 7(4a)	18. 9(8w)
19.	20.
21.	22.
23.	24.
25.	26.
27.	28.

In the following exercises, find the additive inverse of each number.

29. a) b) 4.3 c) d)	30. a) b) 2.1 c) d)
31. a) b) c) 23 d)	32. a) b) c) 52 d)

33. a) 6 b) c) 0.7	34. a) 12 b) c) 0.13
35. a) b) c)	36. a) b) c)

Use the Properties of Zero

In the following exercises, simplify.

37.	38.
39.	40.
41.	42.
43.	44.

Mixed Practice

In the following exercises, simplify.

45.	46.
47.	48.
49. , where	50. , where
51. , where	52. , where
53. , where	54. , where
55. where	56. where
57.	58.

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

59.	60.
61.	62.
63.	64.
65.	66.
67.	68.
69.	70.
71.	72.
73.	74.
75.	76.
77.	78.
79.	80.
81.	82.
83.	84.
85.	86.
87.	88.
89.	90.
91.	92.
93.	94.

Everyday Math

99. In your own words, state the commutative property of addition.	100. What is the difference between the additive inverse and the multiplicative inverse of a number?
101. Simplify using the distributive property and explain each step.	102. Explain how you can multiply 4($5.97) without paper or calculator by thinking of $5.97 as and then using the distributive property.

Answers

1. 12x	3.	5.
7.	9.	11.
13. 17	15. 14.88	17. 28a
19.	21. 10p	23.
25.	27.	29. a) b) c) 8 d)
31. a) b) 0.075 c) d)	33. a) b) c)	35. a) b) c)
37. 0	39. 0	41. 0
43. 0	45. 44	47. d
49. 0	51. 0	53. undefined
55. undefined	57.	59.
61.	63.	65.
67.	69.	71.
73.	75.	77.
79.	81.	83.
85.	87.	89.
91.	93.	95. a) $80 b) $80 c) answers will vary
97. a) $23.88 b) no, the price is the same	99. Answers may vary	101. Answers may vary

Attributions

This chapter has been adapted from “Properties of Real Numbers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Review Exercises

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1.	2.
3.	4.

Simplify Fractions

In the following exercises, simplify.

5.	6.
7.	8.
9.	10.
11.	12.

In the following exercises, multiply.

13.	14.
15.	16.
17.	18.
19.	20.

In the following exercises, divide.

21. ÷	22. ÷
23. ÷	24. ÷
25. ÷	26. ÷
27. ÷	28. ÷
29. ÷	30. ÷

31.	32.
33.	34.
35.	36.

In the following exercises, simplify.

37.	38.
39.	40.
41.	42.
43.	44.
45.	46.
47.	48.

In the following exercises, translate each English phrase into an algebraic expression.

49. the quotient of c and the sum of d and 9.

50. the quotient of the difference of h and k, and .

In the following exercises, add.

51.	52.
53.	54.
55.	56.

In the following exercises, subtract.

57.	58.
59.	60.
61.	62.

In the following exercises, add or subtract.

63.	64.
65.	66.
67.	68.
69.	70.
71.	72.
73.	74.

In the following exercises, simplify.

75.	76.
77.	78.