Whole Numbers, Integers, and an Introduction to Algebra

As we begin our study of intermediate algebra, we need to refresh some of our skills and vocabulary. This chapter and the next will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

We can visualize counting numbers and whole numbers on a number line .See Figure 1.

The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. While this number line shows only the whole numbers 0 through 6, the numbers keep going without end.

Figure 1

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 2 shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

The number 5,278,194 is shown in the chart. The digit 5 is in the millions place. The digit 2 is in the hundred-thousands place. The digit 7 is in the ten-thousands place. The digit 8 is in the thousands place. The digit 1 is in the hundreds place. The digit 9 is in the tens place. The digit 4 is in the ones place.

Figure 2

EXAMPLE 1

In the number 63,407,218, find the place value of each digit:

1. 7
2. 0
3. 1
4. 6
5. 3

Solution

Place the number in the place value chart:

a) The 7 is in the thousands place.
b) The 0 is in the ten thousands place.
c) The 1 is in the tens place.
d) The 6 is in the ten-millions place.
e) The 3 is in the millions place.

TRY IT 1.1

For the number 27,493,615, find the place value of each digit:

a) 2 b) 1 c) 4 d) 7 e) 5

Show answer

a) ten millions b) tens c) hundred thousands d) millions e) ones

TRY IT 1.2

For the number 519,711,641,328, find the place value of each digit:

a) 9 b) 4 c) 2 d) 6 e) 7

Show answer

a) billions b) ten thousands c) tens d) hundred thousands e) hundred millions

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 3). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Figure 3

In 2016, Statistics Canada estimated the population of Toronto as 13,448,494. We could say the population of Toronto was approximately 13.4 million. In many cases, you don’t need the exact value; an approximate number is good enough.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of Toronto is approximately 13.4 million means that we rounded to the hundred thousands place.

EXAMPLE 5

Round to the nearest:

1. hundred
2. thousand
3. ten thousand

Solution

a)

Locate the hundreds place in 103,978.
Underline the digit to the right of the hundreds place.
Since 7 is greater than or equal to 5, add 1 to the 9. Replace all digits to the right of the hundreds place with zeros.
	So, 104,000 is 103,978 rounded to the nearest hundred.

b)

Locate the thousands place and underline the digit to the right of the thousands place.
Since 9 is greater than or equal to 5, add 1 to the 3. Replace all digits to the right of the hundreds place with zeros.
	So, 104,000 is 103,978 rounded to the nearest thousand.

c)

Locate the ten thousands place and underline the digit to the right of the ten thousands place.
Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros.
	So, 100,000 is 103,978 rounded to the nearest ten thousand.

TRY IT 5.1

Round 206,981 to the nearest: a) hundred b) thousand c) ten thousand.

Show answer

a) 207,000 b) 207,000 c) 210,000

TRY IT 5.2

Round 784,951 to the nearest: a) hundred b) thousand c) ten thousand.

Show answer

a) 785,000 b) 785,000 c) 780,000

Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2

Similarly, a multiple of 3 would be the product of a counting number and 3

We could find the multiples of any number by continuing this process.

The Table 1 below shows the multiples of 2 through 9 for the first 12 counting numbers.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, is 5, so 15 is .

Look at the multiples of 5 in Table 1. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Divisibility Tests

A number is divisible by:

• 2 if the last digit is 0, 2, 4, 6, or 8.
• 3 if the sum of the digits is divisible by 3.
• 5 if the last digit is 5 or 0.
• 6 if it is divisible by both 2 and 3.
• 10 if it ends with 0.

EXAMPLE 6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Solution

Is 5,625 divisible by 2?
Does it end in 0,2,4,6, or 8?	No. 5,625 is not divisible by 2.
Is 5,625 divisible by 3?
What is the sum of the digits?
Is the sum divisible by 3?	Yes. 5,625 is divisble by 3.
Is 5,625 divisible by 5 or 10?
What is the last digit? It is 5.	5,625 is divisble by 5 but not by 10.
Is 5,625 divisible by 6?
Is it divisible by both 2 or 3?	No, 5,625 is not divisible by 2, so 5,625 is not divisible by 6.

TRY IT 6.1

Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10

Show answer

by 2, 3, and 6

TRY IT 6.2

Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10

Show answer

by 3 and 5

Find Prime Factorization and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since , we say that 8 and 9 are factors of 72. When we write , we say we have factored 72

Other ways to factor 72 are . Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72

Figure 4

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

EXAMPLE 8

Find the prime factorization of 252

Solution

Step 1. Find two factors whose product is 252. 12 and 21 are not prime. Break 12 and 21 into two more factors. Continue until all primes are factored.
Step 2. Write 252 as the product of all the circled primes.

TRY IT 8.1

Find the prime factorization of 126

Show answer

TRY IT 8.2

Find the prime factorization of 294

Show answer

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18

EXAMPLE 9

Find the least common multiple of 15 and 20 by listing multiples.

Solution

Make lists of the first few multiples of 15 and of 20, and use them to find the least common multiple.
Look for the smallest number that appears in both lists.	The first number to appear on both lists is 60, so 60 is the least common multiple of 15 and 20.

Notice that 120 is in both lists, too. It is a common multiple, but it is not the least common multiple.

TRY IT 9.1

Find the least common multiple by listing multiples: 9 and 12

Show answer

36

TRY IT 9.2

Find the least common multiple by listing multiples: 18 and 24

Show answer

72

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM . So 36 is the least common multiple of 12 and 18

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

EXAMPLE 11

Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

Solution

Find the primes of 24 and 36. Match primes vertically when possible.Bring down all columns.
Multiply the factors.
	The LCM of 24 and 36 is 72.

TRY IT 11.1

Find the LCM using the prime factors method: 21 and 28

Show answer

84

TRY IT 11.2

Find the LCM using the prime factors method: 24 and 32

Show answer

96

Key Concepts

• Place Value as in Figure 2.
• Name a Whole Number in Words
  1. Start at the left and name the number in each period, followed by the period name.
  2. Put commas in the number to separate the periods.
  3. Do not name the ones period.
• Write a Whole Number Using Digits
  1. Identify the words that indicate periods. (Remember the ones period is never named.)
  2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
  3. Name the number in each period and place the digits in the correct place value position.
• Round Whole Numbers
  1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
  2. Underline the digit to the right of the given 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. Replace all digits to the right of the given place value with zeros.
• Divisibility Tests: A number is divisible by:
  • 2 if the last digit is 0, 2, 4, 6, or 8.
  • 3 if the sum of the digits is divisible by 3.
  • 5 if the last digit is 5 or 0.
  • 6 if it is divisible by both 2 and 3.
  • 10 if it ends with 0.
• Find the Prime Factorization of a Composite Number
  1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
  3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. Write the composite number as the product of all the circled primes.
• Find the Least Common Multiple by Listing Multiples
  1. List several multiples of each number.
  2. Look for the smallest number that appears on both lists.
  3. This number is the LCM.
• Find the Least Common Multiple Using the Prime Factors Method
  1. Write each number as a product of primes.
  2. List the primes of each number. Match primes vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.

Glossary

composite number

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

counting numbers

The counting numbers are the numbers 1, 2, 3, …

divisible by a number

If a number is a multiple of , then is divisible by . (If 6 is a multiple of 3, then 6 is divisible by 3.)

factors

If , then are factors of . Since 3 · 4 = 12, then 3 and 4 are factors of 12.

least common multiple

The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

multiple of a number

A number is a multiple of n if it is the product of a counting number and n.

number line

A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.

origin

The origin is the point labeled 0 on a number line.

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

prime number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

whole numbers

The whole numbers are the numbers 0, 1, 2, 3, ….

Practice Makes Perfect

Use Place Value with Whole Numbers

In the following exercises, find the place value of each digit in the given numbers.

1. 51,493 a) 1 b) 4 c) 9 d) 5 e) 3	2. 87,210 a) 2 b) 8 c) 0 d) 7 e) 1
3. 164,285 a) 5 b) 6 c) 1 d) 8 e) 2	4. 395,076 a) 5 b) 3 c) 7 d) 0 e) 9
5. 93,285,170 a) 9 b) 8 c) 7 d) 5 e) 3	6. 36,084,215 a) 8 b) 6 c) 5 d) 4 e) 3
7. 7,284,915,860,132 a) 7 b) 4 c) 5 d) 3 e) 0	8. 2,850,361,159,433 a) 9 b) 8 c) 6 d) 4 e) 2

In the following exercises, name each number using words.

9. 1,078	10. 5,902
11. 364,510	12. 146,023
13. 5,846,103	14. 1,458,398
15. 37,889,005	16. 62,008,465

In the following exercises, write each number as a whole number using digits.

17. four hundred twelve	18. two hundred fifty-three
19. thirty-five thousand, nine hundred seventy-five	20. sixty-one thousand, four hundred fifteen
21. eleven million, forty-four thousand, one hundred sixty-seven	22. eighteen million, one hundred two thousand, seven hundred eighty-three
23. three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen	24. eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

In the following, round to the indicated place value.

25. Round to the nearest ten. a) 386 b) 2,931	26. Round to the nearest ten. a) 792 b) 5,647
27. Round to the nearest hundred. a) 13,748 b) 391,794	28. Round to the nearest hundred. a) 28,166 b) 481,628
29. Round to the nearest ten. a) 1,492 b) 1,497	30. Round to the nearest ten. a) 2,791 b) 2,795
31. Round to the nearest hundred. a) 63,994 b) 63,940	32. Round to the nearest hundred. a) 49,584 b) 49,548

In the following exercises, round each number to the nearest a) hundred, b) thousand, c) ten thousand.

33. 392,546	34. 619,348
35. 2,586,991	36. 4,287,965

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10

37. 84	38. 9,696
39. 75	40. 78
41. 900	42. 800
43. 986	44. 942
45. 350	46. 550
47. 22,335	48. 39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

49. 86	50. 78
51. 132	52. 455
53. 693	54. 400
55. 432	56. 627
57. 2,160	58. 2,520

In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.

59. 8, 12	60. 4, 3
61. 12, 16	62. 30, 40
63. 20, 30	64. 44, 55

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

65. 8, 12	66. 12, 16
67. 28, 40	68. 84, 90
69. 55, 88	70. 60, 72

Everyday Math

71. Writing a Check Jorge bought a car for $24,493. He paid for the car with a check. Write the purchase price in words.	72. Writing a Check Marissa’s kitchen remodeling cost $18,549. She wrote a check to the contractor. Write the amount paid in words.
73. Buying a Car Jorge bought a car for $24,493. Round the price to the nearest a) ten b) hundred c) thousand; and d) ten-thousand.	74. Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549, Round the cost to the nearest a) ten b) hundred c) thousand and d) ten-thousand.
75. Population The population of China was 1,339,724,852 on November 1, 2010. Round the population to the nearest a) billion b) hundred-million; and c) million.	76. Astronomy The average distance between Earth and the sun is 149,597,888 kilometres. Round the distance to the nearest a) hundred-million b) ten-million; and c) million.
77. Grocery Shopping Hot dogs are sold in packages of 10, but hot dog buns come in packs of eight. What is the smallest number that makes the hot dogs and buns come out even?	78. Grocery Shopping Paper plates are sold in packages of 12 and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

Writing Exercises

79. What is the difference between prime numbers and composite numbers?	80. Give an everyday example where it helps to round numbers.
81. Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.

Answers

1. a) thousands b) hundreds c) tens d) ten thousands e) ones	3. a) ones b) ten thousands c) hundred thousands d) tens e) hundreds	5. a) ten millions b) ten thousands c) tens d) thousands e) millions
7. a) trillions b) billions c) millions d) tens e) thousands	9. one thousand, seventy-eight	11. three hundred sixty-four thousand, five hundred ten
13. five million, eight hundred forty-six thousand, one hundred three	15. thirty-seven million, eight hundred eighty-nine thousand, five	17. 412
19. 35,975	21. 11,044,167	23. 3,226,512,017
25. a) 390 b) 2,930	27. a) 13,700 b) 391,800	29. a) 1,490 b) 1,500
31. a) 64,000 b) 63,900	33. a) 392,500 b) 393,000 c) 390,000	35. a) 2,587,000 b) 2,587,000 c) 2,590,000
37. divisible by 2, 3, and 6	39. divisible by 3 and 5	41. divisible by 2, 3, 5, 6, and 10
43. divisible by 2	45. divisible by 2, 5, and 10	47. divisible by 3 and 5
49.	51.	53.
55.	57.	59. 24
61. 48	63. 60	65. 24
67. 420	69. 440	71. twenty-four thousand, four hundred ninety-three dollars
73. a) $24,490 b) $24,500 c) $24,000 d) $20,000	75. a) 1,000,000,000 b) 1,300,000,000 c) 1,340,000,000	77. 40
79. Answers may vary.	81. Answers may vary.

Attributions

This chapter has been adapted from “Introduction to Whole 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 Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is years old and Alex is , so Alex is years older than Greg. When Greg was , Alex was . When Greg is , Alex will be . No matter what Greg’s age is, Alex’s age will always be years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age . Then we could use to represent Alex’s age. See the table below.

Greg’s age	Alex’s age

Letters are used to represent variables. Letters often used for variables are .

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In 1.1 Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Operation	Notation	Say:	The result is…
Addition			the sum of and
Subtraction			the difference of and
Multiplication			The product of and
Division		divided by	The quotient of and

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

• The sum of and means add plus , which we write as .
• The difference of and means subtract minus , which we write as .
• The product of and means multiply times , which we can write as .
• The quotient of and means divide by , which we can write as .

EXAMPLE 1

Translate from algebra to words:

1. 
2. 
3. 
4. 

Solution

a.

12 plus 14

the sum of twelve and fourteen

b.

30 times 5

the product of thirty and five

c.

64 divided by 8

the quotient of sixty-four and eight

d.

minus

the difference of and

When two quantities have the same value, we say they are equal and connect them with an equal sign.

When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic Notation	Say
	is equal to
	is not equal to
<	is less than
>	is greater than
	is less than or equal to
	is greater than or equal to

Symbols < and >

The symbols < and > each have a smaller side and a larger side.

smaller side < larger side
larger side > smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

EXAMPLE 2

Translate from algebra to words:

1. 
2. 
3. >
4. <

Solution

a.

20 is less than or equal to 35

b.

11 is not equal to 15 minus 3

c.

>

9 is greater than 10 divided by 2

d.

<

plus 2 is less than 10

TRY IT 2.1

Translate from algebra to words.

1. 
2. 
3. >
4. <

Show Answer

1. fourteen is less than or equal to twenty-seven
2. nineteen minus two is not equal to eight
3. twelve is greater than four divided by two
4. x minus seven is less than one

EXAMPLE 3

The information in (Figure 1) compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol < >. in each expression to compare the fuel economy of the cars.

Figure 1

1. MPG of Prius_____ MPG of Mini Cooper
2. MPG of Versa_____ MPG of Fit
3. MPG of Mini Cooper_____ MPG of Fit
4. MPG of Corolla_____ MPG of Versa
5. MPG of Corolla_____ MPG of Prius

Solution

a.
	MPG of Prius____MPG of Mini Cooper
Find the values in the chart.	48____27
Compare.	48 > 27
	MPG of Prius > MPG of Mini Cooper

b.
	MPG of Versa____MPG of Fit
Find the values in the chart.	26____27
Compare.	26 < 27
	MPG of Versa < MPG of Fit

c.
	MPG of Mini Cooper____MPG of Fit
Find the values in the chart.	27____27
Compare.	27 = 27
	MPG of Mini Cooper = MPG of Fit

d.
	MPG of Corolla____MPG of Versa
Find the values in the chart.	28____26
Compare.	28 > 26
	MPG of Corolla > MPG of Versa

e.
	MPG of Corolla____MPG of Prius
Find the values in the chart.	28____48
Compare.	28 < 48
	MPG of Corolla < MPG of Prius

TRY IT 3.1

Use Figure 1 to fill in the appropriate < >.

1. MPG of Prius_____MPG of Versa
2. MPG of Mini Cooper_____ MPG of Corolla

Show Answer

1. >
2. <

TRY IT 3.2

Use Figure 1 to fill in the appropriate < >.

1. MPG of Fit_____ MPG of Prius
2. MPG of Corolla _____ MPG of Fit

Show Answer

1. <
2. <

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.

Common Grouping Symbols

parentheses
brackets
braces

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:

Expression	Words	Phrase
		the sum of three and five
	minus one	the difference of and one
		the product of six and seven
	divided by	the quotient of and

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

Equation	Sentence
	The sum of three and five is equal to eight.
	minus one equals fourteen.
	The product of six and seven is equal to forty-two.
	is equal to fifty-three.
	plus nine is equal to two minus three.

Expressions and Equations

An expression is a number, a variable, or a combination of numbers and variables and operation symbols.

An equation is made up of two expressions connected by an equal sign.

EXAMPLE 4

Determine if each is an expression or an equation:

1. 
2. 
3. 
4. 

Solution

a.	This is an equation—two expressions are connected with an equal sign.
b.	This is an expression—no equal sign.
c.	This is an expression—no equal sign.
d.	This is an equation—two expressions are connected with an equal sign.

TRY IT 4.1

Determine if each is an expression or an equation:

1. 
2. 

Show Answer

1. equation
2. expression

TRY IT 4.2

Determine if each is an expression or an equation:

1. 
2. 

Show Answer

1. expression
2. equation

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify we’d first multiply to get and then add the to get . A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

Suppose we have the expression . We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write as and as . In expressions such as , the is called the base and the is called the exponent. The exponent tells us how many factors of the base we have to multiply.

We say is in exponential notation and is in expanded notation.

Exponential Notation

For any expression is a factor multiplied by itself times if is a positive integer.

The expression is read to the power.

For powers of and , we have special names.

The table below lists some examples of expressions written in exponential notation.

Exponential Notation	In Words
	to the second power, or squared
	to the third power, or cubed
	to the fourth power
	to the fifth power

EXAMPLE 5

Write each expression in exponential form:

1. 
2. 
3. 
4. 

Solution

a. The base 16 is a factor 7 times.
b. The base 9 is a factor 5 times.
c. The base is a factor 4 times.
d. The base is a factor 8 times.

TRY IT 5.1

Write each expression in exponential form:

Show Answer

41⁵

TRY IT 5.2

Write each expression in exponential form:

Show Answer

7⁹

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

EXAMPLE 7

Simplify: .

Solution

Expand the expression.
Multiply left to right.
Multiply.

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

EXAMPLE 8

Simplify the expressions:

1. 
2. 

Solution

a.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.
Add.

b.
Are there any parentheses? Yes.
Simplify inside the parentheses.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.

TRY IT 8.1

Simplify the expressions:

1. 
2. 

Show Answer

1. 2
2. 14

TRY IT 8.2

Simplify the expressions:

1. 
2. 

Show Answer

1. 35
2. 99

EXAMPLE 9

Simplify:

1. 
2. 

Solution

a.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right. Divide.
Multiply.

b.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right.
Multiply.
Divide.

TRY IT 9.1

Simplify:

Show Answer

18

TRY IT 9.2

Simplify:

Show Answer

9

EXAMPLE 10

Simplify: .

Solution

Parentheses? Yes, subtract first.
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
Any other multiplication or division? Yes.
Multiply.
Any other multiplication or division? No.
Any addition or subtraction? Yes.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

EXAMPLE 11

.

Solution

Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets.
Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition or subtraction? Yes.
Add.
Add.

EXAMPLE 12

Simplify: .

Solution

If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.
Divide.
Add.
Subtract.

TRY IT 12.1

Simplify:

Show Answer

81

TRY IT 12.2

Simplify:

Show Answer

75

ACCESS ADDITIONAL ONLINE RESOURCES

Key Concepts

Operation	Notation	Say:	The result is…
Addition			the sum of and
Multiplication			The product of and
Subtraction			the difference of and
Division		divided by	The quotient of and

• Equality Symbol
  • is read as is equal to
  • The symbol is called the equal sign.
• Inequality
  • < is read is less than
  • is to the left of on the number line
    
  • > is read is greater than
  • is to the right of on the number line
    

Algebraic Notation	Say
	is equal to
	is not equal to
<	is less than
>	is greater than
	is less than or equal to
	is greater than or equal to

• Exponential Notation
  • For any expression is a factor multiplied by itself times, if is a positive integer.
  • means multiply factors of
    
  • The expression of is read to the power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

• Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
• Exponents: Simplify all expressions with exponents.
• Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
• Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

Glossary

Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

1.	2.
3.	4.
5.	6.
7.	8.
9.	10.
11. <	12. <
13.	14.
15.	16.
17. >	18. >
19.	20.
21.	22.

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

23.	24.
25.	26.
27.	28.
29.	30.

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

31.	32.
33.	34.

In the following exercises, write in expanded form.

35.	36.
37.	38.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

39. a. b.	40. a. b.
41.	42.
43.	44.
45.	46.
47.	48.
49.	50.
51.	52.
53.	54.
55.	56.
57.	58.
59.	60.
61.	62.
63.	64.

Everyday Math

65. Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol ( = ,<,  >). Spurs Height Heat Height Tim Duncan Rashard Lewis Boris Diaw LeBron James Kawhi Leonard Chris Bosh Tony Parker Dwyane Wade Danny Green Ray Allen Height of Tim Duncan____Height of Rashard Lewis Height of Boris Diaw____Height of LeBron James Height of Kawhi Leonard____Height of Chris Bosh Height of Tony Parker____Height of Dwyane Wade Height of Danny Green____Height of Ray Allen

66. Elevation In Colorado there are more than mountains with an elevation of over The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol. Mountain Elevation Mt. Elbert Mt. Massive Mt. Harvard Blanca Peak La Plata Peak Uncompahgre Peak Crestone Peak Mt. Lincoln Grays Peak Mt. Antero Elevation of La Plata Peak____Elevation of Mt. Antero Elevation of Blanca Peak____Elevation of Mt. Elbert Elevation of Gray’s Peak____Elevation of Mt. Lincoln Elevation of Mt. Massive____Elevation of Crestone Peak Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

Writing Exercises

Answers

1. 16 minus 9, the difference of sixteen and nine	3. 5 times 6, the product of five and six	5. 28 divided by 4, the quotient of twenty-eight and four
7. x plus 8, the sum of x and eight	9. 2 times 7, the product of two and seven	11. fourteen is less than twenty-one
13. thirty-six is greater than or equal to nineteen	15. 3 times n equals 24, the product of three and n equals twenty-four	17. y minus 1 is greater than 6, the difference of y and one is greater than six
19. 2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six	21. a is not equal to 7 times 4, a is not equal to the product of seven and four	23. equation
25. expression	27. expression	29. equation
31. 37	33. x5	35. 125
37. 256	39. a. 43 b. 55	41. 5
43. 34	45. 58	47. 6
49. 13	51. 4	53. 35
55. 10	57. 41	59. 81
61. 149	63. 50	65. a. > b. c. < d. < e. >
67. Answer may vary.

Attributions

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

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

EXAMPLE 1

Evaluate when

1.  
2.  

Solution

a. To evaluate, substitute for in the expression, and then simplify.

Substitute.
Add.

When , the expression has a value of .

b. To evaluate, substitute for in the expression, and then simplify.

Substitute.
Add.

When , the expression has a value of .

Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for were different. When we evaluate an expression, the value varies depending on the value used for the variable.

EXAMPLE 2

Evaluate

1.  
2.  

Solution

Remember means times , so means times .

a. To evaluate the expression when , we substitute for , and then simplify.

Multiply.
Subtract.

b. To evaluate the expression when , we substitute for , and then simplify.

Multiply.
Subtract.

Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.

EXAMPLE 3

Evaluate when .

Solution

We substitute for , and then simplify the expression.

Use the definition of exponent.
Multiply.

When , the expression has a value of .

TRY IT 3.1

Evaluate:

.

Show Answer

64

TRY IT 3.2

Evaluate:

.

Show Answer

216

EXAMPLE 4

.

Solution

In this expression, the variable is an exponent.

Use the definition of exponent.
Multiply.

When , the expression has a value of .

EXAMPLE 5

.

Solution

This expression contains two variables, so we must make two substitutions.

Multiply.
Add and subtract left to right.

When and , the expression has a value of .

TRY IT 5.1

Evaluate:

Show Answer

33

TRY IT 5.2

Evaluate:

Show Answer

10

EXAMPLE 6

.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .

Simplify .
Multiply.
Add.

TRY IT 6.1

Evaluate:

.

Show Answer

40

TRY IT 6.2

Evaluate:

.

Show Answer

9

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are .

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term is . When we write , the coefficient is , since . The table below gives the coefficients for each of the terms in the left column.

Term	Coefficient

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression	Terms

Which of these terms are like terms?

• The terms and are both constant terms.
• The terms and are both terms with .
• The terms and both have .

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms ,

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think would simplify to? If you thought , you would be right!

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what is. If you have of something and add more of the same thing, the result is of them. For example, oranges plus oranges is oranges. We will discuss the mathematical properties behind this later.

The expression has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

Now it is easier to see the like terms to be combined.

EXAMPLE 9

Simplify the expression: .

Solution

Identify the like terms.
Rearrange the expression, so the like terms are together.
Add the coefficients of the like terms.
The original expression is simplified to…

TRY IT 9.1

Simplify:

Show Answer

16x + 17

TRY IT 9.2

Simplify:

Show Answer

17y + 7

EXAMPLE 10

Simplify the expression: .

Solution

Identify the like terms.
Rearrange the expression so like terms are together.
Add the coefficients of the like terms.

These are not like terms and cannot be combined. So is in simplest form.

Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in the table below.

Operation	Phrase	Expression
Addition	plus the sum of and increased by more than the total of and added to
Subtraction	minus the difference of and subtracted from decreased by less than
Multiplication	times the product of and	, , ,
Division	divided by the quotient of and the ratio of and divided into	, , ,

Look closely at these phrases using the four operations:

• the sum of and
• the difference of and
• the product of and
• the quotient of and

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

How old will you be in eight years? What age is eight more years than your age now? Did you add to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract from your present age. Seven less than means seven subtracted from your present age.

EXAMPLE 14

The height of a rectangular window is inches less than the width. Let represent the width of the window. Write an expression for the height of the window.

Solution

Write a phrase about the height.	less than the width
Substitute for the width.	less than
Rewrite ‘less than’ as ‘subtracted from’.	subtracted from
Translate the phrase into algebra.

EXAMPLE 15

Blanca has dimes and quarters in her purse. The number of dimes is less than times the number of quarters. Let represent the number of quarters. Write an expression for the number of dimes.

Solution

Write a phrase about the number of dimes.	two less than five times the number of quarters
Substitute for the number of quarters.	less than five times
Translate times .	less than
Translate the phrase into algebra.

TRY IT 15.1

Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let represent the number of quarters. Write an expression for the number of dimes.

Show Answer

6q − 7

TRY IT 15.2

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let represent the number of nickels. Write an expression for the number of dimes.

Show Answer

4n + 8

Key Concepts

• Combine like terms.
  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add the coefficients of the like terms

Glossary

Practice Makes Perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

1.	2.
3.	4.
5.	6.
7.	8.
9.	10.
11.	12.
13.	14.
15.	16.
17.	18.
19.	20.

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

21.	22.
23.	24.

In the following exercises, identify the coefficient of the given term.

25.	26.
27.	28.

In the following exercises, identify all sets of like terms.

29.	30.
31.	32.

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

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

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

49. The sum of 8 and 12	50. The sum of 9 and 1
51. The difference of 14 and 9	52. 8 less than 19
53. The product of 9 and 7	54. The product of 8 and 7
55. The quotient of 36 and 9	56. The quotient of 42 and 7
57. The difference of and	58. less than
59. The product of and	60. The product of and
61. The sum of and	62. The sum of and
63. The quotient of and	64. The quotient of and
65. Eight times the difference of and nine	66. Seven times the difference of and one
67. Five times the sum of and	68.  times five less than twice

In the following exercises, write an algebraic expression.

69. Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let represent the cost of the blouse. Write an expression for the cost of the skirt.	70. Eric has rock and classical CDs in his car. The number of rock CDs is more than the number of classical CDs. Let represent the number of classical CDs. Write an expression for the number of rock CDs.
71. The number of girls in a second-grade class is less than the number of boys. Let represent the number of boys. Write an expression for the number of girls.	72. Marcella has fewer male cousins than female cousins. Let represent the number of female cousins. Write an expression for the number of boy cousins.
73. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let represent the number of nickels. Write an expression for the number of pennies.	74. Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let represent the number of tens. Write an expression for the number of fives.

Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

Writing Exercises

77. Explain why “the sum of x and y” is the same as “the sum of y and x,” but “the difference of x and y” is not the same as “the difference of y and x.” Try substituting two random numbers for and to help you explain.

78. Explain the difference between times the sum of and and “the sum of times and

Answers

1. 22	3. 26	5. 144
7. 32	9. 27	11. 21
13. 41	15. 9	17. 73
19. 54	21. 15x2, 6x, 2	23. 10y3, y, 2
25. 8	27. 5	29. x3, 8x3 and 14, 5
31. 16ab and 4ab; 16b2 and 9b2	33. 13x	35. 26a
37. 7c	39. 12x + 8	41. 10u + 3
43. 12p + 10	45. 22a + 1	47. 17x2 + 20x + 16
49. 8 + 12	51. 14 − 9	53. 9 ⋅ 7
55. 36 ÷ 9	57. x − 4	59. 6y
61. 8x + 3x	63.	65. 8 (y − 9)
67. 5 (x + y)	69. b + 15	71. b − 4
73. 2n − 7	75. He will pay $750. His insurance company will pay $1350.	77. Answers will vary.

Attributions

This chapter has been adapted from “Evaluate, Simplify, and Translate Expressions” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

Use Negatives and Opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than . The negative numbers are to the left of zero on the number line. See Figure 1.

Figure 1 The number line shows the location of positive and negative numbers.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See Figure 2.

Figure 2 The numbers on a number line increase in value going from left to right and decrease in value going from right to left.

Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in Figure 3 are called the integers. The integers are the numbers

Figure 3 All the marked numbers are called integers.

EXAMPLE 1

Order each of the following pairs of numbers, using < or >: a) ___ b) ___ c) ___ d) ___.

Solution

It may be helpful to refer to the number line shown.

a) 14 is to the right of 6 on the number line.	___  >
b) −1 is to the left of 9 on the number line.	___
c) −1 is to the right of −4 on the number line.	___
d) 2 is to the right of −20 on the number line.	___ >

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and are the same distance from zero, they are called opposites. The opposite of 2 is , and the opposite of is 2

(Figure 4) illustrates the definition.

The opposite of 3 is .

Figure 4

Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.

	Between two numbers, it indicates the operation of subtraction. We read as “10 minus 4.”
	In front of a number, it indicates a negative number. We read −8 as “negative eight.”
	In front of a variable, it indicates the opposite. We read as “the opposite of .”
	Here there are two “−” signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2. We read as “the opposite of negative two.”

Opposite Notation

means the opposite of the number a.

The notation is read as “the opposite of a.”

EXAMPLE 2

Find: a) the opposite of 7 b) the opposite of c) .

Solution

a) −7 is the same distance from 0 as 7, but on the opposite side of 0.	The opposite of 7 is −7.
b) 10 is the same distance from 0 as −10, but on the opposite side of 0.	The opposite of −10 is 10.
c) −(−6)	The opposite of −(−6) is −6.

Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers. The integers are the numbers

Integers

The whole numbers and their opposites are called the integers.

The integers are the numbers

When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether is positive or negative. We can see this in Example 3.

Simplify: Expressions with Absolute Value

We saw that numbers such as are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

• units away from , so .
• units away from , so .

Figure 5 illustrates this idea.

The integers units away from .

Figure 5

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from on the number line is zero units.

Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.

In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!

EXAMPLE 5

Fill in <, >, for each of the following pairs of numbers:

a) ___ b) ___c) ___ d) –___

Solution

	___ –
a) Simplify. Order.	5 ___ -5
	5  > -5
	> –
b) Simplify. Order.	___ –
	8 ___ -8
	8 > -8
	8 > –
c) Simplify. Order.	9 ___ – -9 ___ -9 -9 = -9 -9 = –
d) Simplify. Order.	– ___ – 16 ____ -16 16 > -16 – > –

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

EXAMPLE 6

Simplify: .

Solution

Work inside parentheses first: subtract 2 from 6.
Multiply 3(4).
Subtract inside the absolute value bars.
Take the absolute value.
Subtract.

EXAMPLE 7

Evaluate: a) b) c) d) .

Solution

a)

Take the absolute value.	35

b)

Simplify.
Take the absolute value.	20

c)

Take the absolute value.

d)

Take the absolute value.

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two colour counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one colour (blue) represent positive. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add the four addition facts using the numbers and .

To add , we realize that means the sum of 5 and 3

We start with 5 positives.
And then we add 3 positives.
We now have 8 positives. The sum of 5 and 3 is 8.

Now we will add . Watch for similarities to the last example .

To add , we realize this means the sum of .

We start with 5 negatives.
And then we add 3 negatives.
We now have 8 negatives. The sum of −5 and −3 is −8.

In what ways were these first two examples similar?

• The first example adds 5 positives and 3 positives—both positives.
• The second example adds 5 negatives and 3 negatives—both negatives.

In each case we got 8—either 8 positives or 8 negatives.

When the signs were the same, the counters were all the same color, and so we added them.

EXAMPLE 8

Add: a)  b) .

Solution

a)

1 positive plus 4 positives is 5 positives.

 

b)

1 negative plus 4 negatives is 5 negatives.

So what happens when the signs are different? Let’s add . We realize this means the sum of and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other.

	−5 + 3 means the sum of −5 and 3.
We start with 5 negatives.
And then we add 3 positives.
We remove any neutral pairs.
We have 2 negatives left.
The sum of −5 and 3 is −2.	−5 + 3 = −2

Notice that there were more negatives than positives, so the result was negative.

Let’s now add the last combination, .

	5 + (−3) means the sum of 5 and −3.
We start with 5 positives.
And then we add 3 negatives.
We remove any neutral pairs.
We have 2 positives left.
The sum of 5 and −3 is 2.	5 + (−3) = 2

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

EXAMPLE 9

Add: a) b) .

Solution

a)

	−1 + 5
There are more positives, so the sum is positive.	4

b)

	1 + (−5)
There are more negatives, so the sum is negative.	−4

Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.

When you need to add numbers such as , you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because , there are 16 more red counters.

Therefore, the sum of is .

Let’s try another one. We’ll add . Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is .

Let’s look again at the results of adding the different combinations of and .

Addition of Positive and Negative Integers

When the signs are the same, the counters would be all the same color, so add them.

When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left.

Visualize the model as you simplify the expressions in the following examples.

The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!

EXAMPLE 11

Simplify: .

Solution

Simplify inside the parentheses.
Multiply.
Add left to right.

Subtract Integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read as take away When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers and .

To subtract , we restate the problem as take away

We start with 5 positives.
We ‘take away’ 3 positives.
We have 2 positives left.
The difference of 5 and 3 is 2.	2

Now we will subtract . Watch for similarities to the last example .

To subtract , we restate this as take away

We start with 5 negatives.
We ‘take away’ 3 negatives.
We have 2 negatives left.
The difference of −5 and −3 is −2.	−2

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

EXAMPLE 12

Subtract: a)  b) .

Solution

a) Take 5 positive from 7 positives and get 2 positives.
b) Take 5 negatives from 7 negatives and get 2 negatives.

What happens when we have to subtract one positive and one negative number? We’ll need to use both white and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

• To subtract , we restate it as take away 3.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

	−5 − 3 means −5 take away 3.
We start with 5 negatives.
We now add the neutrals needed to get 3 positives.
We remove the 3 positives.
We are left with 8 negatives.
The difference of −5 and 3 is −8.	−5 − 3 = −8

And now, the fourth case, . We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.

	5 − (−3) means 5 take away −3.
We start with 5 positives.
We now add the needed neutrals pairs.
We remove the 3 negatives.
We are left with 8 positives.
The difference of 5 and −3 is 8.	5 − (−3) = 8

EXAMPLE 13

Subtract: a)  b) .

Solution

a)

Take 1 positive from the one added neutral pair.

−3 − 1 −4

b)

Take 1 negative from the one added neutral pair.

3 − (−1) 4

TRY IT 13.1

Subtract: a)  b) .

Show answer

a) b) 10

TRY IT 13.2

Subtract: a)  b) .

Show answer

a) b) 11

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In Example 13, is the same as and is the same as . You will often see this idea, the subtraction property, written as follows:

Subtraction Property

Subtracting a number is the same as adding its opposite.

Look at these two examples.

.

Of course, when you have a subtraction problem that has only positive numbers, like , you just do the subtraction. You already knew how to subtract long ago. But knowing that gives the same answer as helps when you are subtracting negative numbers. Make sure that you understand how and give the same results!

EXAMPLE 14

Simplify: a)  and  b)  and .

Solution

a) Subtract.
b) Subtract.

Look at what happens when we subtract a negative.

Subtracting a negative number is like adding a positive!

You will often see this written as .

Does that work for other numbers, too? Let’s do the following example and see.

EXAMPLE 15

Simplify: a)  and  b)  and .

Solution

a)

b)

a) Subtract.
b) Subtract.

Let’s look again at the results of subtracting the different combinations of and .

What happens when there are more than three integers? We just use the order of operations as usual.

EXAMPLE 16

Simplify: .

Solution

Simplify inside the parentheses first.
Subtract left to right.
Subtract.

Key Concepts

• Addition of Positive and Negative Integers
  
• Property of Absolute Value: for all numbers. Absolute values are always greater than or equal to zero!
• Subtraction of Integers
  
• Subtraction Property: Subtracting a number is the same as adding its opposite.

Glossary

Practice Makes Perfect

Use Negatives and Opposites of Integers

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

1. a) ___ b) ___ c) ___ d) ___

2. a) ___ b) ___ c) ___ d) ___

In the following exercises, find the opposite of each number.

3. a) 2 b)

4. a) 9 b)

In the following exercises, simplify.

5.	6.
7.	8.

In the following exercises, evaluate.

9. when a) b)

10. when a) b)

Simplify Expressions with Absolute Value

In the following exercises, simplify.

11. a) b) c)

12. a) b) c)

In the following exercises, fill in <, >, or for each of the following pairs of numbers.

13. a) ___ b) ___

14. a) ___ b) ___

In the following exercises, simplify.

15.	16.
17.	18.
19.	20.
21.	22.

In the following exercises, evaluate.

23. a) b)

24. a) b)

Add Integers

In the following exercises, simplify each expression.

25.	26.
27.	28.
29.	30.
31.	32. 6
33.	34.

Subtract Integers

In the following exercises, simplify.

35.	36.
37.	38.
39.	40.
41. a) b)	42. a) b)
43. a) b)	44. a) b)

In the following exercises, simplify each expression.

45.	46.
47.	48.
49.	50.
51.	52.
53.	54.
55.	56.
57.	58.
59.	60.
61.	62.
63.	64.
65.	66.
67.	68.
69.	70.

Everyday Math

71. Elevation The highest elevation in North America is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level. Use integers to write the elevation of: a) Mount McKinley. b) Death Valley.	72. Extreme temperatures The highest recorded temperature on Earth was ° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was ° below ° Celsius, recorded in Antarctica in 1983 Use integers to write the: a) highest recorded temperature. b) lowest recorded temperature.
73. Provincial budgets For 2019 the province of Quebec estimated it would have a budget surplus of $5.6 million. That same year, Alberta estimated it would have a budget deficit of $7.5 million. Use integers to write the budget of: a) Quebec. b) Alberta.	74. University enrolmentsThe number of international students enrolled in Canadian postsecondary institutions has been on the rise for two decades, with their numbers increasing at a higher rate than that of Canadian students. Enrolments of international students rose by 24,315 from 2015 to 2017. Meanwhile, there was a slight decline in the number of Canadian students, by 912 for the same fiscal years. Use integers to write the change: a) in International Student enrolment from Fall 2015 to Fall 2017. b) in Canadian student enrolment from Fall 2015 to Fall 2017.
75. Stock Market The week of September 15, 2008 was one of the most volatile weeks ever for the US stock market. The closing numbers of the Dow Jones Industrial Average each day were: Monday Tuesday Wednesday Thursday Friday What was the overall change for the week? Was it positive or negative?	76. Stock Market During the week of June 22, 2009, the closing numbers of the Dow Jones Industrial Average each day were: Monday Tuesday Wednesday Thursday Friday What was the overall change for the week? Was it positive or negative?

Writing Exercises

77. Give an example of a negative number from your life experience.	78. What are the three uses of the sign in algebra? Explain how they differ.
79. Explain why the sum of and 2 is negative, but the sum of 8 and is positive.	80. Give an example from your life experience of adding two negative numbers.

Answers

1. a) > b) < c) < d) >	3. a)b) 6	5. 4
7. 15	9. a)b) 12	11. a) 32 b) 0 c) 16
13. a) < b)	15.	17. 56
19. 0	21. 8	23. a) b)
25.	27. 32	29.
31. 108	33. 29	35. 6
37.	39. 12	41. a) 16 b) 16
43. a) 45 b) 45	45. 27	47.
49.	51.	53. -99
55.	57.	59. 22
61.	63. 0	65. 6
67.	69.	71. a) 20,320 b)
73. a) $5.6 million b) million	75.	77. Answers may vary
79. Answers may vary

Attributions

This chapter has been adapted from “Add and Subtract Integers” 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.

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that means add a, b times. Here, we are using the model just to help us discover the pattern.

The next two examples are more interesting.

What does it mean to multiply 5 by It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

In summary:

Notice that for multiplication of two signed numbers, when the:

• signs are the same, the product is positive.
• signs are different, the product is negative.

We’ll put this all together in the chart below

Multiplication of Signed Numbers

For multiplication of two signed numbers:

Same signs	Product	Example
Two positives Two negatives	Positive Positive

Different signs	Product	Example
Positive \cdot negative Negative \cdot positive	Negative Negative

EXAMPLE 1

Multiply: a) b) c) d) .

Solution

a) Multiply, noting that the signs are different so the product is negative.
b) Multiply, noting that the signs are the same so the product is positive.
c) Multiply, with different signs.
d) Multiply, with same signs.

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by ? Let’s multiply a positive number and then a negative number by to see what we get.

Each time we multiply a number by , we get its opposite!

Multiplication by

Multiplying a number by gives its opposite.

EXAMPLE 2

Multiply: a) b) .

Solution

a) Multiply, noting that the signs are different so the product is negative.
b) Multiply, noting that the signs are the same so the product is positive.

TRY IT 2.1

Multiply: a) b) .

Show answer

a) b) 17

TRY IT 2.2

Multiply: a) b) .

Show answer

a) b) 16

Divide Integers

What about division? Division is the inverse operation of multiplication. So, because . In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

• signs are the same, the quotient is positive.
• signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

Multiplication and Division of Signed Numbers

For multiplication and division of two signed numbers:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.

Same signs	Result
Two positives	Positive
Two negatives	Positive

If the signs are the same, the result is positive.

Different signs	Result
Positive and negative	Negative
Negative and positive	Negative

If the signs are different, the result is negative.

EXAMPLE 3

Divide: a) b) .

Solution

a) Divide. With different signs, the quotient is negative.
b) Divide. With signs that are the same, the quotient is positive.

TRY IT 3.1

Divide: a) b) .

Show answer

a) b) 39

TRY IT 3.2

Divide: a) b) .

Show answer

a) b) 23

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

EXAMPLE 4

Simplify: .

Solution

Multiply first.
Add.
Subtract.