1.1 Integers

Integers are positive and negative numbers that can be written without a fraction or decimal point. Zero is also an integer.

There are many situations in everyday life that use integers. Temperatures above freezing can be expressed with a positive integer, and temperatures below freezing can be expressed with a negative integer. When you have money in your bank account, your balance is positive, but when you go into debt, your balance is negative. As well, land elevations represented with positive integers indicate above sea level, and negative integers indicate below sea level.

In this lesson, we will review how to multiply, divide, add, and subtract integers.

Multiplying Integers

There are many ways that we can represent the multiplication of numbers. Multiplying can be represented in each of these ways:

3 × 4 = 12 3 \bullet 4 = 12 (3) × (4) = 12 (3)(4) = 12 3(4) = 12 (3)4 = 12

The result of multiplying two or more numbers together is called the product. In the calculation 3 \times 4 = 12, the product is 12. The following chart outlines the resulting sign when multiplying integers together.

Table 1.1.1
Multiplication Rules for Integers Examples
Positive × Positive = Positive

Negative × Negative = Positive

Negative × Positive = Negative

Positive × Negative = Negative

 3 \times 4 = 12

(–3) \times (–4) = 12

(–3) \times 4 = –12

3 \times (–4) = –12
Why does a negative × negative = positive? Work through the following example to explain this.
Imagine you owe $2 to several people. Look for a pattern to help you fill in the blanks.
3 × (–2) = –6 You owe 3 people $2, so $6 is owed (–6).
2 × (–2) = –4 You owe 2 people $2, so $4 is owed (–4).
1 × (–2) = –2 You owe 1 person $2, so $2 owed (–2).
0 × (–2) =   0 You owe 0 people $2, so $0 is owed (0).
(–1) × (–2) =            
            × (–2) =            
            ×             =            
Practice 1

Find the product.

a) 8\times3 b) (-3)(-3) c) 20(–3) d) –5 \bullet 4 e) –1(–2) f) –(–10)
g) (–4) \times (2) \times (–3) k) (–3) \times (5) \times (–2) \times (–1)

Dividing Integers

There are many ways that we can represent the division of numbers. Dividing can be represented in each of these ways:

12\div3=4 12 divided by 3 equal 4 using the long division symbol. \frac{12}{3}=4

The result of dividing two numbers is called the quotient. In the calculation 12\div3=4, the quotient is 4. The following chart outlines the resulting sign when dividing integers.

Table 1.1.2
Division Rules for Integers Examples
Positive \div Positive = Positive

Negative \div Negative = Positive

Negative \div Positive = Negative

Positive \div Negative = Negative

 12 \div 3 = 4

(–12) \div (–3) = 4

(–12) \div 3 = –4

12 \div (–3) = –4
Practice 2

Find the quotient.

a) 8\div(-2) b) \frac{-16}{4} c) -30\div-5 d) \frac{0}{-6} e) \frac{-12}{0} f) \frac{-8}{-8}

Adding and Subtracting Integers

We will use the example of digging holes at the beach to learn the rules of adding and subtracting integers. A pile of sand will represent a positive number and a hole will represent a negative number. Considered the beach to be equivalent to zero. When adding and subtracting integers there are four possible cases. The first two cases are shown.

Table 1.1.3
Case 1 Case 2
10 + 4 = 14

A pile of sand with a height of 10 feet (ft) plus a pile of sand with a height of 4 ft creates a pile of sand with a height of 14 ft. This means the answer is positive 14.

A pile of sand at the beach labelled with 10 and on top of that pile a pile of sand labelled with 4

 –4 – 10 = –14

A hole with a depth of 4 ft and a hole with a depth of 10 ft creates a hole with a depth of 14 ft. This means the answer is negative 14.

A hole at the beach labelled with -4 and under that hole a hole labelled with -10.
Practice 3

Evaluate.

a) 9 + 37 b) –4 – 5 c) –12 – 12 d) –5 – 3 – 2 e) –7 – 1 – 10

Before moving onto Case 3 and Case 4, we need to understand what happens when we evaluate –1 + 1 or –10 + 10 or –4 + 4.

Table 1.1.4
–4 + 4 = 0
A pile of sand at the beach labelled with 4 and to the left of the pile of sand there is a hole labelled with -4		 Fill the hole with the sand to figure out the answer. The pile is the same size as the hole, so it will fill the hole exactly to the level of the beach, which makes the answer 0.
Practice 4

Evaluate.

a) –1 + 1 b) –12 + 12 c) –2 + 2 + 5 d) –6 – 100 + 6

There are two other cases that you may encounter when adding and subtracting integers:

Table 1.1.5
Case 3 Case 4
10 – 4 = 6

If you fill the hole with sand, you are left with a pile of sand with a height of 6 ft. This means your answer is positive 6.

A pile of sand at the beach labelled with 10 and to the left of the pile of sand there is a hole labelled with -4

 –10 + 4 = –6

If you fill the hole with sand, you are left with a hole with a depth of 6 ft. This means your answer is negative 6.

A pile of sand at the beach labelled with 4 and to the left of the pile of sand there is a hole labelled with -10

To sum up, when the signs are different (one positive and one negative), subtract the numbers without their signs, and choose the sign of the number with the greater absolute value. That is, choose the sign of the number farther from the zero line (the beach).

Practice 5

Evaluate.

a) 11 – 5 b) –2 + 5 c) –2 + 12 d) 3 – 8
e) 4 – 14 f) –16 + 5 g) –6 + 5 + 2 h) 5 + 85 – 3

When adding and subtracting integers, you will sometimes encounter questions that have an addition or subtraction sign and a positive or negative sign between the two numbers. There are two possible ways this may show up in a question.

Table 1.1.6
Adding a Negative Number Subtracting a Negative Number
Example: 10 + (–4)

Adding a negative number is the same as subtracting the two numbers, so you can rewrite the question like so:

10 + (–4) = 10 – 4 = 6

 Example: 10 – (–4)

Subtracting a negative number is the same as adding the two numbers, so you can rewrite the question like so:

10 – (–4) = 10 + 4 = 14
Practice 6

Evaluate.

a) –8 + (–5) b) 12 – (–6) c) –6 – (–5) + 2 d) 16 + –50 + 10

Homework 

  1. Find the product.

    a) –5 \bullet 2 b) (–2)(8) c) (–7) \bullet (–7)
    d) –(–7)  e) 6(–8) f) 70 \times 8 =
    g) (–90)30 h) –5 \bullet –6 \bullet –8 i) (–7)(4)(–2)
    j) 4\times10\times–8

     

    		 k) (–9)(–6)(4)

     

    		 l) –2\times–3\times–1\times1\times–2\times3

     
    m) 2 \bullet –3 \bullet –1 \bullet 1 \bullet –2 \bullet 3

     

     

    		 n) (–2)(–3)(–2)(1)(–2)(–3)

     

     

    		 o) (–10)(–1)(–1)(–1)(–1)(–10)

     

     

  2. Find the quotient.

    a) \frac{-1}{1} b) 24\div-4 c) \frac{0}{50}
    d) -12\div-4 e) -560\div8 f) \frac{-6}{0}
    g) \frac{-2400}{-30} h) \frac{-12}{-12} i) 0\div-16
    j) \frac{4500}{-90} k) \frac{-700}{10} l) -5400\div(-9)
    m) -17\div1 n) -368\div0 \frac{102}{-1}

  3. Evaluate.

    a) 6 + 2 b) –8 + 6 c) 7 – 10
    d) –5 + –3 e) –9 + 9 f) –2 + (–14)
    g) –19 – 5 h) 100 – (–6) i) –35 – 6
    j) –6 – (–10) k) –16 + 3 + 16 l) –5 + 7
    m) –2 + 22 + 5 n) –7 + –1 o) 16 – 12
    p) –4 – 110 + (–5)

     

    		 q) –15 – (–2) – (–13)

     

    		 r) 5 + 15 + (–3)

     
    s) –12 – (–3) + 13

     

    		 t) 29 + (–37) + 13

     
    u) –8 – 5 – 5 – 7

     

    		 v) –12 + 15 + –34 + 12

     
    w) –22 – (–17) + –34 + 12 – 8 – (+34)

     

     

    		 x) 12 – 15 + –27 – (–3) + –11 + 13 + 14

     

     
    y) 24 + –15 + –30 + 17

     

     

    		 z) –20 – (–23) – 44 + 5 – 66 + 7

     

     
    aa) 12 + –3 + (–6) + 7

     

    		 bb) 13 – 8 + 6

     

  4. What is the result of −4 + 9 − 15 + 6?

    a) –4
    b) 6
    c) –8
    d) 0

  5. What is the result of −4 + 9 − 4 + 4 – 9 + 4 – 4 + 4?

    a) –4
    b) 4
    c) –9
    d) 0

  6. Which of the following is true when multiplying two integers with the same sign?

    a) The product is always positive.
    b) The product is always negative.
    c) The product is zero.
    d) The product depends on the size of the numbers.

  7. When dividing integers, which rule can be used to determine the sign of the quotient?

    a) If the integers that are being divided have the same sign, the quotient is negative.
    b) If one integer is positive and the other is negative, the quotient is positive.
    c) If the integers that are being divided have different signs, the quotient is positive.
    d) If the integers that are being divided have the same sign, the quotient is positive.

  8. Which of the following statements is true?

    a) Dividing any number by zero results in zero.
    b) Multiplying any number by zero results in zero.
    c) Dividing zero by any number results in zero.
    d) Dividing zero by zero is always a valid operation.

Answers

1.

a) –10 b) –16 c) 49 d) 7 e) –48
f) 560 g) –2700 h) –240 i) 56 j) –320
k) 216 l) 36 m) –36 n) –72 o) 100

2.

a) –1 b) –6 c) 0 d) 3 e) –70
f) undefined g) 80 h) 1 i) 0 j) –50
k) –70 l) 600 m) –17 n) undefined o) –102

3.

a) 8 b) –2 c) –3 d) –8 e) 0 f) –16 g) –24
h) 106 i) –41 j) 4 k) 3 l) 2 m) 25 n) –8
o) 4 p) –119 q) 0 r) 17 s) 4 t) 5 u) –25
v) –19 w) –69 x) –11 y) –4 z) –95 aa) 10 bb) 11
4. a 5. d 6. a 7. d 8. b

 

 

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

 

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