1. Operations with Real Numbers

1.2 Integers

Learning Objectives

By the end of this section it is expected that you will be able to:

  • Use negatives and opposites
  • Simplify: expressions with absolute value
  • Add ans subtract integers
  • Multiply and divide integers
  • Simplify Expressions with Integers

Use Negatives and Opposites

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 0. The negative numbers are to the left of zero on the number line. See Figure 1.

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.
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.

A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.
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.

Remember that we use the notation:

a < b (read “a is less than b”) when a is to the left of b on the number line.

a > b (read “a is greater than b”) when a is to the right of b on the number line.

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 \text{…}-3,-2,-1,0,1,2,3\text{…}

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.
Figure 3 All the marked numbers are called integers.

EXAMPLE 1

Order each of the following pairs of numbers, using < or >: a) 14___6 b) -1___9 c) -1___-4 d) 2___-20.

Solution

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

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.

a) 14 is to the right of 6 on the number line. 14___6  14 > 6
b) −1 is to the left of 9 on the number line. -1___9  -1<9
c) −1 is to the right of −4 on the number line. -1___-4
d) 2 is to the right of −20 on the number line. 2___-20 2 > -20

TRY IT 1

Order each of the following pairs of numbers, using < or > \text{:} a) 15___7 b) -2___5 c) -3___-7 d) 5___-17.

Show answer

a) > b) < c) > d) >

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 -2 are the same distance from zero, they are called opposites. The opposite of 2 is -2, and the opposite of -2 is 2

Opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

(Figure 4) illustrates the definition.

The opposite of 3 is -3.

A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.
Figure 4

 

Opposite Notation

-a means the opposite of the number a.

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

The whole numbers and their opposites are called the integers. The integers are the numbers \text{…}-3,-2,-1,0,1,2,3\text{…}

Integers

The whole numbers and their opposites are called the integers.

The integers are the numbers

\text{…}-3,-2,-1,0,1,2,3\text{…}

Simplify: Expressions with Absolute Value

We saw that numbers such as 3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-3 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.

Absolute Value

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number n is written as |n|.

For example,

  • -5\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}5 units away from 0, so |-5|=5.
  • \phantom{\rule{0.65em}{0ex}}5\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}5 units away from 0, so |5|=5.

Figure 5 illustrates this idea.

The integers 5\phantom{\rule{0.2em}{0ex}}\text{and are}\phantom{\rule{0.2em}{0ex}}5 units away from 0.

A number line is shown ranging from negative 5 to 5. A bracket labeled “5 units” lies above the points negative 5 to 0. An arrow labeled “negative 5 is 5 units from 0, so absolute value of negative 5 equals 5.” is written above the labeled bracket. A bracket labeled “5 units” lies above the points “0” to “5”. An arrow labeled “5 is 5 units from 0, so absolute value of 5 equals 5.” and is written above the labeled bracket.
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 0\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0 on the number line is zero units.

Property of Absolute Value

|n|\ge 0 for all numbers

Absolute values are always greater than or equal to zero!

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

EXAMPLE 2

Simplify: a) |3| b) |-44| c) |0|.

Solution

The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.

a) |3|
\phantom{\rule{1.2em}{0ex}}3

b) |-44|
\phantom{\rule{1.5em}{0ex}}44

c) |0|
0\phantom{\rule{1.5em}{0ex}}

TRY IT 2

Simplify: a) |4| b) |-28| c) |0|.

Show answer

a) 4 b) 28 c) 0

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

EXAMPLE 3

Fill in <, >, \phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}= for each of the following pairs of numbers:

a) |-5|___-|-5| b) 8___-|-8|c) -9___-|-9| d) –-16___-|-16|

Solution
|-5|___ –|-5|
a) Simplify.
Order.
5 ___ -5
5  > -5
|-5|  > –|-5|
b) Simplify.
Order.
8 ___ –|-8|
8 ___ -8
8 > -8
8 > –|-8|
c) Simplify.
Order.
9 ___ –|-9|

-9 ___ -9

-9 = -9

-9 = –|-9|

d) Simplify.
Order.
(-16) ___ –|-16|

16 ____ -16

16 > -16

(-16) > –|-16|

TRY IT 3

Fill in <, >, or = for each of the following pairs of numbers: a) |-9|___-|-9| b) 2___-|-2| c) -8___|-8|
d) \-(-9___-|-9|.

Show answer

a) > b) > c) < d) >

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.

Grouping Symbols
Parentheses ( )
Brackets [ ]
Braces { }
Absolute value | |

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

EXAMPLE 4

Simplify: 24-|19-3\left(6-2\right)|.

Solution
24-|19-3\left(6-2\right)|
Work inside parentheses first: subtract 2 from 6. 24-|19-3\left(4\right)|
Multiply 3(4). 24-|19-12|
Subtract inside the absolute value bars. 24-|7|
Take the absolute value. 24-7
Subtract. 17

TRY IT 4

Simplify: 19-|11-4\left(3-1\right)|.

Show answer

16

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.

In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.

We will use the counters to show how to add the four addition facts using the numbers 5,-5 and 3,-3.

\begin{array}{cccccccccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill & & & \hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \end{array}

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.

This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text “8 positives.” Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase “8 negatives”. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.

So what happens when the signs are different? Let’s add -5+3 and 5+\left(-3\right).

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.

Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “negative 5 plus 3” and below is written “More negatives – the sum is negative.” The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “5 plus negative 3” and below is written “More positives – the sum is positive.”

EXAMPLE 5

Add: a) -1+5 b) 1+\left(-5\right).

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

TRY IT 5

Add: a) -2+4 b) 2+\left(-4\right).

Show answer

a) 2 b) -2

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 37+\left(-53\right), 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 53-37=16, there are 16 more red counters.

Therefore, the sum of 37+\left(-53\right) is -16.

37+\left(-53\right)=-16

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

-74+\left(-27\right)=-101

Let’s look again at the results of adding the different combinations of 5,-5 and 3,-3.

Addition of Positive and Negative Integers

\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill \\ \hfill 8\hfill & & & \hfill -8\hfill \\ \hfill \text{both positive, sum positive}\hfill & & & \hfill \text{both negative, sum negative}\hfill \end{array}

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

\begin{array}{cccc}\hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \\ \hfill -2\hfill & & & \hfill 2\hfill \\ \hfill \text{different signs, more negatives, sum negative}\hfill & & & \hfill \text{different signs, more positives, sum positive}\hfill \end{array}

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.

EXAMPLE 6

Simplify: a) 19+\left(-47\right) b) -14+\left(-36\right).

Solution
  1.  Since the signs are different, we subtract \text{19 from 47}\text{.} The answer will be negative because there are more negatives than positives.
    \begin{array}{cccc}& & & \hfill \phantom{\rule{0.3em}{0ex}}19+\left(-47\right)\hfill \\ \text{Add.}\hfill & & & \hfill \phantom{\rule{0.3em}{0ex}}-28\hfill \end{array}
  2.  Since the signs are the same, we add. The answer will be negative because there are only negatives.
    \begin{array}{cccc}& & & \hfill -14+\left(-36\right)\hfill \\ \text{Add.}\hfill & & & \hfill -50\hfill \end{array}

TRY IT 6

Simplify: a) -31+\left(-19\right) b) 15+\left(-32\right).

Show answer

a)-50 b)-17

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 7

Simplify: -5+3\left(-2+7\right).

Solution
-5+3\left(-2+7\right)
Simplify inside the parentheses. -5+3\left(5\right)
Multiply. -5+15
Add left to right. 10

TRY IT 7

Simplify: -2+5\left(-4+7\right).

Show answer

13

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 \text{``}5-3\text{''} as \text{``}5 take away 3.\text{''} When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers 5 and 3.

\begin{array}{cccccccccc}\hfill 5-3\hfill & & & \hfill -5-\left(-3\right)\hfill & & & \hfill -5-3\hfill & & & \hfill 5-\left(-3\right)\hfill \end{array}

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.

Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation “5 minus 3 equals 2.” The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation “negative 5, minus, negative 3, equals negative 2.”

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 -5-3, we restate it as -5 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, 5-\left(-3\right). 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 8

Subtract: a) -3-1 b) 3-\left(-1\right).

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 8

Subtract: a) -6-4 b) 6-\left(-4\right).

Show answer

a) -10 b) 10

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In Example 8, -3-1 is the same as -3+\left(-1\right) and 3-\left(-1\right) is the same as 3+1. You will often see this idea, the subtraction property, written as follows:

Subtraction Property

a-b=a+\left(-b\right)
Subtracting a number is the same as adding its opposite.

Look at these two examples.

Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as “6 minus 4” and below as “2.” The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as “6 plus, open parenthesis, negative 4, close parenthesis” and below as “2.”

6-4\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}6+\left(-4\right).

Of course, when you have a subtraction problem that has only positive numbers, like 6-4, you just do the subtraction. You already knew how to subtract 6-4 long ago. But knowing that 6-4 gives the same answer as 6+\left(-4\right) helps when you are subtracting negative numbers. Make sure that you understand how 6-4 and 6+\left(-4\right) give the same results!

Look at what happens when we subtract a negative.

This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.

8-\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}8+5

Subtracting a negative number is like adding a positive!

You will often see this written as a-\left(-b\right)=a+b.

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

EXAMPLE 9

Simplify: 7-\left(-4-3\right)-9.

Solution
7-\left(-4-3\right)-9
Simplify inside the parentheses first. 7-\left(-7\right)-9
Subtract left to right. 14-9
Subtract. 5

TRY IT 9

Simplify: 8-\left(-3-1\right)-9.

Show answer

3

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 a \cdot b means add a, b times. Here, we are using the model just to help us discover the pattern.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

The next two examples are more interesting.

What does it mean to multiply 5 by -3? 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 work space. Then we take away 5 three times.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

\begin{array}{cccccccc}\hfill 5 \cdot 3& =\hfill & 15\hfill & & & \hfill -5\left(3\right)& =\hfill & -15\hfill \\ \hfill 5\left(-3\right)& =\hfill & -15\hfill & & & \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\hfill \end{array}

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
\begin{array}{ccc}\hfill 7\cdot4& =\hfill & 28\hfill \\ \hfill -8\left(-6\right)& =\hfill & 48\hfill \end{array}
Different signs Product Example
Positive \cdot negative
Negative \cdot positive
Negative
Negative
\begin{array}{ccc}\hfill 7\left(-9\right)& =\hfill & -63\hfill \\ \hfill -5\cdot10& =\hfill & -50\hfill \end{array}

EXAMPLE 10

Multiply: a) -9\cdot3 b) -2\left(-5\right) c) 4\left(-8\right) d) 7\cdot6.

Solution
a)
Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-9\cdot3\\ -27\end{array}
b)
Multiply, noting that the signs are the same so the product is positive.
\begin{array}{c}-2\left(-5\right)\\ 10\end{array}
c)
Multiply, with different signs.
\begin{array}{c}4\left(-8\right)\\ -32\end{array}
d)
Multiply, with same signs.
\begin{array}{c}7\cdot6\\ 42\end{array}

TRY IT 10

Multiply: a) -6\cdot8 b) -4\left(-7\right) c) 9\left(-7\right) d) 5\cdot12.

Show answer

a)-48 b) 28 c) -63 d) 60

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

\begin{array}{ccccccc}& & & \hfill -1\cdot 4\hfill & & & \hfill -1\left(-3\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill -4\hfill & & & \hfill 3\hfill \\ & & & \hfill -4\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}4.\hfill & & & \hfill 3\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-3.\hfill \end{array}

Each time we multiply a number by -1, we get its opposite!

Multiplication by -1

-1a=\text{-}a

Multiplying a number by -1 gives its opposite.

EXAMPLE 11

Multiply: a) -1\cdot7 b) -1\left(-11\right).

Solution
a)
Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-1\cdot7\\ -7\\ -7\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}7.\end{array}
b)
Multiply, noting that the signs are the same so the product is positive.
\begin{array}{c}-1\left(-11\right)\\ 11\\ 11\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-11.\end{array}

TRY IT 11

Multiply: a) -1\cdot9 b) -1 \cdot \left(-17\right).

Show answer

a)-9 b) 17

Divide Integers

What about division? Division is the inverse operation of multiplication. So, 15\div 3=5 because 5\cdot3=15. 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.

\begin{array}{cccccccccccccc}\hfill 5\cdot3& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div 3\hfill & =\hfill & 5\hfill & & & & & \hfill -5\left(3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div3\hfill & =\hfill & -5\hfill \\ \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div \left(-3\right)\hfill & =\hfill & -5\hfill & & & & & \hfill 5\left(-3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div \left(-3\right)\hfill & =\hfill & 5\hfill \end{array}

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 12

Divide: a) -27\div 3 b) -100\div \left(-4\right).

Solution
a)
Divide. With different signs, the quotient is negative.
\begin{array}{c}-27\div 3\\ -9\end{array}
b)
Divide. With signs that are the same, the quotient is positive.
\begin{array}{c}-100\div\left(-4\right)\\ 25\end{array}

TRY IT 12

Divide: a) -42\div 6 b) -117\div\left(-3\right).

Show answer

a)-7 b) 39

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 13

Simplify: 7\left(-2\right)+4\left(-7\right)-6.

Solution
7\left(-2\right)+4\left(-7\right)-6
Multiply first. -14+\left(-28\right)-6
Add. -42-6
Subtract. -48

TRY IT 13

Simplify: 8\left(-3\right)+5\left(-7\right)-4.

Show answer

-63

EXAMPLE 14

Simplify: a) {\left(-2\right)}^{4} b) \text{-}{2}^{4}.

Solution
a)
Write in expanded form.
Multiply.
Multiply.
Multiply.
\begin{array}{c}{\left(-2\right)}^{4}\\ \left(-2\right)\left(-2\right)\left(-2\right)\left(-2\right)\\ 4\left(-2\right)\left(-2\right)\\ -8\left(-2\right)\\ 16\end{array}
b)
Write in expanded form. We are asked to find the opposite of\phantom{\rule{0.2em}{0ex}}{2}^{4}.
Multiply.
Multiply.
Multiply.
\begin{array}{c}\text{-}{2}^{4}\\ \text{-}\left(2\cdot2\cdot2\cdot2\right)\\ \text{-}\left(4\cdot2\cdot2\right)\\ \text{-}\left(8\cdot2\right)\\ 16\end{array}

Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the \left(-2\right) to the {4}^{\text{th}} power. In part b), the exponent means to raise just the 2 to the {4}^{\text{th}} power and then take the opposite.

TRY IT 14

Simplify: a) {\left(-3\right)}^{4} b) \text{-}{3}^{4}.

Show answer

a) 81 b) -81

The next example reminds us to simplify inside parentheses first.

EXAMPLE 15

Simplify: 12-3\left(9-12\right).

Solution
12-3\left(9-12\right)
Subtract in parentheses first. 12-3\left(-3\right)
Multiply. 12-\left(-9\right)
Subtract. 21

TRY IT 15

Simplify: 17-4\left(8-11\right).

Show answer

29

EXAMPLE 16

Simplify: 8\left(-9\right)\div{\left(-2\right)}^{3}.

Solution
8\left(-9\right)\div{\left(-2\right)}^{3}
Exponents first. 8\left(-9\right)\div\left(-8\right)
Multiply. -72\div \left(-8\right)
Divide. 9

TRY IT 16

Simplify: 12\left(-9\right)\div{\left(-3\right)}^{3}.

Show answer

4

EXAMPLE 17

Simplify: -30\div 2+\left(-3\right)\left(-7\right).

Solution
-30\div 2+\left(-3\right)\left(-7\right)
Multiply and divide left to right, so divide first. -15+\left(-3\right)\left(-7\right)
Multiply. -15+21
Add. 6

TRY IT 17

Simplify: -27\div 3+\left(-5\right)\left(-6\right).

Show answer

21

Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications.

Key Concepts

  • Addition of Positive and Negative Integers
    \begin{array}{cccc}5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}-5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}8\hfill & & & \phantom{\rule{3.45em}{0ex}}-8\hfill \\ \text{both positive,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{both negative,}\hfill \\ \text{sum positive}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum negative}\hfill \\ \\ \\ -5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-2\hfill & & & \phantom{\rule{3.45em}{0ex}}2\hfill \\ \text{different signs,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{different signs,}\hfill \\ \text{more negatives}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{more positives}\hfill \\ \text{sum negative}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum positive}\hfill \end{array}
  • Property of Absolute Value: |n|\ge 0 for all numbers. Absolute values are always greater than or equal to zero!
  • Subtraction of Integers
    \begin{array}{cccc}5-3\hfill & & & -5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}2\hfill & & & \phantom{\rule{0.95em}{0ex}}-2\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{2 positives}\hfill & & & \text{2 negatives}\hfill \\ \\ \\ -5-3\hfill & & & 5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-8\hfill & & & \phantom{\rule{0.95em}{0ex}}8\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{negatives, want to}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{positives, want to}\hfill \\ \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{need neutral pairs}\hfill & & & \text{need neutral pairs}\hfill \end{array}
  • Subtraction Property: Subtracting a number is the same as adding its opposite.
  • Multiplication and Division of Two Signed Numbers
    • Same signs—Product is positive
    • Different signs—Product is negative

Glossary

absolute value
The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as |n|.
integers
The whole numbers and their opposites are called the integers: …−3, −2, −1, 0, 1, 2, 3…
opposite
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: -a means the opposite of the number. The notation -a is read “the opposite of a.”

1.2 Exercise Set

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

    1. 9___4
    2. -3___6
    3. -8___-2
    4. 1___-10

In the following exercises, simplify.

    1. |-32|
    2. |0|
    3. |16|

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

    1. -6___|-6|
    2. -|-3|___-3

In the following exercises, simplify.

  1. -\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-|-5|
  2. 8|-7|
  3. |15-7|-|14-6|
  4. 18-|2\left(8-3\right)|

In the following exercises, simplify each expression.

  1. -21+\left(-59\right)
  2. 48+\left(-16\right)
  3. -14+\left(-12\right)+4
  4. 135+\left(-110\right)+83
  5. 19+2\left(-3+8\right)

In the following exercises, simplify.

  1. 8-2
  2. -5-4
  3. 8-\left(-4\right)
    1. 44-28
    2. 44+\left(-28\right)
    1. 27-\left(-18\right)
    2. 27+18

In the following exercises, simplify each expression.

  1. 15-\left(-12\right)
  2. 48-87
  3. -17-42
  4. -103-\left(-52\right)
  5. -45-\left(54\right)
  6. 8-3-7
  7. -5-4+7
  8. -14-\left(-27\right)+9
  9. \left(2-7\right)-\left(3-8\right)\left(2\right)
  10. -\left(6-8\right)-\left(2-4\right)
  11. 25-\left[10-\left(3-12\right)\right]
  12. 6.3-4.3-7.2
  13. {5}^{2}-{6}^{2}

In the following exercises, multiply.

  1. -4\cdot8
  2. -1\cdot6
  3. 9\left(-7\right)
  4. -1\left(-14\right)

In the following exercises, divide.

  1. -24\div6
  2. -180\div15
  3. -52\div\left(-4\right)

In the following exercises, simplify each expression.

  1. 5\left(-6\right)+7\left(-2\right)-3
  2. {\left(-2\right)}^{6}
  3. \text{-}{4}^{2}
  4. -3\left(-5\right)\left(6\right)
  5. \left(8-11\right)\left(9-12\right)
  6. 26-3\left(2-7\right)
  7. 65\div\left(-5\right)+\left(-28\right)\div\left(-7\right)
  8. 9-2\left[3-8\left(-2\right)\right]
  9. {\left(-3\right)}^{2}-24\div\left(8-2\right)

In the following exercises, solve.

  1. Temperature On January 15, the high temperature in Lytton, British Columbia, was 84° . That same day, the high temperature in Fort Nelson, British Columbia was -12°. What was the difference between the temperature in Lytton and the temperature in Embarrass?
  2. Checking Account Ester has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?
  3. Checking Account Kevin has a balance of -\$38 in his checking account. He deposits $225 to the account. What is the new balance?
  4. 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:

    1. Quebec
    2. Alberta

Answers:

    1. >
    2. <
    3. <
    4. >
    1. 32
    2. 0
    3. 16
    1. <
    2. =
  1. 5,-5
  2. 56
  3. 0
  4. 8
  5. -80
  6. 32
  7. -22
  8. 108
  9. 29
  10. 6
  11. -9
  12. 12
    1. 16
    2. 16
    1. 45
    2. 45
  13. 27
  14. -39
  15. -59
  16. -51
  17. -99
  18. -2
  19. -2
  20. 22
  21. -15
  22. 4
  23. 6
  24. -5.2
  25. -11
  26. -32
  27. -6
  28. -63
  29. 14
  30. -4
  31. -12
  32. 13
  33. -47
  34. 64
  35. -16
  36. 90
  37. 9
  38. 41
  39. -9
  40. -29
  41. 5
  42. 96°
  43. -\$28
  44. $187
    1. $5.6 million
    2. -\$7.5 million

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.

License

Icon for the Creative Commons Attribution 4.0 International License

Business/Technical Mathematics by Izabela Mazur and Kim Moshenko is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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