Section 1.4: Integers

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Be Prepared

We will rely heavily on these skills throughout this section.

Plot 0, 2, and 5 on a number line
Fill in the appropriate symbol $=$, $<$, or $>$: $2$_____$5$
Simplify $3+4-5$
Simplify $\sqrt{144}$

Learning Objectives

Motivating Problem

You check your bank account and see that you have $15. Then you accidentally use your debit card to buy a $25 pair of shoes. What does your account balance show now? How can a number be less than zero, and what does that mean?

Fun Fact

Although ancient Chinese and Indian mathematicians employed negative numbers for trade and accounting, Europeans only began accepting them during the Renaissance. Today, negative numbers are used in everything from banking to data science.

The Goal

In this section, we will discover that numbers don’t stop at zero. We'll explore the world of positive and negative integers and learn how to add, subtract, multiply, and divide them. With the help of number lines and real-life examples like temperatures and money, we will understand opposites, absolute value, and how to make sense of operations with signed numbers.

Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line below.

$Figure shows a horizontal line marked with numbers at equal distances. At the center of the line is 0. To the right of this, starting from the number closest to 0 are 1, 2, 3 and 4. These are labeled positive numbers. To the left of 0, starting from the number closest to 0 are minus 1, minus 2, minus 3 and minus 4. These are labeled negative numbers.$

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, each one is called the opposite of the other. 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.

$Figure shows a number line with the numbers 3 and minus 3 highlighted. These are equidistant from 0, both being 3 numbers away from 0.$
The opposite of 3 is $−3$.

OPPOSITE NOTATION

\[ \begin{align*} & -a \text{ means the opposite of the number } a \\ & \text{The notation } -a \text{ is read as “the opposite of } a \text{.”} \end{align*} \]

We saw that numbers such as 3 and −3 are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

Definition: 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 $|n|$ and $|n|≥0$ for all numbers.

Absolute values are always greater than or equal to zero.

For example,

\[\begin{align*} & -5 \text{ is } 5 \text{ units away from 0, so } |-5|=5. \\ & 5 \text{ is }5\text{ units away from 0, so }|5|=5. \end{align*}\]

The Figure below illustrates this idea.

Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.

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 to 0 on the number line is zero units.

In the next example, we’ll order expressions with absolute values.

Example 1

Fill in $<,\,>,$ or $=$ for each of the following pairs of numbers:

$\mathrm{|−5|}\_\_\mathrm{−|−5|}$
$\text{8__−|−8|}$
$\text{−9__−|−9|}$
$\text{−(−16)__|−16|}$.

Solution

$\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {|−5| \\ 5 \\ 5 \\ |−5|} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−5| \\ −5 \\ −5 \\ −|−5|} \end{array}$

$\begin{array}{llcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {8 \\ 8 \\ 8 \\ 8} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−8| \\ −8 \\ −8 \\ −|−8|} \end{array}$

$\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−9 \\ −9 \\ −9 \\ −9} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−9| \\ −9 \\ −9 \\ −|−9|} \end{array}$

$\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−(−16) \\ 16 \\ 16 \\ −(−16)} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−16| \\ 16 \\ 16 \\ |−16|} \end{array}$

Try It 1

Fill in $<,\,>,$ or $=$ for each of the following pairs of numbers:

a. $−|−10|\_\_−10$

b. $2 \_\_−|−2|$

c. $−8 \_\_|−8|$

d. $−(−9) \_\_|−9|.$

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

\[\begin{array}{lclc} \text{Parentheses} & () & \text{Braces} & \{ \} \\ \text{Brackets} & [] & \text{Absolute value} & ||\end{array}\nonumber\]

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

Example 2

Simplify: $\mathrm{24−|19−3(6−2)|}$.

Solution

$\begin{array}{lc} \text{} & 24−|19−3(6−2)| \\ \text{Work inside parentheses first:} & \text{} \\ \text{subtract 2 from 6.} & 24−|19−3(4)| \\ \text{Multiply 3(4).} & 24−|19−12| \\ \text{Subtract inside the absolute value bars.} & 24−|7| \\ \text{Take the absolute value.} & 24−7 \\ \text{Subtract.} & 17 \end{array}$

Try It 2

Simplify: $19−|11−4(3−1)|$.

Answer: 16

Add and Subtract Integers

So far in our examples, we have only used the counting numbers and the whole numbers.

\[\begin{array}{ll} \text{Counting numbers} & 1,2,3… \\ \text{Whole numbers} & 0,1,2,3…. \end{array}\nonumber\]

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 $…−3,−2,−1,0,1,2,3…$

Definition: INTEGERS

The whole numbers and their opposites are called the integers.

The integers are the numbers

\[…-3,-2,-1,0,1,2,3…,\nonumber\]

Most students are comfortable with the addition and subtraction facts for positive numbers. However, adding or subtracting with both positive and negative numbers may be more challenging.

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

We let one color (blue) represent positive. The other color (red) will represent the negatives.

$Figure show two circles labeled positive blue and negative red.$

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.

$Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.$

We will use the counters to show how to add:

\[5+3 \; \; \; \; \; \; −5+(−3) \; \; \; \; \; \; −5+3 \; \; \; \; \; \; \; 5+(−3)\nonumber\]

The first example, $5+3,$ adds 5 positives and 3 positives—both positives.

The second example, $−5+(−3),$ adds 5 negatives and 3 negatives—both negatives.

When the signs are the same, the counters are all the same color, so we add them. In each case, we get 8—either 8 positives or 8 negatives.

$Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.$

So what happens when the signs are different? Let’s add $−5+3$ and $5+(−3)$.

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.

$Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.$

Example 3

Perform the following addition:

a. $−1+(−4)$

b. $−1+5$

c. $1+(−5)$

Solution

	$alt$
	$alt$
1 negative plus 4 negatives is 5 negatives	$alt$

	$alt$
	$alt$
There are more positives, so the sum is positive.	$alt$

	$alt$
	$alt$
There are more negatives, so the sum is negative.	$alt$

Try It 3

Add:

a. $−2+(−4)$

b. $−2+4$

c. $2+(−4)$

Answer

a. $−6$

b. $2$

c. $−2$

We will continue to use counters to model the subtraction. Perhaps when you were younger, you read $“5−3”$ as “5 take away 3.” When you use counters, you can think of subtraction the same way!

We will use the counters to show to subtract:

\[5−3 \; \; \; \; \; \; −5−(−3) \; \; \; \; \; \; −5−3 \; \; \; \; \; \; 5−(−3) \nonumber\]

The first example, $5−3$, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, $−5−(−3),$ 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.

$Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.$

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add 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.

Let’s look at $−5−3$ and $5−(−3)$.

	$alt$	$alt$
Model the first number.	$alt$	$alt$
We now add the needed neutral pairs.	$alt$	$alt$
We remove the number of counters modeled by the second number.	$alt$	$alt$
Count what is left.	$alt$	$alt$
	$alt$	$alt$
	$alt$	$alt$

Example 4

Perform the following subtraction:

a. $3−1$

b. $−3−(−1)$

c. $−3−1$

d. $3−(−1)$

Solution

	$alt$	$alt$
Take 1 positive from 3 positives and get 2 positives.		$alt$

	$alt$	$alt$
Take 1 positive from 3 negatives and get 2 negatives.		$alt$

	$alt$	$alt$
Take 1 positive from the one added neutral pair.	$alt$	$alt$

	$alt$	$alt$
Take 1 negative from the one added neutral pair.	$alt$	$alt$

Try It 4

Perform the following Subtraction:

a. $7−4$

b. $−7−(−4)$

c. $−7−4$

d. $7−(−4)$

Answer

a. $3$

b. $−3$

c. $−11$

d. $11$

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example, $−3−1$ is the same as $−3+(−1)$ and $3−(−1)$ is the same as $3+1$. You will often see this idea, the Subtraction Property, written as follows:

Definition: SUBTRACTION PROPERTY

\[a−b=a+(−b)\nonumber\]

Subtracting a number is the same as adding its opposite.

Example 5

Simplify:

a. $13−8$ and $13+(−8)$

b. $−17−9$ and $−17+(−9)$

c. $9−(−15)$ and $9+15$

d. $−7−(−4)$ and $−7+4$

Solution

a. $\begin{array}{lccc} \text{} & 13−8 & \text{and} & 13+(−8) \\ \text{Subtract.} & 5 & \text{} & 5 \end{array}$

b. $\begin{array}{lccc} \text{} & −17−9 & \text{and} & −17+(−9) \\ \text{Subtract.} & −26 & \text{} & −26 \end{array}$

c. $\begin{array}{lccc} \text{} & 9−(−15) & \text{and} & 9+15 \\ \text{Subtract.} & 24 & \text{} & 24 \end{array}$

d. $\begin{array}{lccc} \text{} & −7−(−4) & \text{and} & −7+4 \\ \text{Subtract.} & −3 & \text{} & −3 \end{array}$

Try It 5

Simplify:

a. $15−7$ and $15+(−7)$

b. $−14−8$ and $−14+(−8)$

c. $4−(−19)$ and $4+19$

d. $−4−(−7)$ and $−4+7$.

Answer

a. $8,8$

b. $−22,−22$

c. $23,23$

d. $3,3$

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

Example 6

Simplify: $7−(−4−3)−9.$

Solution

$\begin{array}{lc} \text{} & 7−(−4−3)−9 \\ \text{Simplify inside the parentheses first.} & 7−(−7)−9 \\ \text{Subtract left to right.} & 14−9 \\ \text{Subtract.} & 5 \end{array}$

Try It 6

Simplify: $8−(−3−1)−9$.

Answer: $3$

Multiply and Divide 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 are using the model just to help us discover the pattern.

We remember that a⋅b means add a, b times.

$The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.$

The next two examples are more interesting. What does it mean to multiply 5 by −3? It means subtract 5 from itself three 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.

$The figure on the left is labeled 5 open parentheses minus 3 close parentheses. We need to take away 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 negatives. Hence, 5 times minus 3 is minus 15. The figure on the right is labeled open parentheses minus 5 close parentheses open parentheses minus 3 close parentheses. We need to take away minus 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 positives. Hence, minus 5 times minus 3 is 15.$

In summary:

\[\begin{array}{ll} 5·3=15 & −5(3)=−15 \\ 5(−3)=−15 & (−5)(−3)=15 \end{array}\nonumber\]

Notice that for multiplication of two signed numbers, when the

\[ \text{signs are the } \textbf{same} \text{, the product is } \textbf{positive.} \\ \text{signs are } \textbf{different} \text{, the product is } \textbf{negative.} \nonumber\]

What about division? Division is the inverse operation of multiplication. So, $15÷3=5$ because $15·3=15$. In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

\[\begin{array}{lclrccl} 5·3=15 & \text{so} & 15÷3=5 & \text{ } −5(3)=−15 & \text{so} & −15÷3=−5 \\ (−5)(−3)=15 & \text{so} & 15÷(−3)=−5 & \text{ } 5(−3)=−15 & \text{so} & −15÷(−3)=5 \end{array}\nonumber\]

Division follows the same rules as multiplication with regard to signs.

MULTIPLICATION AND DIVISION OF SIGNED NUMBERS

For multiplication and division of two signed numbers:

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 7

Multiply or divide:

a. $−100÷(−4)$

b. $7⋅6$

c. $4(−8)$

d. $−27÷3$

Solution

a. $\begin{array}{lc} \text{} & −100÷(−4) \\ \text{Divide, with signs that are} \\ \text{the same the quotient is positive.} & 25 \end{array}$

b. $\begin{array} {lc} \text{} & 7·6 \\ \text{Multiply, with same signs.} & 42 \end{array}$

c. $\begin{array} {lc} \text{} & 4(−8) \\ \text{Multiply, with different signs.} & −32 \end{array}$

d. $\begin{array}{lc} \text{} & −27÷3 \\ \text{Divide, with different signs,} \\ \text{the quotient is negative.} & −9 \end{array}$

Try It 7

Multiply or divide:

a. $−117÷(−3)$

b. $3⋅13$

c. $7(−4)$

d. $−42÷6$.

Answer

a. $39$

b. $39$

c. $−28$

d. $−7$

When we multiply a number by 1, the result is the same. Each time we multiply a number by −1, we get its opposite!

MULTIPLICATION BY −1

\[−1a=−a\nonumber\]

Multiplying a number by $−1$ gives its opposite.

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 Please Excuse 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 8

Simplify:

a. $(−2)^4$

b. $−2^4$.

Solution

Notice the difference in parts a and b. In a, the exponent applies to all of $(−2)$, while in b, it only applies to the 2.

a. $\begin{array}{lc} \text{} & (−2)^4 \\ \text{Write in expanded form.} & (−2)(−2)(−2)(−2) \\ \text{Multiply.} & 4(−2)(−2) \\ \text{Multiply.} & −8(−2) \\ \text{Multiply.} & 16 \end{array}$

b. $\begin{array}{lc} \text{} & −2^4 \\ \text{Write in expanded form.} & −(2·2·2·2) \\ \text{Multiply.} & −(4·2·2) \\ \text{Multiply.} & −(8·2) \\ \text{Multiply.} & −16 \end{array}$

Try It 8

Simplify:

a. $(−3)^4$

b. $−3^4$.

Answer

a. $81$

b. $−81$

The last example showed us the difference between $(−2)^4$ and $−2^4$. This distinction is important to prevent future errors. The following example reminds us to multiply and divide in order left to right.

Example 9

Simplify:

a. $8(−9)÷(−2)^3$

b. $−30÷2+(−3)(−7)$

Solution

a. $\begin{array}{lc} \text{} & 8(−9)÷(−2)^3 \\ \text{Exponents first.} & 8(−9)÷(−8) \\ \text{Multiply.} & −72÷(−8) \\ \text{Divide.} & 9 \end{array}$

b. $\begin{array}{lc} \text{} & −30÷2+(−3)(−7) \\ \text{Divide first.} & −15+(−3)(−7) \\ \text{Multiply.} & −15+21 \\ \text{Add.} & 6 \end{array}$

Try It 9

Simplify:

a. $18(−4)÷(−2)^3$

b. $−32÷4+(−2)(−7)$.

Answer

a. $9$

b. $6$

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Example 10

Evaluate $4x^2−2xy+3y^2$ when $x=2, y=−1$.

Solution

$\begin{array}{lc} \text{} & 4x^2 − 2xy + 3y^2 \\ \text{Substitute.} & 4(2)^2 − 2(2)(−1) + 3(−1)^2 \\ \text{Simplify exponents.} & 4(4) − 2(2)(−1) + 3(1) \\ \text{Multiply.} & 16 + 4 + 3 \\ \text{Add.} & 23 \end{array}$

Try It 10

Evaluate: $3x^2−2xy+6y^2$ when $x=1, y=−2$.

Answer: $31$

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Example 11

Translate and simplify: the sum of 8 and −12, increased by 3.

Solution

$\begin{array}{lc} \text{Translate.} & [8 + (−12)] + 3 \\ \text{Simplify inside the brackets.} & −4 + 3 \\ \text{Add.} & −1 \end{array}$

Try It 11

Translate and simplify the sum of 9 and −16, increased by 4.

Answer: $(9+(−16))+4 = −3$

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Example 12

The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9 degrees. What was the difference in the morning and afternoon temperatures?

$Glass thermometer showing -9°C and 11°C.$

Solution

$\begin{array}{lc} \text{Identify the problem.} & \text{We are asked to find the difference between 11 and −9.} \\ \text{Translate.} & 11 - (−9) \\ \text{Subtracting a negative becomes addition.} & 11 + 9 \\ \text{Add.} & 20 \end{array}$

The difference in temperatures was 20 degrees.

Try It 12

The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

Answer: The difference in temperatures was 45 degrees Fahrenheit.