A: Operations with Integers

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Feb 2, 2020
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- Chapter 10: Appendix
- B: Working with Fractions

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SUMMARY

By the end of this section, you will be able to:

Add and subtract integers
Multiply and divide integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate phrases to expressions with integers
Use integers in applications

Add and Subtract Integers

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. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model 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.

We will use two color counters to model 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:

$5+3 \quad −5+(−3) \quad −5+3 \quad 5+(−3) \nonumber$

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, and 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 $−5+3 \text{ and } 5+(−3). \nonumber$

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 $\PageIndex{1}$

Add: a) $-1+(-4)−1+(-4) \quad$ b) $-1+5−1+5 \quad$ c) $1+(-5)1+(-5).$

Solution

	$CNX_IntAlg_Figure_01_02_008b_img.jpg$
	$CNX_IntAlg_Figure_01_02_008a_img.jpg$
1 negative plus 4 negatives is 5 negatives	$CNX_IntAlg_Figure_01_02_008c_img.jpg$

	$CNX_IntAlg_Figure_01_02_009a_img.jpg$
	$CNX_IntAlg_Figure_01_02_009b_img.jpg$
There are more positives, so the sum is positive.	$CNX_IntAlg_Figure_01_02_009c_img.jpg$

	$CNX_IntAlg_Figure_01_02_010a_img.jpg$
	$CNX_IntAlg_Figure_01_02_010b_img.jpg$
There are more negatives, so the sum is negative.	$CNX_IntAlg_Figure_01_02_010c_img.jpg$

EXAMPLE ��ℎ��10.8Chapter10.8

Add: ⓐ −2+(−4)−2+(−4) ⓑ −2+4−2+4 ⓒ 2+(−4)2+(−4).

Answer: ⓐ −6−6 ⓑ 22 ⓒ −2−2

EXAMPLE ��ℎ��10.9Chapter10.9

Add: ⓐ −2+(−5)−2+(−5) ⓑ −2+5−2+5 ⓒ 2+(−5)2+(−5).

Answer: ⓐ −7−7 ⓑ 33 ⓒ −3−3

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 $5-(-3).$

	$CNX_IntAlg_Figure_01_02_012a_img.jpg$	$CNX_IntAlg_Figure_01_02_012h_img.jpg$
Model the first number.	$CNX_IntAlg_Figure_01_02_012b_img.jpg$	$CNX_IntAlg_Figure_01_02_012i_img.jpg$
We now add the needed neutral pairs.	$CNX_IntAlg_Figure_01_02_012c_img.jpg$	$CNX_IntAlg_Figure_01_02_012j_img.jpg$
We remove the number of counters modeled by the second number.	$CNX_IntAlg_Figure_01_02_012d_img.jpg$	$CNX_IntAlg_Figure_01_02_012k_img.jpg$
Count what is left.	$CNX_IntAlg_Figure_01_02_012e_img.jpg$	$CNX_IntAlg_Figure_01_02_012l_img.jpg$
	$CNX_IntAlg_Figure_01_02_012f_img.jpg$	$CNX_IntAlg_Figure_01_02_012m_img.jpg$
	$CNX_IntAlg_Figure_01_02_012g_img.jpg$	$CNX_IntAlg_Figure_01_02_012n_img.jpg$

EXAMPLE ��ℎ��10.10Chapter10.10

Subtract: ⓐ 3−13−1 ⓑ −3−(−1)−3−(−1) ⓒ −3−1−3−1 ⓓ 3−(−1)3−(−1).

Answer

ⓐ

	$CNX_IntAlg_Figure_01_02_018a_img.jpg$	$CNX_IntAlg_Figure_01_02_018b_img.jpg$
Take 1 positive from 3 positives and get 2 positives.		$CNX_IntAlg_Figure_01_02_018c_img.jpg$

ⓑ

	$CNX_IntAlg_Figure_01_02_019a_img.jpg$	$CNX_IntAlg_Figure_01_02_019b_img.jpg$
Take 1 positive from 3 negatives and get 2 negatives.		$CNX_IntAlg_Figure_01_02_019c_img.jpg$

ⓒ

	$CNX_IntAlg_Figure_01_02_013a_img.jpg$	$CNX_IntAlg_Figure_01_02_013c_img.jpg$
Take 1 positive from the one added neutral pair.	$CNX_IntAlg_Figure_01_02_013b_img.jpg$	$CNX_IntAlg_Figure_01_02_013d_img.jpg$

ⓓ

	$CNX_IntAlg_Figure_01_02_014a_img.jpg$	$CNX_IntAlg_Figure_01_02_014c_img.jpg$
Take 1 negative from the one added neutral pair.	$CNX_IntAlg_Figure_01_02_014b_img.jpg$	$CNX_IntAlg_Figure_01_02_014d_img.jpg$

EXAMPLE ��ℎ��10.11Chapter10.11

Subtract: ⓐ 6−46−4 ⓑ −6−(−4)−6−(−4) ⓒ −6−4−6−4 ⓓ 6−(−4)6−(−4).

Answer: ⓐ 22 ⓑ −2−2 ⓒ −10−10 ⓓ 1010

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 ��ℎ��10.13Chapter10.13

Simplify: ⓐ 13−813−8 and 13+(−8)13+(−8) ⓑ −17−9−17−9 and −17+(−9)−17+(−9) ⓒ 9−(−15)9−(−15) and 9+159+15 ⓓ −7−(−4)−7−(−4) and −7+4−7+4.

Answer

ⓐ

Subtract.13−85and13+(−8)513−8and13+(−8)Subtract.55

ⓑ

Subtract.−17−9−26and−17+(−9)−26−17−9and−17+(−9)Subtract.−26−26

ⓒ

Subtract.9−(−15)24and9+15249−(−15)and9+15Subtract.2424

ⓓ

Subtract.−7−(−4)−3and−7+4−3−7−(−4)and−7+4Subtract.−3−3

EXAMPLE ��ℎ��10.14Chapter10.14

Simplify: ⓐ 21−1321−13 and 21+(−13)21+(−13) ⓑ −11−7−11−7 and −11+(−7)−11+(−7) ⓒ 6−(−13)6−(−13) and 6+136+13 ⓓ −5−(−1)−5−(−1) and −5+1−5+1.

Answer

ⓐ 8,88,8 ⓑ −18,−18−18,−18

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

EXAMPLE ��ℎ��10.16Chapter10.16

Simplify: 7−(−4−3)−9.7−(−4−3)−9.

Answer: Simplify inside the parentheses first.Subtract left to right.Subtract.7−(−4−3)−97−(−7)−914−957−(−4−3)−9Simplify inside the parentheses first.7−(−7)−9Subtract left to right.14−9Subtract.5

EXAMPLE ��ℎ��10.17Chapter10.17

Simplify: 8−(−3−1)−9.8−(−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, 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.

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

In summary:

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

Notice that for multiplication of two signed numbers, when the

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

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 $\PageIndex{19}$

Answer

ⓐ

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

ⓑ

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

ⓒ

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

ⓓ

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

EXAMPLE $\PageIndex{20}$

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

MULTIPLICATION BY −1

$−1a=−a$

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

When an expression has many numbers and many operations in it, remember to multiply and divide in order from left to right.

EXAMPLE $\PageIndex{25}$

Simplify: ⓐ $8(−9)÷(−2)^3$ ⓑ $−30÷2+(−3)(−7)$ .

Solution

ⓐ

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

ⓑ

$\begin{array}{lc} \text{} & −30÷2+(−3)(−7) \\ \text{Multiply and divide} \\ \text{left to right, so divide first.} & −15+(−3)(−7) \\ \text{Multiply.} & −15+21 \\ \text{Add.} & 6 \end{array}$

EXAMPLE $\PageIndex{26}$

Simplify: ⓐ $12(−9)÷(−3)^3$ ⓑ $−27÷3+(−5)(−6).$

Answer: ⓐ 4 ⓑ 21

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Add and Subtract Integers

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

Multiply and Divide Integers

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