9.2.1: Adding Integers

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Learning Objectives

Add two or more integers with the same sign.
Add two or more integers with different signs.

Introduction

On an extremely cold day, the temperature may be -10. If the temperature rises 8 degrees, how will you find the new temperature? Knowing how to add integers is important here and in much of algebra.

Adding Integers with the Same Sign

Since positive integers are the same as natural numbers, adding two positive integers is the same as adding two natural numbers. To add positive integers on the number line, you move to the right (in the positive direction).

Supplemental Interactive Activity

Try out the interactive number line below. Choose a few pairs of positive integers to add. Click and drag the blue and red dots, and watch how the addition works.

$Screen Shot 2021-05-20 at 8.29.58 PM.png$

To add negative integers on a number line, you move to the left (in the negative direction).

Supplemental Interactive Activity

Try out the interactive number line below. Choose a few pairs of negative integers to add. Click and drag the blue and red dots, and watch how the subtraction works.

$Screen Shot 2021-05-20 at 8.32.46 PM.png$

In both cases, the total number of units moved is the total distance moved. Since the distance of a number from 0 is the absolute value of that number, then the absolute value of the sum of the integers is the sum of the absolute values of the addends.

When both numbers are negative, you move left in a negative direction, and the sum is negative. When both numbers are positive, you move right in a positive direction, and the sum is positive.

To add two numbers with the same sign (both positive or both negative):

Add their absolute values and give the sum the same sign.

Example

Find -23+(-16).

Solution

Both addends have the same sign (negative).

So, add their absolute values:

|-23|=23 and |-16|=16

The sum of those numbers is 23+16=39.

Since both addends are negative, the sum is negative.

-23+(-16)=-39

With more than two addends that have the same sign, use the same process with all addends.

Example

Find -27+(-138)+(-55).

Solution

All addends have the same sign (negative).

So, add their absolute values:

|-27|=27, |-138|=138, and |-55|=55

The sum of those numbers is 27+138+55=220.

Since all addends are negative, the sum is negative.

-27+(-138)+(-55)=-220

Exercise

Find -32+(-14).

Answer

Incorrect. You found the sum of the absolute values, but the sum must be negative, as you are adding two negative numbers. The correct answer -46
Incorrect. You subtracted 14 from 32. The sum is found by first adding the absolute values of the addends: |-32|+|-14|=32+14=46. Then you must give the sum the same sign as the two addends. The correct answer is -46.
Incorrect. The sum is found by first adding the absolute values of the addends: |-32|+|-14|=32+14=46. Then you must give the sum the same sign as the two addends. The correct answer is -46.
Correct. The sum is found by first adding the absolute values of the addends: |-32|+|-14|=32+14=46. Then you must give the sum the same sign as the two addends, so the answer is -46.

Adding Integers with Different Signs

What happens when the addends have different signs, like in the temperature problem in the introduction? If it’s -10 degrees, and then the temperature rises 8 degrees, the new temperature is -10+8. How can you calculate the new temperature?

When you add a positive integer and a negative integer, you move in the positive (right) direction to the first number, and then move in the negative (left) direction to add the negative integer.

Since the distances overlap, the absolute value of the sum is the difference of their distances. So to add a positive number and a negative number, you subtract their absolute values (their distances from 0).

Supplemental Interactive Activity

Try adding integers with different signs with this interactive number line. Move in a positive direction (right) to add a positive number, and move in a negative direction (left) to add a negative number. See if you can find a rule for adding numbers without using the number line.

$Screen Shot 2021-05-21 at 12.48.24 PM.png$

What is the sign of the sum of a positive and a negative integer? It’s pretty easy to figure out. If you moved further to the right than you did to the left, you ended to the right of 0, and the answer is positive; and if you move further to the left, the answer is negative.

$Screen Shot 2021-05-21 at 12.49.31 PM.png$

If you didn’t have the number line to refer to, you could find the sum of -1+4 by

subtracting the distances from zero (the absolute values) 4-1=3 and then
applying the sign of the one furthest from zero (the largest absolute value). In this case, 4 is further from 0 than -1, so the answer is positive: -1+4=3

Look at the illustration below.

$Screen Shot 2021-05-21 at 12.53.40 PM.png$

If you didn’t have the number line to refer to, you can find the sum of -3+2 by

subtracting the distances from zero (the absolute values) 3-2=1 and then
applying the sign of the one furthest from zero (the largest absolute value). In this case, |-3|>|2|, so the answer is negative: -3+2=-1

To add two numbers with different signs (one positive and one negative):

Find the difference of their absolute values.
Give the sum the same sign as the number with the greater absolute value.

Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one. The example below shows you how to solve the temperature question that you considered earlier.

Example

Find 8+(-10).

Solution

The addends have different signs.

So find the difference of their absolute values.

|-10|=10 and |8|=8

The difference of the absolute values is 10-8=2.

Since 10>8, the sum has the same sign as -10.

8+(-10)=-2

Example

Evaluate $\ x+37$ when $\ x=-22$.

Solution

$\ x+37$

$\ -22+37$

Substitute $\ -22$ for $\ x$ in the expression.

$\ |-22|=22 \text { and }|37|=37$

$\ 37-22=15$

The addends have different signs. So find the difference of their absolute values.

Since |37|>|-22|, the sum has the same sign as 37.

-22+37=15

With more than two addends, you can add the first two, then the next one, and so on.

Example

Find -27+(-138)+55.

Solution

	Add two at a time, starting with -27+(-138).
$\ \begin{array}{c} \|-27\|=27 \text { and }\|-138\|=138 \\ 27+138=165 \\ -27+-138=-165 \end{array}$	Since they have the same signs, you add their absolute values and use the same sign to get -165.
$\ \begin{array}{c} -165+55 \\ \|-165\|=165 \text { and }\|55\|=55 \\ 165-55=110 \end{array}$	Now add -165+55. Since -165 and 55 have different signs, you add them by subtracting their absolute values.
$\ -165+55=-110$	Since 165>55, the sign of the final sum is the same as the sign of -165.

$\ -27+(-138)+55=-110$

Exercise

Find 32+(-14).

Answer

Incorrect. You found the sum of the absolute values. Since the addends have different signs, you must find the difference of the absolute values. |32|=32 and |-14|=14. The difference is 32-14=18. The sign of the sum is the same as the addend with the greater absolute value. Since |32|>|-14|, the sum is positive, the same sign as 32. The correct answer is 18.
Correct. Since the addends have different signs, you must find the difference of the absolute values. |32|=32 and |-14|=14. The difference is 32-14=18. The sign of the sum is the same as the addend with the greater absolute value. Since |32|>|-14|, the sum is positive.
Incorrect. You found the difference of the absolute values of the addends. However, because |32|>|-14|, the sign of the sum should be the same sign as 32. The correct answer is 18.
Incorrect. You added the absolute values of the addends, and gave the sum the wrong sign. Since the addends have different signs, you must find the difference of the absolute values. |32|=32 and |-14|=14. The difference is 32-14=18. The sign of the sum is the same as the addend with the greater absolute value. Since |32|>|-14|, the sum is positive, the same sign as 32. The correct answer is 18.

Summary

There are two cases to consider when adding integers. When the signs are the same, you add the absolute values of the addends and use the same sign. When the signs are different, you find the difference of the absolute values and use the same sign as the addend with the greater absolute value.

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To add two numbers with the same sign (both positive or both negative):

Example

Example

Exercise

To add two numbers with different signs (one positive and one negative):

Example

Example

Example

Exercise

	Add two at a time, starting with -27+(-138).
\(\ \begin{array}{c} \|-27\|=27 \text { and }\|-138\|=138 \\ 27+138=165 \\ -27+-138=-165 \end{array}\)	Since they have the same signs, you add their absolute values and use the same sign to get -165.
\(\ \begin{array}{c} -165+55 \\ \|-165\|=165 \text { and }\|55\|=55 \\ 165-55=110 \end{array}\)	Now add -165+55. Since -165 and 55 have different signs, you add them by subtracting their absolute values.
\(\ -165+55=-110\)	Since 165>55, the sign of the final sum is the same as the sign of -165.