# 3.3 Addition of signed numbers

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be able to add numbers with like signs and unlike signs, understand addition with zero.

## Overview

• Addition of Numbers with Like Signs
• Addition with Zero
• Addition of Numbers with Unlike Signs

## Addition of numbers with like signs

Let us add the two positive numbers 2 and 3. We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 3 is positive, we move 3 more units to the right.
We are now located at 5.
Thus, $2+3=5$ .

Summarizing, we have

$\left(2\text{\hspace{0.17em}}\text{positive}\text{\hspace{0.17em}}\text{units}\right)+\left(3\text{\hspace{0.17em}}\text{positive}\text{\hspace{0.17em}}\text{units}\right)=\left(5\text{\hspace{0.17em}}\text{positive}\text{\hspace{0.17em}}\text{units}\right)$

Now let us add the two negative numbers $-2$ and $-3$ . We perform this addition on the number line as follows.

We begin at 0, the origin.
Since $-2$ is negative, we move 2 units to the left.
Since $-3$ is negative, we move 3 more units to the left.
We are now located at $-5$ .

Thus, $\left(-2\right)+\left(-3\right)=-5$ .

Summarizing, we have

$\left(2\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{units}\right)+\left(3\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{units}\right)=\left(5\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{units}\right)$

These two examples suggest that

$\begin{array}{l}\left(\text{positive}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\text{positive}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\text{positive}\text{\hspace{0.17em}}\text{number}\right)\\ \left(\text{negative}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\text{negative}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\text{negative}\text{\hspace{0.17em}}\text{number}\right)\end{array}$

## Adding numbers with the same sign

To add two real numbers that have the same sign, add the absolute values of the numbers and associate the common sign with the sum.

## Sample set a

Find the sums.

$3+7$

$\begin{array}{l}\begin{array}{cc}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Add}\text{\hspace{0.17em}}\text{these}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{values}\text{.}\end{array}\\ \begin{array}{cc}\begin{array}{l}|3|=3\\ |7|=7\end{array}\right\}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3+7=10\end{array}\\ \begin{array}{cc}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}\text{common}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{"+}\text{."}\end{array}\end{array}$

$3+7=+10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3+7=10$

$\left(-4\right)+\left(-9\right)$

$\begin{array}{l}\begin{array}{cc}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Add}\text{\hspace{0.17em}}\text{these}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{values}\text{.}\end{array}\\ \begin{array}{cc}\begin{array}{l}|-4|=4\\ |-9|=9\end{array}\right\}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4+9=13\end{array}\\ \begin{array}{cc}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}\text{common}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{"}-\text{."}\end{array}\end{array}$

$\left(-4\right)+\left(-9\right)=-13$

## Practice set a

Find the sums.

$8+6$

14

$41+11$

52

$\left(-4\right)+\left(-8\right)$

$-12$

$\left(-36\right)+\left(-9\right)$

$-45$

$-14+\left(-20\right)$

$-34$

$-\frac{2}{3}+\left(-\frac{5}{3}\right)$

$-\frac{7}{3}$

$-2.8+\left(-4.6\right)$

$-7.4$

Notice that

## Addition with 0

$\begin{array}{l}\left(\text{0}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\text{a}\text{\hspace{0.17em}}\text{positive}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\text{that}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{​}\text{positive}\text{\hspace{0.17em}}\text{number}\right)\\ \left(\text{0}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\text{a}\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{number}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\text{that}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{​}\text{negative}\text{\hspace{0.17em}}\text{number)}\end{array}$

## The additive identity is 0

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity .

## Addition of numbers with unlike signs

Now let us perform the addition $2+\left(-6\right)$ . These two numbers have unlike signs. This type of addition can also be illustrated using the number line.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since $-6$ is negative, we move, from the 2, 6 units to the left.
We are now located at $-4$ .

A rule for adding two numbers that have unlike signs is suggested by noting that if the signs are disregarded, 4 can be obtained from 2 and 6 by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and $-6$ . Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.

## Adding numbers with unlike signs

To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate the sign of the number with the larger absolute value with this difference.

## Sample set b

Find the following sums.

$7+\left(-2\right)$

$\begin{array}{cc}\underset{\begin{array}{l}\text{Larger}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}\\ \text{Sign}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}"+".\end{array}}{\underbrace{|7|=7}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}& \underset{\text{Smaller}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}}{\underbrace{|-2|=2}}\end{array}$

$\begin{array}{ll}\text{Subtract}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{values:}\hfill & 7-2=5.\hfill \\ \text{Attach}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{proper}\text{\hspace{0.17em}}\text{sign:}\hfill & "+".\hfill \end{array}$

$7+\left(-2\right)=+5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}7+\left(-2\right)=5$

$3+\left(-11\right)$

$\begin{array}{cc}\underset{\text{Smaller}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}}{\underbrace{|3|=3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}& \underset{\begin{array}{l}\text{Larger}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}\\ \text{Sign}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}"-".\end{array}}{\underbrace{|-11|=11}}\end{array}$

$\begin{array}{ll}\text{Subtract}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{values:}\hfill & 11-3=8.\hfill \\ \text{Attach}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{proper}\text{\hspace{0.17em}}\text{sign:}\hfill & "-".\hfill \end{array}$

$3+\left(-11\right)=-8$

The morning temperature on a winter's day in Lake Tahoe was $-12$ degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?

We need to find $-12+25$ .

$\begin{array}{cc}\underset{\text{Smaller}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}}{\underbrace{|-12|=12}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}& \underset{\begin{array}{l}\text{Larger}\text{​}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{value}\text{.}\\ \text{Sign}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{"+"}\text{.}\end{array}}{\underbrace{|25|=25}}\end{array}$

$\begin{array}{ll}\text{Subtract}\text{\hspace{0.17em}}\text{absolute}\text{\hspace{0.17em}}\text{values:}\hfill & 25-12=13.\hfill \\ \text{Attach}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{proper}\text{\hspace{0.17em}}\text{sign:}\hfill & "+".\hfill \end{array}$

$-12+25=13$

Thus, the afternoon temperature is 13 degrees.

Add $-147+84$ .                                                      Display Reads

$\begin{array}{l}\begin{array}{ccc}\text{Type}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}147& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}147\end{array}\\ \begin{array}{ccc}\text{Press}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{||}\hline +/-\\ \hline\end{array}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{147}\end{array}\\ \begin{array}{ccc}\text{Press}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{||}\hline +\\ \hline\end{array}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{147}\end{array}\\ \begin{array}{ccc}\text{Type}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}84& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}84\end{array}\\ \begin{array}{ccc}\text{Press}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{||}\hline =\\ \hline\end{array}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-63\end{array}\end{array}$

## Practice set b

Find the sums.

$4+\left(-3\right)$

1

$-3+5$

2

$15+\left(-18\right)$

$-3$

$0+\left(-6\right)$

$-6$

$-26+12$

$-14$

$35+\left(-78\right)$

$-43$

$15+\left(-10\right)$

5

$1.5+\left(-2\right)$

$-0.5$

$-8+0$

$-8$

$0+\left(0.57\right)$

$0.57$

$-879+454$

$-425$

$-1345.6+\left(-6648.1\right)$

$-7993.7$

## Exercises

Find the sums for the the following problems.

$4+12$

16

$8+6$

$6+2$

8

$7+9$

$\left(-3\right)+\left(-12\right)$

$-15$

$\left(-6\right)+\left(-20\right)$

$\left(-4\right)+\left(-8\right)$

$-12$

$\left(-11\right)+\left(-8\right)$

$\left(-16\right)+\left(-8\right)$

$-24$

$\left(-2\right)+\left(-15\right)$

$14+\left(-3\right)$

11

$21+\left(-4\right)$

$14+\left(-6\right)$

8

$18+\left(-2\right)$

$10+\left(-8\right)$

2

$40+\left(-31\right)$

$\left(-3\right)+\left(-12\right)$

$-15$

$\left(-6\right)+\left(-20\right)$

$10+\left(-2\right)$

8

$8+\left(-15\right)$

$-2+\left(-6\right)$

$-8$

$-11+\left(-14\right)$

$-9+\left(-6\right)$

$-15$

$-1+\left(-1\right)$

$-16+\left(-9\right)$

$-25$

$-22+\left(-1\right)$

$0+\left(-12\right)$

$-12$

$0+\left(-4\right)$

$0+\left(24\right)$

24

$-6+1+\left(-7\right)$

$-5+\left(-12\right)+\left(-4\right)$

$-21$

$-5+5$

$-7+7$

0

$-14+14$

$4+\left(-4\right)$

0

$9+\left(-9\right)$

$84+\left(-61\right)$

23

$13+\left(-56\right)$

$452+\left(-124\right)$

328

$636+\left(-989\right)$

$1811+\left(-935\right)$

876

$-373+\left(-14\right)$

$-1221+\left(-44\right)$

$-1265$

$-47.03+\left(-22.71\right)$

$-1.998+\left(-4.086\right)$

$-6.084$

$\left[\left(-3\right)+\left(-4\right)\right]+\left[\left(-6\right)+\left(-1\right)\right]$

$\left[\left(-2\right)+\left(-8\right)\right]+\left[\left(-3\right)+\left(-7\right)\right]$

$-20$

$\left[\left(-3\right)+\left(-8\right)\right]+\left[\left(-6\right)+\left(-12\right)\right]$

$\left[\left(-8\right)+\left(-6\right)\right]+\left[\left(-2\right)+\left(-1\right)\right]$

$-17$

$\left[4+\left(-12\right)\right]+\left[12+\left(-3\right)\right]$

$\left[5+\left(-16\right)\right]+\left[4+\left(-11\right)\right]$

$-18$

$\left[2+\left(-4\right)\right]+\left[17+\left(-19\right)\right]$

$\left[10+\left(-6\right)\right]+\left[12+\left(-2\right)\right]$

14

$9+\left[\left(-4\right)+7\right]$

$14+\left[\left(-3\right)+5\right]$

16

$\left[2+\left(-7\right)\right]+\left(-11\right)$

$\left[14+\left(-8\right)\right]+\left(-2\right)$

4

In order for a small business to break even on a project, it must have sales of $21,000$ . If the amount of sales was $15,000$ , how much money did this company fall short?

Suppose a person has $56.00$ in his checking account. He deposits $100.00$ into his checking account by using the automatic teller machine. He then writes a check for $84.50$ . If an error causes the deposit not to be listed into this person's account, what is this person's checking balance?

$-28.50$

A person borrows $7.00$ on Monday and then $12.00$ on Tuesday. How much has this person borrowed?

A person borrows $11.00$ on Monday and then pays back $8.00$ on Tuesday. How much does this person owe?

$3.00$

## Exercises for review

( [link] ) Simplify $\frac{4\left({7}^{2}-6\cdot {2}^{3}\right)}{{2}^{2}}$ .

( [link] ) Simplify $\frac{35{a}^{6}{b}^{2}{c}^{5}}{7{b}^{2}{c}^{4}}$ .

$5{a}^{6}c$

( [link] ) Simplify ${\left(\frac{12{a}^{8}{b}^{5}}{4{a}^{5}{b}^{2}}\right)}^{3}$ .

( [link] ) Determine the value of $|-8|$ .

8

( [link] ) Determine the value of $\left(|2|+{|4|}^{2}\right)+{|-5|}^{2}$ .

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