# 4.2 Real numbers: subtraction

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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: understand the definition of subtraction, be able to subtract signed numbers.

## Overview

• Definition of Subtraction
• Subtraction of Signed Numbers

## Definition of subtraction

We know from our experience with arithmetic that the subtraction $5-2$ produces 3, that is, $5-2=3$ . Illustrating this process on the number line suggests a rule for subtracting signed numbers.

We begin at 0, the origin.
Since 5 is positive, we move 5 units to the right.
Then, we move 2 units to the left to get to 3. (This reminds us of addition with a negative number.)

This illustration suggests that $5-2$ is the same as $5+\left(-2\right)$ .
This leads us directly to the definition of subtraction.

## Definition of subtraction

If $a$ and $b$ are real numbers, $a-b$ is the same as $a+\left(-b\right)$ , where $-b$ is the opposite of $b$ .

## Subtraction of signed numbers

The preceding definition suggests the rule for subtracting signed numbers.

## Subtraction of signed numbers

To perform the subtraction $a-b$ , add the opposite of $b$ to $a$ , that is, change the sign of $b$ and add.

## Sample set a

Perform the subtractions.

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

$4-9=4+\left(-9\right)=-5$

$-4-6=-4+\left(-6\right)=-10$

$-3-\left(-12\right)=-3+12=9$

$0-\left(-15\right)=0+15=15$

The high temperature today in Lake Tahoe was ${26}^{\circ }\text{F}$ . The low temperature tonight is expected to be $-{7}^{\circ }\text{F}$ . How many degrees is the temperature expected to drop?
We need to find the difference between 26 and $-7$ .

$26-\left(-7\right)=26+7=33$

Thus, the expected temperature drop is ${33}^{\circ }\text{F}$ .

$\begin{array}{lll}-6-\left(-5\right)-10\hfill & =\hfill & -6+5+\left(-10\right)\hfill \\ \hfill & =\hfill & \left(-6+5\right)+\left(-10\right)\hfill \\ \hfill & =\hfill & -1+\left(-10\right)\hfill \\ \hfill & =\hfill & -11\hfill \end{array}$

## Practice set a

Perform the subtractions.

$9-6$

3

$6-9$

$-3$

$0-7$

$-7$

$1-14$

$-13$

$-8-12$

$-20$

$-21-6$

$-27$

$-6-\left(-4\right)$

$-2$

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

18

$1-\left(-12\right)$

13

$86-\left(-32\right)$

118

$0-16$

$-16$

$0-\left(-16\right)$

16

$0-\left(8\right)$

$-8$

$5-\left(-5\right)$

10

$24-\left(-\left(-24\right)\right)$

0

## Exercises

For the following exercises, perform the indicated operations.

$8-3$

5

$12-7$

$5-6$

$-1$

$14-30$

$2-15$

$-13$

$5-18$

$1-7$

$-6$

$4-11$

$-6-5$

$-11$

$-8-14$

$-1-12$

$-13$

$-4-4$

$-6-8$

$-14$

$-1-12$

$-5-\left(-3\right)$

$-2$

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

$-7-\left(-12\right)$

5

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

$-4-\left(-15\right)$

11

$-11-\left(-16\right)$

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

5

$-8-\left(-14\right)$

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

$-5$

$-11-\left(-4\right)$

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

$-8$

$-12-\left(-11\right)$

$0-6$

$-6$

$0-15$

$0-\left(-7\right)$

7

$0-\left(-10\right)$

$67-38$

29

$142-85$

$816-1140$

$-324$

$105-421$

$-550-\left(-121\right)$

$-429$

$-15.016-\left(4.001\right)$

$-26+7-52$

$-71$

$-15-21-\left(-2\right)$

$-104-\left(-216\right)-\left(-52\right)$

164

$-0.012-\left(-0.111\right)-\left(0.035\right)$

$\left[5+\left(-6\right)\right]-\left[2+\left(-4\right)\right]$

1

$\left[2+\left(-8\right)\right]-\left[5+\left(-7\right)\right]$

$\left[4+\left(-11\right)\right]-\left[2+\left(-10\right)\right]$

1

$\left[9+\left(-6\right)\right]-\left[4+\left(-12\right)\right]$

$\left(11-8\right)-\left(1-6\right)$

8

$\left(5-12\right)-\left(4-10\right)$

$\left(1-10\right)-\left(2-15\right)$

4

$\left(0-8\right)-\left(4-12\right)$

$\left(-4+7\right)-\left(2-5\right)$

6

$\left(-6+2\right)-\left(5-11\right)$

$\left[-8+\left(-5+3\right)\right]-\left[9-\left(-3-5\right)\right]$

$-27$

$\left[-4+\left(-1+6\right)\right]-\left[7-\left(-6-1\right)\right]$

$\left[2-\left(-6+10\right)\right]-\left[1-\left(2-11\right)\right]$

$-12$

$\left[5-\left(-2-5\right)\right]-\left[2-\left(-1-4\right)\right]$

When a particular machine is operating properly, its meter will read 34. If a broken bearing in the machine causes the meter reading to drop by 45 units, what is the meter reading?

$-11$

The low temperature today in Denver was $-{4}^{\circ }\text{F}$ and the high was ${42}^{\circ }\text{F}$ . What is the temperature difference?

## Exercises for review

( [link] ) Use the distributive property to expand $4x\left(5y+11\right)$ .

$20xy+44x$

( [link] ) Simplify $\frac{2{\left(3{x}^{2}{y}^{2}\right)}^{3}{\left(2{x}^{4}{y}^{3}\right)}^{0}}{27{x}^{4}{y}^{3}}$ . Assume $x\ne 0,y\ne 0$ .

( [link] ) Simplify $|-\left({4}^{2}+{2}^{2}-{3}^{2}\right)|$ .

11

( [link] ) Find the sum. $-8+\left(-14\right)$ .

( [link] ) Find the sum. $3+\left(-6\right)$ .

$-3$

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