# 3.2: Signed Numbers

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## Overview

• Positive and Negative Numbers
• Opposites

## Positive and Negative Numbers

When we studied the number line in Section 2.3 we noted that

Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line.

Positive and Negative Numbers
Each real number has a sign inherently associated with it. A real number is said to be a positive number if it is located to the right of 0 on the number line. It is a negative number if it is located to the left of 0 on the number line.

THE NOTATION OF SIGNED NUMBERS
A number is denoted as positive if it is directly preceded by a "+" sign or no sign at all.
A number is denoted as negative if it is directly preceded by a "−" sign.

The "+" and "−" signs now have two meanings:

+ can denote the operation of addition or a positive number.
− can denote the operation of subtraction or a negative number.

Read the "−" Sign as "Negative"
To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative."

## Sample Set A

##### Example $$\PageIndex{1}$$

−8 should be read as "negative eight" rather than "minus eight."

##### Example $$\PageIndex{2}$$

$$4+(−2)$$ should be read as "four plus negative two" rather than "four plus minus two."

##### Example $$\PageIndex{3}$$

$$−6+(−3)$$ should be read as "negative six plus negative three" rather than "minus six plus minus three."

##### Example $$\PageIndex{4}$$

$$−15−(−6)$$ should be read as "negative fifteen minus negative six" rather than "minus fifteen minus minus six."

##### Example $$\PageIndex{5}$$

$$−5+7$$ should be read as "negative five plus seven" rather than "minus five plus seven."

##### Example $$\PageIndex{6}$$

$$0−2$$ should be read as "zero minus two."

## Practice Set A

Write each expression in words.

##### Practice Problem $$\PageIndex{1}$$

$$4 + 10$$

four plus ten

##### Practice Problem $$\PageIndex{2}$$

$$7 + (-4)$$

seven plus negative four

##### Practice Problem $$\PageIndex{3}$$

$$-9 + 2$$

negative nine plus two

##### Practice Problem $$\PageIndex{4}$$

$$-16 - (+8)$$

negative sixteen minus positive eight

##### Practice Problem $$\PageIndex{5}$$

$$-1 -(-9)$$

negative one minus negative nine

##### Practice Problem $$\PageIndex{6}$$

$$0 + (-7)$$

zero plus negative seven

## Opposites

Opposites
On the number line, each real number has an image on the opposite side of $$0$$. For this reason we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.

The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if $$a$$ is any real number, then $$−a$$ is its opposite. Notice that the letter $$a$$ is a variable. Thus, "$$a$$" need not be positive, and "$$−a$$" need not be negative.

If $$a$$ is a real number, $$−a$$ is opposite $$a$$ on the number line and $$a$$ is opposite $$−a$$ on the number line.

$$−(−a)$$ is opposite $$−a$$ on the number line. This implies that $$−(−a)=a$$.

This property of opposites suggests the double-negative property for real numbers.

##### The Double-Negative Property

If a is a real number, then
$$−(−a)=a$$

## Sample Set B

##### Example $$\PageIndex{7}$$

If $$a=3$$, then $$−a=−3$$ and $$−(−a)=−(−3)=3$$.

##### Example $$\PageIndex{8}$$

If $$a=−4$$, then $$−a=−(−4)=4$$ and $$−(−a)=a=−4$$.

## Practice Set B

Find the opposite of each real number.

##### Practice Problem $$\PageIndex{7}$$

$$8$$

$$-8$$

##### Practice Problem $$\PageIndex{8}$$

$$17$$

$$-17$$

##### Practice Problem $$\PageIndex{9}$$

$$-6$$

$$6$$

##### Practice Problem $$\PageIndex{10}$$

$$-15$$

$$15$$

##### Practice Problem $$\PageIndex{11}$$

$$-(-1)$$

$$-1$$, since$$-(-1) = 1$$

##### Practice Problem $$\PageIndex{12}$$

$$-[-(-7)]$$

$$7$$

##### Practice Problem $$\PageIndex{13}$$

Suppose that a is a positive number. What type of number is $$−a$$ ?

If $$a$$ is positive, $$-a$$ is negative.

##### Practice Problem $$\PageIndex{14}$$

Suppose that $$a$$ is a negative number. What type of number is $$−a$$ ?

If $$a$$ is negative, $$-a$$ is positive.

##### Practice Problem $$\PageIndex{15}$$

Suppose we do not know the sign of the number $$m$$. Can we say that $$−m$$ is positive, negative, or that we do not know ?

We must say that we do not know.

## Exercises

##### Exercise $$\PageIndex{1}$$

A number is denoted as positive if it is directly preceded by ____________________ .

a plus sign or no sign at all

##### Exercise $$\PageIndex{2}$$

A number is denoted as negative if it is directly preceded by ____________________ .

For the following problems, how should the real numbers be read ? (Write in words.)

##### Exercise $$\PageIndex{1}$$

$$-5$$

a negative five

##### Exercise $$\PageIndex{2}$$

$$-3$$

##### Exercise $$\PageIndex{3}$$

$$12$$

twelve

##### Exercise $$\PageIndex{4}$$

$$10$$

##### Exercise $$\PageIndex{5}$$

$$-(-4)$$

negative negative four

##### Exercise $$\PageIndex{6}$$

$$-(-1)$$

For the following problems, write the expressions in words.

##### Exercise $$\PageIndex{7}$$

$$5 + 7$$

five plus seven

##### Exercise $$\PageIndex{8}$$

$$2 + 6$$

##### Exercise $$\PageIndex{9}$$

$$11 + (-2)$$

eleven plus negative two

##### Exercise $$\PageIndex{10}$$

$$1 + (-5)$$

##### Exercise $$\PageIndex{11}$$

$$6 - (-8)$$

six minus negative eight

##### Exercise $$\PageIndex{12}$$

$$0 - (-15)$$

Rewrite the following problems in a simpler form.

##### Exercise $$\PageIndex{13}$$

$$−(−8)$$

$$−(−8)=8$$

##### Exercise $$\PageIndex{14}$$

$$−(−5)$$

##### Exercise $$\PageIndex{15}$$

$$−(−2)$$

$$2$$

##### Exercise $$\PageIndex{16}$$

$$−(−9)$$

##### Exercise $$\PageIndex{17}$$

$$−(−1)$$

$$1$$

##### Exercise $$\PageIndex{18}$$

$$−(−4)$$

##### Exercise $$\PageIndex{19}$$

$$−[−(−3)])$$

$$-3$$

##### Exercise $$\PageIndex{20}$$

$$−[−(−10)]$$

##### Exercise $$\PageIndex{21}$$

$$−[−(−6)]$$

$$-6$$

##### Exercise $$\PageIndex{22}$$

$$−[−(−15)]$$

##### Exercise $$\PageIndex{23}$$

$$−\{−[−(−26)]\}$$

$$26$$

##### Exercise $$\PageIndex{24}$$

$$−\{−[−(−11)]\}$$

##### Exercise $$\PageIndex{25}$$

$$−\{−[−(−31)]\}$$

$$31$$

##### Exercise $$\PageIndex{26}$$

$$−\{−[−(−14)]\}$$

##### Exercise $$\PageIndex{27}$$

$$−[−(12)]$$

$$12$$

##### Exercise $$\PageIndex{28}$$

$$−[−(2)]$$

##### Exercise $$\PageIndex{29}$$

$$−[−(17)]$$

$$17$$

##### Exercise $$\PageIndex{30}$$

$$−[−(42)]$$

##### Exercise $$\PageIndex{31}$$

$$5−(−2)$$

$$5−(−2)=5+2=7$$

##### Exercise $$\PageIndex{32}$$

$$6−(−14)$$

##### Exercise $$\PageIndex{33}$$

$$10−(−6)$$

$$16$$

##### Exercise $$\PageIndex{34}$$

$$18−(−12)$$

##### Exercise $$\PageIndex{35}$$

$$31−(−1)$$

$$32$$

##### Exercise $$\PageIndex{36}$$

$$54−(−18)$$

##### Exercise $$\PageIndex{37$$

$$6−(−3)−(−4)$$

$$13$$

##### Exercise $$\PageIndex{38}$$

$$2−(−1)−(−8)$$

##### Exercise $$\PageIndex{39}$$

$$15−(−6)−(−5)$$

$$26$$

##### Exercise $$\PageIndex{40}$$

$$24−(−8)−(−13)$$

## Exercises for Review

##### Exercise $$\PageIndex{41}$$

There is only one real number for which $$(5a)^2=5a^2$$. What is the number?

$$0$$

##### Exercise $$\PageIndex{42}$$

Simplify $$(3xy)(2x^2y^3)(4x^2y^4)$$.

##### Exercise $$\PageIndex{43}$$

Simplify $$x^{n + 3} \cdot x^5$$

$$x^{n + 8}$$

##### Exercise $$\PageIndex{44}$$

Simplify $$(a^3b^2c^4)^4$$.

##### Exercise $$\PageIndex{45}$$

Simplify $$(\dfrac{4a^2b}{3xy^3})^2$$

$$\dfrac{16a^4b^2}{9x^2y^6}$$