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10.2: Signed Numbers

Last updated

Jun 8, 2024
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- 10.1: Variables, Constants, and Real Numbers
- 10.3: Absolute Value

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Learing Objectives

be able to distinguish between positive and negative real numbers
be able to read signed numbers
understand the origin and use of the double-negative product property

Positive and Negative Numbers

Definition: Positive and Negative Numbers

Each real number other than zero has a sign associated with it. A real number is said to be a positive number if it is to the right of 0 on the number line and negative if it is to the left of 0 on the number line.

THE NOTATION OF SIGNED NUMBERS

+ and − Notation
A number is denoted as positive if it is directly preceded by a plus sign or no sign at all.
A number is denoted as negative if it is directly preceded by a minus sign.

Reading Signed Numbers

The plus and minus signs now have two meanings:

The plus sign can denote the operation of addition or a positive number.

The minus sign can denote the operation of subtraction or a negative number.

To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative." When "+" is used as an operation sign, it is read as "plus." When "-" is used as an operation sign, it is read as "minus."

Sample Set A

Read each expression so as to avoid confusion between "operation" and "sign."

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

Sample Set A

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

Sample Set A

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

Sample Set A

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

Sample Set A

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

Sample Set A

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

Practice Set A

Write each expression in words.

$6 + 1$

Answer: six plus one

Practice Set A

$2 + (-8)$

Answer: two plus negative eight

Practice Set A

$-7 + 5$

Answer: negative seven plus five

Practice Set A

$-10 - (+3)$

Answer: negative ten minus three

Practice Set A

$-1 - (-8)$

Answer: negative one minus negative eight

Practice Set A

$0 + (-11)$

Answer: zero plus negative eleven

Opposites

Opposites
On the number line, each real number, other than zero, 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.

Note

The letter " $a$ " is a variable. Thus, " $a$ " need not be positive, and " $-a$ " need not be negative.

If \9a\) is any real number, $-a$ is opposite $a$ on the number line.

$Two number lines. One number line with hash marks from left to right, -a, 0, and a. This number line is titled a positive. A second number line with hash marks from left to right, a, 0, and -a. This number line is titled a negative.$

The Double-Negative Property

The number $a$ is opposite $-a$ on the number line. Therefore, $-(-a)$ is opposite $-a$ on the number line. This means that $-(-a) = a$

From this property of opposites, we can suggest the double-negative property for real numbers.

Double-Negative Property: $-(-a) = a$
If $a$ is real number, then
$-(-a) = a$

Sample Set B

Find the opposite of each number.

If $a = 2$ , then $-a = -2$ , Also, $-(-a) = -(-2) = 2$ .

$A number line with hash marks from left to right, -2, 0, and 2. Below the -2 is -a, and below the 2 is a, or -(-a).$

Sample Set B

If $a = -4$ , then $-a = -(-4) = 4$ , Also, $-(-a) = a = -4$ .

$A number line with hash marks from left to right, -4, 0, and 4. Below the -4 is a, or -(-a), and below the 2 is -a.$

Practice Set B

Find the opposite of each number.

Answer: -8

Practice Set B

Answer: -17

Practice Set B

-6

Answer: 6

Practice Set B

-15

Answer: 15

Practice Set B

-(-1)

Answer: -1

Practice Set B

−[−(−7)]

Answer: 7

Practice Set B

Suppose $a$ is a positive number. Is $-a$ positive or negative?

Answer: $-a$ is negative

Practice Set B

Suppose $a$ is a negative number. Is $-a$ positive or negative?

Answer: $-a$ is positive

Practice Set B

Suppose we do not know the sign of the number $k$ . Is $-k$ positive, negative, or do we not know?

Answer: -17

We must say that we do not know.

Exercises

Exercise $\PageIndex{1}$

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

Answer: + (or no sign)

Exercise $\PageIndex{2}$

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

How should the number in the following 6 problems be read? (Write in words.)

Exercise $\PageIndex{3}$

-7

Answer: negative seven

Exercise $\PageIndex{4}$

-5

Exercise $\PageIndex{5}$

Answer: fifteen

Exercise $\PageIndex{6}$

Exercise $\PageIndex{7}$

-(-1)

Answer: negative negative one, or opposite negative one

Exercise $\PageIndex{8}$

-(-5)

For the following 6 problems, write each expression in words.

Exercise $\PageIndex{9}$

5 + 3

Answer: five plus three

Exercise $\PageIndex{10}$

3 + 8

Exercise $\PageIndex{11}$

15 + (-3)

Answer: fifteen plus negative three

Exercise $\PageIndex{12}$

1 + (-9)

Exercise $\PageIndex{13}$

-7 - (-2)

Answer: negative seven minus negative two

Exercise $\PageIndex{14}$

0 - (-12)

For the following 6 problems, rewrite each number in simpler form.

Exercise $\PageIndex{15}$

-(-2)

Answer: 2

Exercise $\PageIndex{16}$

-(-16)

Exercise $\PageIndex{17}$

-[-(-8)]

Answer: -8

Exercise $\PageIndex{18}$

-[-(-20)]

Exercise $\PageIndex{19}$

7 - (-3)

Answer: 7 + 3 = 10

Exercise $\PageIndex{20}$

6 - (-4)

Exercises for Review

Exercise $\PageIndex{21}$

Find the quotient; $8 \div 27$ .

Answer: $0.\overline{296}$

Exercise $\PageIndex{22}$

Solve the proportion: $\dfrac{5}{9} = \dfrac{60}{x}$

Exercise $\PageIndex{23}$

Use the method of rounding to estimate the sum: $5829 + 8767$

Answer: $6,000 + 9,000 = 15,000$ $(5,829 + 8,767 = 14,596)$ or $5,800 + 8,800 + 14,600$

Exercise $\PageIndex{24}$

Use a unit fraction to convert 4 yd to feet.

Exercise $\PageIndex{25}$

Convert 25 cm to hm.

Answer: 0.0025 hm

Learing Objectives

Positive and Negative Numbers

Definition: Positive and Negative Numbers

THE NOTATION OF SIGNED NUMBERS

Reading Signed Numbers

Sample Set A

Sample Set A

Sample Set A

Sample Set A

Sample Set A

Sample Set A

Opposites

Note

The Double-Negative Property

Sample Set B

Sample Set B

Exercises

Exercises for Review

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