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Basic Operations with Real Numbers: Signed Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 familiar with positive and negative numbers and with the concept of opposites.

Overview

  • Positive and Negative Numbers
  • Opposites

Positive and Negative Numbers

When we studied the number line in Section (Reference) 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.
A number line with arrows on each end, labeled from negative six to six in increments of one. There are two closed circles at negative two and four, respectively.

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 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Practice Set A

Write each expression in words.

Exercise 1

4 + 10 4 + 10

Solution

four plus ten

Exercise 2

7 + ( − 4 ) 7 + ( − 4 )

Solution

seven plus negative four

Exercise 3

− 9 + 2 − 9 + 2

Solution

negative nine plus two

Exercise 4

− 16 − ( + 8 ) − 16 − ( + 8 )

Solution

negative sixteen minus positive eight

Exercise 5

− 1 − ( − 9 ) − 1 − ( − 9 )

Solution

negative one minus negative nine

Exercise 6

0 + ( − 7 ) 0 + ( − 7 )

Solution

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 a is any real number, then − a − a is its opposite. Notice that the letter a a is a variable. Thus, " a " " a " need not be positive, and " − a " " − a " need not be negative.

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

Two number lines with arrows on each end. The first number line has three labels, zero at the center, negative a to the left of zero and a to the right of zero. Negative a and a are equidistant from zero. The second line has three labels, zero at the center, a to the left of zero and negative a to the right of zero. The points a and negative a are equidistant from zero.

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

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

THE DOUBLE-NEGATIVE PROPERTY

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

Sample Set B

Example 7

If a = 3 a = 3 , then − a = − 3 − a = − 3 and − ( − a ) = − ( − 3 ) = 3 − ( − a ) = − ( − 3 ) = 3 .
A number line with arrows on each end, labeled from negative three to three in increments of three. Negative three is labeled as negative a, and three is labeled as a. There is an additional label for three as the opposite of negative a.

Example 8

If a = − 4 a = − 4 , then − a = − ( − 4 ) = 4 − a = − ( − 4 ) = 4 and − ( − a ) = a = − 4 − ( − a ) = a = − 4 .
A number line with arrows on each end, labeled from negative four to four in increments of three. Negative four is labeled as a, and four is labeled as negative a. There is an additional label for negative four as the opposite of negative a.

Practice Set B

Find the opposite of each real number.

Exercise 7

8

Solution

− 8 − 8

Exercise 8

17

Solution

− 17 − 17

Exercise 9

− 6 − 6

Solution

6

Exercise 10

− 15 − 15

Solution

15

Exercise 11

− ( − 1 ) − ( − 1 )

Solution

− 1 − 1 , since − ( − 1 ) = 1 − ( − 1 ) = 1

Exercise 12

− [ − ( − 7 ) ] − [ − ( − 7 ) ]

Solution

7

Exercise 13

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

Solution

If a a is positive, − a − a is negative.

Exercise 14

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

Solution

If a a is negative, − a − a is positive.

Exercise 15

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

Solution

We must say that we do not know.

Exercises

Exercise 16

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

Solution

a plus sign or no sign at all

Exercise 17

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 18

− 5 − 5

Solution

a negative five

Exercise 19

− 3 − 3

Exercise 20

12

Solution

twelve

Exercise 21

10

Exercise 22

− ( − 4 ) − ( − 4 )

Solution

negative negative four

Exercise 23

− ( − 1 ) − ( − 1 )

For the following problems, write the expressions in words.

Exercise 24

5 + 7 5 + 7

Solution

five plus seven

Exercise 25

2 + 6 2 + 6

Exercise 26

11 + ( − 2 ) 11 + ( − 2 )

Solution

eleven plus negative two

Exercise 27

1 + ( − 5 ) 1 + ( − 5 )

Exercise 28

6 − ( − 8 ) 6 − ( − 8 )

Solution

six minus negative eight

Exercise 29

0 − ( − 15 ) 0 − ( − 15 )

Rewrite the following problems in a simpler form.

Exercise 30

− ( − 8 ) − ( − 8 )

Solution

− ( − 8 ) = 8 − ( − 8 ) = 8

Exercise 31

− ( − 5 ) − ( − 5 )

Exercise 32

− ( − 2 ) − ( − 2 )

Solution

2

Exercise 33

− ( − 9 ) − ( − 9 )

Exercise 34

− ( − 1 ) − ( − 1 )

Solution

1

Exercise 35

− ( − 4 ) − ( − 4 )

Exercise 36

− [ − ( − 3 ) ] − [ − ( − 3 ) ]

Solution

− 3 − 3

Exercise 37

− [ − ( − 10 ) ] − [ − ( − 10 ) ]

Exercise 38

− [ − ( − 6 ) ] − [ − ( − 6 ) ]

Solution

− 6 − 6

Exercise 39

− [ − ( − 15 ) ] − [ − ( − 15 ) ]

Exercise 40

− { − [ − ( − 26 ) ] } − { − [ − ( − 26 ) ] }

Solution

26

Exercise 41

− { − [ − ( − 11 ) ] } − { − [ − ( − 11 ) ] }

Exercise 42

− { − [ − ( − 31 ) ] } − { − [ − ( − 31 ) ] }

Solution

31

Exercise 43

− { − [ − ( − 14 ) ] } − { − [ − ( − 14 ) ] }

Exercise 44

− [ − ( 12 ) ] − [ − ( 12 ) ]

Solution

12

Exercise 45

− [ − ( 2 ) ] − [ − ( 2 ) ]

Exercise 46

− [ − ( 17 ) ] − [ − ( 17 ) ]

Solution

17

Exercise 47

− [ − ( 42 ) ] − [ − ( 42 ) ]

Exercise 48

5 − ( − 2 ) 5 − ( − 2 )

Solution

5−( −2 )=5+2=7 5−( −2 )=5+2=7

Exercise 49

6 − ( − 14 ) 6 − ( − 14 )

Exercise 50

10 − ( − 6 ) 10 − ( − 6 )

Solution

16

Exercise 51

18 − ( − 12 ) 18 − ( − 12 )

Exercise 52

31 − ( − 1 ) 31 − ( − 1 )

Solution

32

Exercise 53

54 − ( − 18 ) 54 − ( − 18 )

Exercise 54

6 − ( − 3 ) − ( − 4 ) 6 − ( − 3 ) − ( − 4 )

Solution

13

Exercise 55

2 − ( − 1 ) − ( − 8 ) 2 − ( − 1 ) − ( − 8 )

Exercise 56

15 − ( − 6 ) − ( − 5 ) 15 − ( − 6 ) − ( − 5 )

Solution

26

Exercise 57

24 − ( − 8 ) − ( − 13 ) 24 − ( − 8 ) − ( − 13 )

Exercises for Review

Exercise 58

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

Solution

0

Exercise 59

((Reference)) Simplify ( 3 x y ) ( 2 x 2 y 3 ) ( 4 x 2 y 4 ) ( 3 x y ) ( 2 x 2 y 3 ) ( 4 x 2 y 4 ) .

Exercise 60

((Reference)) Simplify x n + 3 ⋅ x 5 x n + 3 ⋅ x 5 .

Solution

x n + 8 x n + 8

Exercise 61

((Reference)) Simplify ( a 3 b 2 c 4 ) 4 ( a 3 b 2 c 4 ) 4 .

Exercise 62

((Reference)) Simplify ( 4 a 2 b 3 x y 3 ) 2 ( 4 a 2 b 3 x y 3 ) 2 .

Solution

16 a 4 b 2 9 x 2 y 6 16 a 4 b 2 9 x 2 y 6

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