Download 3.2 Roots and Fractional Exponents

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3.2 Roots and Fractional Exponents
Roots
Roots are the inverse of exponents.
For example, the square of 2 (or raising 2 to the exponent 2) is 4; i.e., 22 = 4. The inverse of squaring a
number is finding the square root of that number. Therefore, the square root of 4 is 2.
Similarly, the square of 3 is 9; i.e., 32 = 9. Therefore, the square root of 9 is 3.
Two types of notations may be used to represent the above. One is using the symbol ‘
represents a radical. The other is using fractional exponents.
For example, the square root of 9 can be represented as
exponent form.
1
9 in radical form, or as 9 2 in fractional
1
9 = 92
Square root of 9 =
We know that 9 = 3 × 3 = 32
Similarly,
’, which
Therefore, the square root (2nd root) of 9 is 3.
8 = 2 × 2 × 2 = 23
Therefore, the cube root (3rd root) of 8 is 2.
4
Therefore, the 4th root of 16 is 2.
16 = 2 × 2 × 2 × 2 = 2
A radical is an indicated root of a number (or expression).
nd
2
9 indicates the 2
3
rd
8 indicates the 3 root (cube root) of 8.
4
n
root (square root) of 9.For square roots, the index 2 does not need to
be written as it is understood to be there, i.e.,
2
9 = 9.
th
16 indicates the 4 root of 16.
th
a refers to the n root of a.
Index of the root
n
Radical sign
a
‘a’ represents any positive number
The index is written as a small number on the left of the radical symbol. It indicates which root is to
be taken. 3 125 indicates the 3rd root or cube root of 125.
Perfect Roots
Roots of a whole number may not always be a whole number. A whole number is a perfect root if its
root is also a whole number.
For example,
4 is a perfect square root of 16 because 42 = 16; i.e., 16 = 4
3 is a perfect cube root of 27 because 33 = 27; i.e., 3 27 = 3
Table 3.2
Examples of Perfect Roots
1
Roots
1
Square Roots
2
3
4
9
3
3
4
5
6
7
8
9
10
16
25
36
49
64
81
100
3
3
Cube Roots
3
1 3 8
Fourth Roots
4
1 4 16 4 81 4 256 4 625 4 1, 296
27
64
125
216
3
4
343
2, 401
3
4
3
512
4, 096
4
729
6, 561
3
4
1, 000
10, 000
The Product Rule and the knowledge of Perfect Roots are used in simplifying square roots, cube roots, etc.
3.2 Roots and Fractional Exponents
90
For Example,
(i) To simplify 12 , 12 may be written as 4 × 3, a combination of two factors, where one of the
factors, 4, is a perfect square.
i.e.,
12 =
4
# 3=2 3
3
(ii) To simplify 54 , may be written as 54 as 27 × 2, a combination of two factors, where one of
the factors, 27, is a perfect cube of 3.
i.e., 3 54 =
Example 3.2-a
3
27 # ^3 2 h = 3^3 2 h
Finding Perfect Roots
Simplify using perfect roots of a number:
Solution
(i)
72
(i)
72
=
(ii)
36 # 2 3
=
40
72 = 36 × 2
= 6 2 (ii)
3
36 is a perfect square of 6.
40
3
=2
8 # ^3 5 h ^3
40 = 8 × 5
5 h 8 is a perfect cube of 2.
Fractional Exponents
Fractional exponents are easier to write than radical notations. As explained earlier, square (or 2nd)
root is written as the power ‘ 1 ’ in fractional exponent notation.
2
1
5 = 52
For example,
Cube (or 3rd) root is written as the power ‘ 1 ’ in fractional exponent notation.
3
1
For example, 3 8 = 8 3
The fourth (4th) root is written as the power ‘ 1 ’ in fractional exponent notation.
4
For example,
4
3 1
3
3
5 = (5 ) 4 = 5 4
An appropriate radical will “undo” an exponent.
For example,
1
52 = (5 2) 2 = 5
3
1
73 = (73 ) 3 = 7
When entering a fractional exponents in a calculator, brackets must be used.
For example,
2
2
In order to evaluate 25 5 brackets around must be used.
5
2
To evalute 25 5 using a calculator, enter it as follows: it would be entered as;
3.623898...
Exponent key
Without the brackets, the operation will mean (25)2 ÷ 5, which is incorrect.
Chapter 3 | Operations with Exponents and Integers
91
Evaluating Expressions with Fractional Exponents Using a Calculator
Example 3.2-b
Solve the following:
1
(i)
3
15 2
(i)
15 2 =
Solution
3
` 5 j4
(ii)
3
(iii) ^2.5h7
3
= 58.09
58.094750...
1
3
` 5 j4 =
(ii)
0.880111...
= 0.88
1.480968...
= 1.48
3
(iii) ^2.5h7 =
Arithmetic Operations with Fractional Exponents
All the rules of exponents Product Rule, Quotient Rule, Power of a Product Rule, Power of a Quotient
Rule, Power of a Power Rule, etc. learned in Section 3.1 and as outlined in Table 3.1-a and Table 3.1-b
are applicable to fractional exponents.
Solving Expressions with Fractional Exponents Using the Product Rule
Example 3.2-c
Simplify the following using the Product Rule to express in exponential form and then evaluate to two
decimal places:
(i)
(i)
Solution
1
1
1
1
1
(ii)
22 # 23
1 1
`2 + 3j
=
22 # 23 = 2
1
1 1
`2 + 3j
=
22 # 23 = 2
1
1
1
1
1 1
`2 + 3j
=
1
1
1 1
`2 + 3j
=
22 # 23 = 2
9
2 6 = 1.781797... = 1.78
5
0
`3
0
`3
+
9
+
0j
12
+
9
+
4
4
32 46 #
# 3 = 3=4 1.78
= 31.781797...
95
12
3 23
0
` 3 + 9 + 0j
` +
63 4j#
= 3 4 4 = 1.78
=34
=324 6#=3 1.781797...
3 2
` +
6 j=
3 5 9
3 2
` +
6 j=
3 42#6 3=4
2
35
7
7
7
2
7
2
(iii) ` 3 j3 # ` 3 j 3
5
5
12
3 2 3
59
0
` 3 + 9 + 0j
` +
6 3j4=#2364 =
# 31.781797...
= 3 4 4= 1.78
=34
2
2
9
2
1 1
`2 + 3j
=
22 # 23 = 2
22 # 23 = 2
3 2
` +
j
6(ii)
=
2
3
3 4 # 3 4 # 30
0j
7
12
9
3
3 3 = 3 ` 7 +3 2 j = 3 3 = 3
3 4 `=3 3j 3 =
# `27.00
=(iii)
`5j
`5j
`5
5
5j
2
9
3 33
3 3 = 3 ` 7 +3 2 j = 3 3 = 3 3 = 0.216 =
=
` 53j =# `27.00
`5j
`5j
`5j
5j
2
9
7+ 2
3
3
3 27.00
# ` 3 j 3 = ` 3 j` 3 j = ` 3 j 3 = ` 3 j = 0.216 = 0.22
=` 33 j=
5
5
5
5
5
2
9
7+ 2
3
=` 33j3 3 # ` 3 j 3 = ` 3 j` 3 j = ` 3 j 3 = ` 3 j = 0.216 = 0.22
5
5
5
5
5
7 3
2
9
3 `7+ 2j
3 3
3 3
3
# 3 = 3 4 = 41.78 = 3 4 3= 33#=327.00
1.781797...
` 5 j ` 5 j = ` 5 j 3 = ` 5 j = ` 5 j = 0.216 = 0.22
3.2 Roots and Fractional Exponents
92
Solving Expressions with Fractional Exponents Using the Quotient Rule
Example 3.2-d
Simplify the following using the Quotient Rule to express the answer in exponential form and then
evaluate to two decimal places:
Solution
(i)
4
23
(i)
4
23
'2
=2
2
3
5
2
'23
` 34
-
=2
4
2
3- 3
=2
4- 2
3 (ii)
=
2
5
1
5
2h21.587401...
.2h` 2
' ^1.2h2 ==^11.59
2^31.=
2
3j
4–2
` 25
(1.2)
5
1
-
4
2
4
2
3- 3
23 '23 = 2
4
2
23 '23 = 2
4
2
3- 3
4- 2
5
3^1.2h
=2
=2
4 - 25
2
1
5
4
1
5
6
3
6- 3
14 4
1 4
1 ` 4 j
1
2 j(iii)
= ^1.`23h2j ='^`13.2jh2 ==`1.44
3j
3
6- 3
4 j
3
3
1 4
= `3j
^1.2h` 2 - 2=j =
^1.2h2 = ^1.2h2 = 1.44
1.587401...
1.59
= 2^1.23 h2 '
=^12.32h=2 =
Solving Expressions with Fractional Exponents Using the Power of a Product Rule
Example 3.2-e
Simplify the following using the Power of a Product Rule to express the answer in exponential form
and then evaluate to two decimal places:
Solution
1
(16 # 9) 2
1
1
1
(16 # 9) 2 = 16 2 # 9 2 =
1
1
1
=
1
(42 × 32) 2
(i)
(42 × 32) 2
1
16 2
1
2
^ 4 h2
1
(i)
1
1
# 92
# `3
1
2 2
j
1
(ii)
b72 ×
1 12
l
32
(iii) (26 × 32) 2
(ii)
b72 ×
1 12
l
32
(iii) (26 × 32) 2
1
1
1
49 2
9 21
^72h2
1
3
3
1
3
3
2 2
2 2
7
1
49 2 ^72h2
6 26
= ^4 h # `3 j b=494## 3l2==12.00
××(3322) 2
(26=×(232)(2
= 2.333333... = 2.33
1 =
1 =
9
1
2
1
= 4 # 3 = 12.00
b49 # l =
9
1
1
1
1
92
=
1
2
1 ^3 h2
3
^32h2
7
= = 2.333333... = 2.33
3
= 29 × 33
= 512 × 27
2
2
1 2 49 2 ^72h2
7
9) 2 = 16 2 # 9 2 = ^4 h2 # `3 j2 = 4 # 3 = 12.00
2.333333... = 2.33
b49 # l = 1 =
1 = 3 =
9
2 2
92
^3 h
= 13,824.00
Solving Expressions with Fractional Exponents Using the Power of a Quotient Rule
Example 3.2-f
Simplify the following using the Power of a Quotient Rule to express the answer in exponential
form and then evaluate to two decimal places:
1
1
42 2
l
32
(i)
b
(i)
b
53 3
l
26
(ii)
b
(ii)
b
1
1
Solution
42 2
l
32
1
1
1
^ 42h 2
16 2 16 2
= 4 = 1.333333... = 1.33
`9j = 1 =
1
3
2
92
^3 h 2
1
1
2
53 3
l
26
=
5
22
=
5
4
1
^4 h2
16 2 16 2
= 4 = 1.333333... = 1.33
`9j = 1 =
1
3
2
1 9 2 2 1^3 h 2
1
2
2
^4 h
16 2 16
= 4 = 1.333333... = 1.33
`9j = 1 =
1
3
2
92
^3 h 2
Chapter 3 | Operations with Exponents and Integers
= 1.25
1
1
1
= ` 1 j 4 = ` 1 j 2 = 3 2 = 1.7
3
3
4
1
3
= ` 1 j4 = ` 1 j 2
3
3
= 0.438691... = 0.44
j = ^1.2h2 = ^1.2h2 = 1.44
2 = ^1.2h` 2=- 2
2'3^1=.2h1.587401...
1.59
2 1
3
1 `
1 4
1 4
` 3 j '` 3 j = ` 3 j
4
1
-
6
1
2j
`
j
2
` - j
= 2 ^13 .2h2 ' ^1.2h2 = ^1.2h 2 2 = ^1.2h2 = ^1.2h = 1.44
5
6
1
1
(iii) ` 3 j 4 ' ` 3 j 4
1
(ii) ^1.2h2 ' ^1.2h2
93
Example 3.2-g
Solving Expressions with Fractional Exponents Using the Power of a Power Rule
Simplify the following using the Power of a Power Rule to express the answer in exponential form
and then evaluate to two decimal places:
(i)
1 3
1
1
2 3 2
(iii) =b l G
3
4
(ii) `18 3 j
b6 2 l
1
1 3
1
3
1 #1
1
1 4
2 3 2
`6 2 j = 6` 2 # 3j = 6 2 = 14.696938...
=
3 j14.7018` 3 4 j
12
b
lG
(ii) `18=
(iii)
=
= 18 = 1.272348...
3 = 1.27
2
1
1 3
1
3
2 3 = 2 3 # 2 = 2 6 = 26 = 64 = 0.08
1
1 4
` 13 # 14 j
`6 2 j = 6` 2 # 3j = 6 2 = 14.696938...
;
`
`3j
`3j
12
`18 3 =
j 14.70
729
3 j E = 1.27
= 18
= 18 = 1.272348...
36
1
2
6
3
3
2
6
3
#
1
1 #3
3
1 #1
1 4
1
2
2
2
2
64
`
j
=` j = 6 =
= 0.0877915... =
`6 2 j = 6` 2 j = 6 2 = 14.696938...
;` 3 j E ==1.27
`18 3 j ==14.70
`3j
18 3 4 = 18 12 = 1.272348...
3
729
3
1
1 3
1
3
1
1
2 32
2 3 # 2 = 2 6 = 26 = 64 = 0.0877915... = 0.09
` 13 # 14 j
`6 2 j = 6` 2 # 3j = 6 2 = 14.696938...
`18 3 j 4 = 18
;` 3 j E ==1.27
14.70
`3j
`3j
= 18 12 = 1.272348...
729
36
1
2
1 3
1 #3
3
1 #1
1
1 4
6
`
j
2 3 = 2 3 # 2 = 2 6 = 2 = 64 = 0.0877915... = 0.09
`6 2 j = 6` 2 j = 6 2 = 14.696938...
`18 3 j ==14.70
18 3 4 = 18 12 = 1.272348... =;` 1.27
`3j
`3j
3j E
729
36
= 0.087791... = 0.09
Solution
Example 3.2-h
(i)
Expressions in Radical Form
Express the following in radical form:
Solution
5
26
(i)
26
(iii) c 2 m4
3
(ii) 3 5
5
3
2
(ii)
(iii) c 2 m4
3
35
= 6 25
Example 3.2-i
3
2
(i)
= 5 32
=
4
c2m
3
3
Solving Expressions with Fractional Exponents and Different Bases
Solve the following to two decimal places:
(i)
1
1
16 2 + 8 2
1
1
(iv) ^5 h2 # ^3 h2
1
1
1
(ii)
– 27 3
25 2 -
(v)
^ 2 h 4 ' ^ 3 h2
3
1
1
1
1
1
1
(iii) ` 7 j4 -– ` 2 j3
8
3
(2 34 )
(vi) 5
7
11
11
7 - 22 33 =
0.967168... –– 0.873580...
0.873580... =
=
25 2 16 2 + 8 2 = 4 + 2.828427...(ii)
= 5 - 3 = 2.00 (iii) `` 88 jj44 – 27=3 6.83
= 6.828427...
– `` 33 jj = 0.967168...
1
1
7
2
= 4 + 2.828427...
=5–3
` 8 j 4 - ` 3 j3 = 0.967168... – 0.873580... = 0.093587
1
1
1
1
1
1
7
2 3 = 0.967168... – 0.873580... = 0.093587... = 0.09
16 2 + 8 2 = 4 + 2.828427... = 6.828427...25
- 27 3 =` 85j-4 3- =` 32.00
j
= 26.83
1
1
1
1
7
2
16 2 + 8 2 = 4 + 2.828427... = 6.828427... = 6.83
` 8 j 4 - ` 3 j3 = 0.967168... – 0.873580... = 0.093587... = 0.09
3
1
1
1
(2 3 )
(vi) 5 4 = 0.970983 = 0.97
(v) ^2 h4=' 3.872979...
(iv) ^5 h2 # ^3 h2 = 2.236067 # 1.732050...
= 3.87 ' 1.732050...
^3 h2 = 1.681792...
Solution
(i)
3
1
11
= 2.236067... × 1.732050...
= 5 4= 0.97
^2 h4 ' ^3 h2 = 1.681792... ' 1.732050... = 0.970983
= 3.872983...
= 0.970983...
= 83.592538...
= 3.87
= 0.97
= 83.59
3.2 Roots and Fractional Exponents
94
Negative Exponents
In the exponential notation of a number, the base of the number may be raised to a negative exponent.
When the exponent is negative, it is represented by a−n. The negative exponent is the reciprocal of a
positive exponent.
Positive Exponent: an = a × a × a × a × a ×…× a
(multiplication of ‘n’ factors of ‘a’)
1
(division of ‘n’ factors of ‘a’)
Negative Exponent: a−n = 1n =
a # a # a # a # a # ... # a
a
−n
a
or
= 1n ,
a
1 = an
a–n
Therefore, an and a−n are reciprocals.
The properties (rules) of exponents in Section 3.1 of this chapter (summarized in Table 3.1-a) also
apply to all negative exponents.
We use these properties to simplify negative exponents and then convert negative exponents to
positive exponents.
Example 3.2-j
Using Properties of Exponents to Simplify Negative Exponents
Simplify the following and express the answer with positive exponents (do not evaluate):
(i)
Solution
–2
–3
2 ×2
–2
–3
(i) 2 × 2
(ii)
3–4
3–2
3
(iii) (3 × 5)–2
Using Product Rule,
(iv) (3–2)
(ii) 3–4
3–2
Using Quotient Rule,
= 2–2 + (–3)
= 3–4 – (–2)
= 2–2 – 3
= 2–5
= 3–4 + 2
= 3–2
Using Negative Exponent Rule,
= 12
3
= 15
2
(iii)(3 × 5)–2
3
Using Power of a Product Rule, (iv) (3–2)
= 3–2 × 5–2 Using Negative Exponent Rule,
= 3–2(3)
= 12 × 12
3
5
4
= 54
3
= 3–6
= 16
3
Chapter 3 | Operations with Exponents and Integers
Using Negative Exponent Rule,
Using Power of a Power Rule,
Using Negative Exponent Rule,
95
Repeated
Multiplication
Positive
Exponential
Notation
5 × 5 × 5 × 5 = 54
5 × 5 × 5 = 53
5 × 5 = 52
5 = 51
1 = 50
Factors are
divided by 5
Repeated
Division
Exhibit 3.2
Exponents are
reduced by 1
If this pattern is continued further...
Negative
Exponential
Notation
1
5
= 5–1
1
5#5
= 5–2
1
5#5#5
= 5–3
Repeated Multiplication of a Number and its Exponential Notation
Any positive number base with a negative exponent will always result in a positive answer.
For example,
Negative
Exponential
Notation
Positive
Exponential
Notation
Repeated
Division
Standard
Notation
5−1
1
51
1
5
1
5
5−2
1
52
1
5#5
1
25
5−3
1
53
1
5#5#5
1
125
Fractions with Negative Exponents
When a fraction has a negative exponent, change the fraction to its reciprocal and drop the sign
of the exponent. After this change, the number in the exponent indicates the number of times the
numerator and denominator should be multiplied.
a -n
b n
b l = c m
b
a
For example,
5 5 5
5 # 5 # 5 125
2 -3 5 3
`5j = `2j = `2j`2j`2j = 2 # 2 # 2 = 8
2 5
Note: The reciprocal of 5 is 2 .
3.2 Roots and Fractional Exponents
96
Solving Fractions with Negative Exponents
Example 3.2-k
Solve the following: (i)
Solution
3 –3
2 –2
(ii) ` 5 j ÷ ` 75j
5 –2
2 –3
`4j × `3j
–3
5 –2
`4j × `2j
3
3 3
4 2
= ` j × ` j
2
5
3
42
3
= 2 × 3
5
2
3 –3
2 –2
(ii) ` 5 j ÷ ` 75j
3 –3
75 – 2
= ` 5 j × ` 2 j
3 –3
2 2
= ` 5 j × ` 75j
27
= 16 × 8
25
2
= 3 3 × 3
75
5
= 125 × 4
27 25
= 5 × 4
27 25
= 20
27
(i)
3
2
27
= 25 × 1
54
= 25
4 = 2.16
= 2 25
3.2 Exercises
3
Answers to odd-numbered problems are available at the end of the textbook.
Express Problems 1 to 4 in their radical form and evaluate.
1
1
1.
a. 64 2
3.
a. 8
25
b. ` 16 j2
27 3
b. ` 64 j
8
b. ` 25 j2
1
125 3
b. ` 8 j
a. 81 2
4.
a. 64 3
1
1
3
1
1
2.
1
Express Problems 5 to 14 in their fractional exponent form and then evaluate and round the answers to two decimal places
wherever applicable.
5.
a. 144
7.
a.
26
9.
a.
8 # 12
11. a.
2
4
25 # 25
2
b. 3 64
6.
a.
81
b. 3 125
b.
40
8.
a.
34
b.
50
b.
7 # 14
10. a. 12 # 10
4
4
12. a. 2 3 # 2
b.
9 # 27
b.
2
4
5 # 25
3
b.
3
6
9 # 27
4
25
24
36
64
b.
14. a.
b.
49
6
64
9
Simplify Problems 15 to 24 by expressing them as single exponents (using the properties of exponents) and then evaluate and
round the answers to two decimal places, wherever applicable.
13. a.
1
3
4
2
1
3
2
3
7
5
1
1
b. 3 8 # 3 9
15. a. 5 2 # 5 4
1
17. a. 8 5 # 8 5 # 8 5
1
19. a. 8 # 8 # 8
b. 5 3 # 5 2 # 5
8
33
b. 2
3
5
7
21. a. 4 2
47
b. _3
Chapter 3 | Operations with Exponents and Integers
1
2 3
i
1
1
4
4
2
3
1
2
3
0
6
18. a. 5 7 # 5 7 # 5 7
1
20. a. 2 # 2 # 2
4
5
22. a. 2 3
25
2
b. 11 4 # 11 3
16. a. 3 2 # 3 4
5
2
b. 9 8 # 9 3 # 9
7
2
b. 6 2
6
0
3
b. ^10 h
0
97
23.
1 8
1 4
b 4l
b. 7
a. `12 2 j
24.
2 6
2 6
b 3l
b. 4
a. b5 3 l
Evaluate Problems 25 to 32 and express the answers rounded to two decimal places wherever applicable.
25.
27.
29.
1
1
1
1
2
a. 5 # 3 + 2
1
2
a. 8 # 9
1
1
2
a.
26.
1
5 2
2 5
b. (2 ) + (5 )
1
2
1
2
b. 45 # 60
1
31.
1
1
b. 16 2 – 9 2
a. 5 2 + 7 2
5 + 42
b.
1
36 2
1
10 2
–
1
28.
1
2
30.
1
a. 12 # 10 + 5
1
2
a. 6 # 3
32.
25 2
a.
1
b. 50 2 – 40 2
1
2
1
1
52
1
a. 125 3 + 64 3
1
1
2
4 3
1
3 4
b. (3 ) + (4 )
1
1
2
1
b. 24 2 # 75 2
1
1
62 + 62
7 72
b. – 1
42
1
92
Simplify Problems 33 to 42 by expressing them as single exponents (using the properties of exponents) and as a radical, then
evaluate and round the answers to two decimal places, wherever applicable.
5
3
33. a. 6- 4 # 6 4
4
2
4
b. 7 3 # 7– 3
2
b.
5
6
2 7 # 2- 7
8
2- 7
37.
5
6- 9 # 6 0
a.
7
6- 9
b.
78 # 7 3
72
39.
a. (5–2) 3
41.
a. (8– 3 )–6
35.
a.
3
4
10- 5 # 10 5
10 5
7
4
2
4
57 # 57
36.
a.
38.
9 5 # 90
a.
3
9- 5
40.
a. (4–2) 2
42.
a. (6– 3 )–3
8
b.
6
5–7
1
1
b. (7– 3 )9
2
4
3 3 # 3- 3
5
33
5
2
b. (6– 2 )–6
2
6
2
b. 3- 7 # 3 7
2
34. a. 5 9 # 5– 9
b.
5
2
56 # 53
52
4
b. (2– 5 )–5
2
4
b. (3– 9 )0
Evaluate Problems 43 to 48 and express the answers rounded to two decimal places, wherever applicable.
43.
–1
a. 3–1
2
b. 3–1 + 2–1
44.
–2
a. 2–1
3
b. 2–2 + 3–1
45.
a. 3–1 × 32 × 3–2
b. [2–3]–1
46.
a. 5–2 × 52 × 53
b. [5–2]–2
b. 2–2 + 1–1
2
48.
–1
–1
a. 3 + 2 ×–13
1
3
b. 3–2 +
47.
2
0
5
× 2 –1
a. 2 + 3 –1
1
2
1
3–1
3.3 Arithmetic Operations with Signed Numbers
Introduction
In the previous chapters, you learned that positive real numbers can be represented by points on a
number line from zero to the right of the zero. That is, whole numbers, positive integers, and positive
rational and irrational numbers can be represented on a number line from zero to the right of the zero.
Every positive number has a negative number known as its opposite. Zero, ‘0’, is neither positive nor negative.
The numbers that are to the left of the zero on the number line represent negative numbers. We use the
negative sign, ‘−’, to represent negative integers, and the positive sign, ‘+’, to represent positive integers.
–6.5
−7 −6
–4
−5
−4
–0.75
−3
−2
−1
Negative real numbers
0.75
0 +1
4
+2
+3
+4
6.5
+5 +6
+7
Positive real numbers
3.3 Arithmetic Operations with Signed Numbers