Download APPENDIX B: Review of Basic Arithmetic

APPENDIX B:
Review of Basic Arithmetic
Personal Trainer
Click Algebra in the Personal Trainer for an interactive review of these concepts.
Algebra
Equality
=
Is equal to
3 = 3
Three equals three.
3 = +3
Three equals positive three.
–4 = –4
Negative four equals negative four.
4 = –(4)
Negative four equals negative the quantity four.
X = X
X equals X.
X = 4
X equals 4.
≈
Is approximately equal to
3.1 ≈ 3.2
3.1 is approximately equal to 3.2.
Not Equality
≠
Is not equal to
4 ≠ 3
4 is not equal to 3.
4 ≠ –4
4 is not equal to negative 4.
X ≠ –X
X is not equal to negative X.
Inequality
>
Is greater than
4 > 3
4 is greater than 3.
555
556 Appendix B Review of Basic Arithmetic
4 > –4
4 is greater than negative 4.
a > b
a is greater than b.
<
Is less than
5 < 7
5 is less than 7.
–1 < 0
Negative 1 is less than 0.
X < 14
X is less than 14.
≥
Is greater than or equal to
4 ≥ 3
4 is greater than or equal to 3.
4 ≥ 4
4 is greater than or equal to 4.
7 ≥ 0
7 is greater than or equal to 0.
≤
Is less than or equal to
2 ≤ 5
2 is less than or equal to 5.
X ≤ 0
X is less than or equal to 0.
–7 ≤ +7
Negative 7 is less than or equal to positive 7.
0 < 3 < 5
3 is greater than 0 and less than 5.
5 > 3 > 03 is less than 5 and greater than 0 (a true statement, but we prefer
the order 0 < 3 < 5).
4 < X ≤ 10
X is greater than 4 and less than or equal to 10.
Absolute Value
| X |The absolute value of X is the value of the number with the negative sign (if any) removed.
| 7 | = 7
The absolute value of 7 is 7.
| –2 | = 2
The absolute value of negative 2 is 2.
| +6 | = 6
The absolute value of positive 6 is 6.
Addition and Subtraction
3 + 5 = 8
3 plus 5 equals 8.
5 – 4 = 1
5 minus 4 equals 1.
3 + 5 = 5 + 3
The order in which numbers are added does not change the result.
3 + 7 + 2 = 7 + 2 + 3
The order in which numbers are added does not change the result.
5 + (–3) = 5 – 3 = 2Adding a negative number is the same as subtracting the same
positive number.
Appendix B Review of Basic Arithmetic
5 + 0 = 5
Anything plus 0 equals itself.
4 – 0 = 4
Anything minus 0 equals itself.
557
Multiplication and Division
2(3) = 6
2 times 3 equals 6.
3(2) = 2(3) = 6
The order in which terms are multiplied is not important.
7(1) = 7
Anything times 1 equals itself.
5(0) = 0
Anything times 0 equals 0.
2(–3) = –6
A positive number times a negative number is always negative.
6
= 6/2 = 3
2
6 divided by 2 equals 3.
6
= 6/1 = 6
1
Anything divided by 1 equals itself.
−6
= (–6)/2 = –3The result of a negative number divided by a positive number is
2
negative.
a
The upper portion of a fraction (here, a) is called the numerator;
b
the lower portion (here, b) is called the denominator.
6
= 6/(–3) = –2The result of a positive number divided by a negative number
−3
is negative; if either the numerator or the denominator (but not
both) is neg­ative, the result is negative.
−6
= (–6)/(–3) = 2If both numerator and denominator are negative, the result is
−3
positive.
0
= 0/6 = 0
6
Zero divided by anything is 0.
6
= 6/0 (undefined)
0
Anything divided by 0 is undefined.
0
= 0/0 (undefined) 0
The result of dividing 0 by 0 is undefined.
(–6)(–2)(–3) = –36When a series of numbers are multiplied, if the total number of
nega­tive terms is odd, the result is negative.
(–6)(2)(–3) = 36When a series of numbers are multiplied, if the total number of
nega­tive terms is even, the result is positive.
(–a)(–b)(–c) = –abc
a(–b)(–c) = abc
The previous two rules hold for both variables and numbers.
(−6)(−3)
= (–6)(–3)/(–2) = –9When a series of numbers are divided, the same rules apply: If
−2
the total number of negative signs is odd, the result is negative; if
the number of negative signs is even, the result is positive.
558 Appendix B Review of Basic Arithmetic
Exponentiation
4 2 = 4(4) = 16
4 squared (or 4 to the second power) equals 16.
a2 = a(a)
Any number squared is that number times itself.
5 1 = 5
Any number to the first power is that number itself.
5 0 = 1
Any number to the zeroth power equals 1.
(–a)2 = (–a)(–a) = a 2
A negative number squared is positive.
(103)(102) = 103 + 2 = 105If two exponential quantities that have the same base are multiplied, the result can be obtained by adding the exponents.
103
= 10 3 + 2 = 105 If two exponential quantities that have the same base are divided,
10 2
the result can be obtained by subtracting the exponents.
103 = 1000
10 raised to the third power is 1 followed by three zeros.
105 = 100,000
10 raised to the fifth power is 1 followed by five zeros.
The square root of
4 = ±2 The square root of 4 is either +2 or –2.
4 ( 4 ) = 4 The definition of the square root: The square root of a number is
that value that multiplied by itself gives the original number.
41 / 2 =
4 = ±2 41 / 2 (41 / 2 )
=
41
The square root can be written as an exponent of 1/2.
= 4 This follows from the rule that states that multiplication is the
addition of exponents, and also from the definition of the square
root.
(103)2 = 103(2) = 10 6 = 1,000,000When a value with an exponent is itself raised to a power, the
exponents are multiplied.
( 3 )2 = (31 / 2 )2 = 31 / 2(1) = 31 = 3
−4 is undefinedThe square root of any negative number is not defined (in the real
number system); if your computation results in taking the square
root of a negative number, you have made a mistake somewhere.
Fractions
a
The upper portion of a fraction (here, a) is called the numerator;
b
the lower portion (here, b) is called the denominator.
2 3 5
+ = = 1 When fractions that are to be added have the same denominator,
5 5 5
add the numerators and divide the sum by the denominator.
1 1 2 3 5
+ = + = When fractions that are to be added do not have the same denom3 2 6 6 6
inator, they must be converted to fractions that do have the same
(“common”) denominator and then added.
Appendix B Review of Basic Arithmetic
2  3
6
=
7  5  35
559
When fractions are to be multiplied, multiply the numerators together and then divide the product by the product of the
denominators.
Fractions may be converted to their decimal equivalents by
4
= .57143
dividing.
7
.57143 = .57Sometimes numbers are rounded (here, to two decimal places)
by pro­cedures described in Chapter 2.
.57 is .57(100) = 57%
2  3 2
=
3  5  5
To convert a decimal fraction to a percent, multiply by 100.
When the same factor appears in both the numerator and the
denomi­nator of two fractions that are being multiplied, they may
be canceled. Here, the 3’s disappear.
Order of Operations
2 + 3 = 3 + 2
The order of addition does not matter.
2(3) = 3(2)
The order of multiplication does not matter.
2(3 + 4) = 2(7) = 14When parentheses (or brackets) are indicated, operations within
the parentheses must be performed first.
2(3) + 4 = 6 + 4 = 10When the order of operations is ambiguous, the following
2 3(4) = 8(4) = 32
sequence must be followed: (1) exponentiation, (2) multiplica4/2 – 5 2 = 4/2 – 25 = 2 – 25 = –23
tion or division, (3) addition or subtraction.
2 + 3(4) = 2 + 12 = 14
Parentheses
2(3 + 4) = 2(7) = 14When parentheses (or brackets) are indicated, operations within
the parentheses must be performed first.
2(3 + 4 + 5) = 2(3) + 2(4) + 2(5) = 24When a sum contained in parentheses is multiplied by a factor,
a(b + c + d) = ab + ac + ad the factor must be multiplied by all the terms in the sum.
2[(3)(4)(5)] = (2)(3)(4)(5) = 120
a(b – c) = ab – ac
(a + b)2 = (a + b)(a + b) = a 2 + 2ab + b 2
(a + b)(a – b) = a 2 – b 2
When a product contained in parentheses is multiplied by a factor,
the factor multiplies the product.
Equations: Solving for Y
We can add the same value (here, 3) to both sides of an equation
Y −3= 4
without altering the equality.
Y = 4+3
Y =7
560 Appendix B Review of Basic Arithmetic
We can subtract the same value (here, 6) from both sides of an
Y + 6 = 12
equation without altering the equality.
Y = 12 − 6
Y =6
We can multiply both sides of an equation by the same value
Y
=5
(here, 3) without altering the equality.
3
Y = (5)3
Y = 15
2Y = 10
We can divide both sides of an equation by the same value
(here, 2) without altering the equality.
10
Y =
2
Y =5
We can perform any sequence of the above four rules without
2Y − 4 = 6
altering the equality as long as we perform the same operation
2Y = 6 + 4
(addition, subtraction, multiplication, or division) on both sides
2Y = 10
of the equal sign.
10
Y =
2
Y =5
5
3Y + 2
5
1.25
−.75
−.25
or Y
=4
=
=
=
=
=
4(3Y + 2)
3Y + 2
3Y
Y
−.25
Self-test for Arithmetic (answers follow)
In questions 1–20, answer true or false.
Equality
1. 7 = 7
5. –X = –(X)
2. 31 = +31
3. +(27) = (+27)
4. +2 = –2
7. –5 ≠ 5
8. 247 ≠ 248
9. a ≠ a
Not Equality
6. +3 ≠ +(3)
Appendix B Review of Basic Arithmetic
Inequality
10. 10 > 9
11. 237 > 1
12. 3 < 2
13. –4 > –3
14. –21 < 21
15. 5 ≤ 5
16. 10 ≥ –9
17. –412 < –399 ≤ – 4
19. | –21| = –| 21 |
20. |–39 | = +39
Absolute Value
18. | 4| = 4
What is the value of X?
Addition and Subtraction
21. X = 214 + 5
22. 214 + (–3) = X
23. X = –47 + 4
24. 33 + 0 = X
26. 14(1) = X
30. X = 229/229
27. 14(0) = X
31. X = 25/2
34. X = 234/1
35.
28. X = 6(–3)
32. −4
= X
2
36. X = (–a) (–b) (c)
Multiplication and Division
25. X = 3(4)
29. X = (5)(–5)
33.
37.
X =
12
−3
X =
−2(24)
2
X =
234
0
Exponentiation
38. 32 = X
42. X = (–4)2
4 –2
46. X =
50. 103(103) = X
39. X = 103
40. X = 71
43. X = Z9
47. X = (102)(103)
51. 103 + 103 = X
44. Z–3
48. X = (641/2)(641/2)
41. X = 290
45. X = 161/2
49. X = (2111/2)2
52. X = Z–9
Fractions
53.
X =
12
6
54.
X =
1 1
+
2 4
55.
X =
4  9
9  4 
56.
X =
1 1
1
+ −
3 9 27
Order of Operations
57. X = 5 + 2(4)
58. X = 5(2) + 4
59. X = 4 2 / 2 + 3
62. X = (3 + 2)(3 – 2)
63. X = (1 + 21)(2 + 2)2
Parentheses
61. X = 14(3 – 2)
In questions 64–66, answer true or false.
64. (5 + 4)2 = (5 + 4) (5 + 4) = 25 + 2(20) + 16
65. (5 – 4)2 = (5 – 4)(5 – 4) = 25 – 2(20) + 16
66. (5 + 4) (5 – 4) = 25 – 16
60. X = (3 + 2)2 + 2
561
562 Appendix B Review of Basic Arithmetic
Equations: Solve for Y
67. Y + 1 = 5
68. 2Y = 16
69. 3Y + 12 = 24
10
=5
4Y − 3
75. 3Y – 8 = 1
72. Y 2 = 9
73.
71.
2Y +
3Y − 2
= 10
2
70. Y
= 30
5
74. Y 2(Y –1) – 5 = 0
Answers to Self-test for Arithmetic
1. True
5. True
9. False
13. False
17. True
21. 219
25. 12
29. –25
33. –4
37. –24
41. 1
2. True
6. False
10. True
14. True
18. True
22. 211
26. 14
30. 1
34. 234
38. 9
42. 16
3. True
7. True
11. True
15. True
19. False
23. – 43
27. 0
31. 12.5
35. Undefined
39. 1000
45. ± 4
49. 211
53. 2
57. 13
61. 14
65. True
69. 4
73. 22/7 = 3.143
46. 1/16
43. ± 3
47. 100,000
50. 1,000,000
54. 3/4 = .75
58. 14
62. 5
66. True
70. 150
74. 5
51. 2000
55. 1
59. 11
63. 352
67. 4
71. 5/4 = 1.25
75. 3
4. False
8. True
12. False
16. True
20. True
24. 33
28. –18
32. – 2
36. abc
40. 7
44. Undefined
48. 64
52. Undefined
56. 11/27 = .407
60. 27
64. True
68. 8
72. +3 or –3