R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Real Numbers
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
Natural Numbers
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Law of Trichotomy: For any two real numbers a and b, precisely one of three relationships is possible.
a < b, a = b, or a > b.
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Absolute Value and Distance
Def. of Absolute Value: Let a be a real number. The absolute value
of a, denoted by | a |, is
| a | = a, if a > 0
­ a, if a < 0
Properties of Absolute Value:
For a and b be real numbers, the following properties are true.
1. | a | > 0
2. | ­ a | = | a |
3. | ab | = | a | | b |
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Distance Between Two Numbers:
Let a and b be real numbers. The distance between a and b is given by
Distance = | b ­­ a | = | a ­­ b |
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Examples
1. Identify all the subsets of the real numbers to which each number in the given set belongs.
{ 3, ­1, 1/3, 6/3, ­1/2 2 , ­7.5 }
2. Order the numbers from least to greatest.
26/15, 3 , 1.7320 , 381/220 , 10 ­­ 2 3. Write the expression without the absolute value sign
| 4 ­ |
4. While traveling on the Pennsylvania Turnpike, you pass milepost 326 near Valley Forge, then milepost 351 near Philadelphia. How far do you travel during that time period?
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Exercises 73 ­ 78 on page R9
Budgeted Expense, Actual Expense, a
b
| a ­­ b |
0.05b
73
$25,000
$24,872.12
$127.88
$1250.00
74
$112,700
$113,356.52
$656.52
$5635.00
75
$9400
$9972.59
$572.59
$470.00
76
$7500
$7104.68 $395.32
$375.00
77
$37,640
$36,968.25
$671.75
$1882.00
78
$2575
$2613.15
$38.15
$128.75
In # 73, 74, 77, and 78, the difference between the budgeted expense and actual expense was less than 5% of the budgeted expense. In # 75 and 76, the difference between the budgeted expense and the actual expense exceeded 5% of the budget.
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
You may hear it said that to take the absolute value of a real number you simply remove any negative sign and make the number positive. Can it ever be true that | a | = ­a for a real number a?
For any real number other than 0, the distance that number is from 0 on the number line can never be expressed as a negative number. If the real number is 0, then it is true that | 0 | = ­0 because ­0 = 0. The real number 0 is neither positive or negative.
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R1.1 Real Numbers, Absolute Value and Distance.notebook
September 08, 2015
Homework: pp. R8 ­ R9 / #5, 9, 11, 17, 19, 29, 35, 43, 45, 59, 61, 67, 71, 81, 85
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