1.2 The Real Number System
Real numbers represent quantities, such as distance, time, speed, area, profit, loss, and temperature.
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The Real Number system can be diagramed in a way to show how different types of numbers are subsets of each other.
Example 1
Classify the following numbers in as specific way as possible: √5, √3, ½, 1/3, 2/7, 0, ­6, 1
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Field ­ the set of real numbers and the two binary operations of addition & multiplication that have the properties of: closure, commutative, associative, identity, inverse, and distributive. To be a field, it must hold true for ALL of these properties.
There are 11 Field Properties of the Set of Real Numbers
Let a, b, and c be real numbers.
Closure
Addition
Multiplication
Commutative
Addition
Multiplication
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Associative
Addition
Multiplication
Identity
Addition
Multiplication
Addition
Multiplication
Inverse
Distributive
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Example 2:
Name the field property of the real number system illustrated in each statement.
a. 2πr = 2rπ
b. 0.625(1.6) = 1
c. 1 + 0 is a real number
d. 2[3.27 + (-3.27)] = 2(0)
Example 4:
Determine which of the subsets of real numbers (natural numbers, whole numbers, integers, rational numbers, irrational numbers) are closed under subtraction.
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Example 5:
Are the following statements true or false? If false, state a
counterexample.
a) The rational numbers are closed under multiplication.
b) The irrational numbers are closed under addition.
Example 6:
Under the operations of addition and multiplication, determine which
field properties DO NOT hold for each set. Get an example.
a) Positive even integers
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b) Integers divisible by 5
The field properties, along with the properties of equality can be used to prove theorems about real numbers.
Properties of Equality
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Example 7
Name the property of equality that illustrated in each statement.
a. If 2.5 = 2 1/2 and 2 1/2 = 5/2, then 2.5 = 5/2
b. 5 = 5
c. If a = b and 9 + a = 15, then 9 + b = 15
d. If 1/2 = 0.5, then 0.5 = 1/2
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