+Properties of Real Numbers

+Properties

Relationships that are always true fro real numbers are called properties. Properties are rules used to rewrite and compare expressions.

Two algebraic expressions are equivalent expressions if they have the same value for all values of the variable(s).

+Properties of Real Numbers:Commutative Properties The Commutative Properties of Addition and

Multiplication state that changing the order of the addends does not change the sum and changing the order of the factors does not change the product.

Addition: For example, 14 + 51 = 51 + 14

Multiplication: For example, 12 × ⅓ = ⅓ × 12

+Properties of Real Numbers:Associative Properties The Associative Properties of Addition and

Multiplication state that changing the grouping of the addends does not change the sum and changing the grouping of the factors does not change the product.

Addition: For example, (23 + 9) +4 = 23 + (9 + 4)

Multiplication: For example, (7 × 9) × 10 = 7 × (9 × 10)

+Properties of Real Numbers:Identity Properties The Identity Properties of Addition and

Multiplication state that the sum of any real number and 0 is the original number and the product of any real number and 1 is the original number.

Addition: For example, 5⅗ + 0 = 5⅗

Multiplication: For example, 67 × 1 = 67

+Properties of Real Numbers:

The Zero Property of Multiplication states that the product of any real number and 0 is 0.

For example, 58 × 0 = 0

The Multiplication Property of – 1 states that the product of – 1 and any real number is the opposite of the original number.

For example, – 1 × 67 = – 67

+Problem 1Identifying Properties What property is illustrated by each of the

following statements?

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A. 42⋅ 0 = 0

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B. y + 2.5( ) + 28 = y + 2.5 + 28( )

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C. 4x⋅1 = 4x

+Problem 2Mental Math A movie ticket costs $7.75. A drink costs

$2.40. Popcorn costs $1.25. What is the total cost for a ticket, a drink and a popcorn? Use mental math.

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Total Cost = 7.75 + 2.40( ) +1.25

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Use Commutative Property 2.40 + 7.75( ) +1.25

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Use Associative Property 2.40 + 7.75 +1.25( )

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Simplify 2.40 + 9.00( )

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11.40

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The total cost is $11.40

+Problem 3Writing Equivalent Expressions Simplify each expression.

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A. 5 3n( )

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Use Associative Property 5⋅ 3( )n

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Simplify 15n

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B. 4 + 7b( ) + 8

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Use Commutative Property

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Use Associative Property

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Simplify

+Problem 3Writing Equivalent Expressions Simplify each expression.

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C. 6xyy

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Use Identity Property of Multiplication

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Multiply fractions

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D. 2.1 4.5x( )€

Use rules for division

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Use Identity Property of Multiplication

+Problem 3Writing Equivalent Expressions Simplify each expression.

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E. 8m

12mn

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F. 6 + 4h + 3( )

+Using Deductive Reasoning and CounterexamplesDeductive Reasoning is the process of

reasoning logically from given facts to a conclusion.

To show a statement is not true, find an example for which the statement is not true. An example showing a statement is false is a counterexample.

+Problem 4Using Deductive Reasoning and Counterexamples Is the statement true or false? If it is false,

give a counterexample.

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A. For all real numbers a and b, a⋅ b = b+ a

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False : 6⋅ 3 ≠ 3 + 6

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B. For all real numbers a,b and c, a+b( ) + c = b+ a( ) + c

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True

+Problem 4Using Deductive Reasoning and Counterexamples Is the statement true or false? If it is false,

give a counterexample.

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C. For all real numbers j and k, j⋅ k = k + 0( )⋅ j

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D. For all real numbers m and n, m n +1( ) = mn +1