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1.3 – AXIOMS FOR THE REAL NUMBERS

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Goals SWBAT apply basic properties of real numbers SWBAT simplify algebraic expressions

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An axiom (or postulate) is a statement that is assumed to be true. The table on the next slide shows axioms of multiplication and addition in the real number system. Note: the parentheses are used to indicate order of operations

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Substitution Principle: Since a + b and ab are unique, changing the numeral by which a number is named in an expression involving sums or products does not change the value of the expression. Example: and Use the substitution principle with the statement above.

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Identity Elements In the real number system: The identity for addition is: 0 The identity for multiplication is: 1

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Inverses For the real number a, The additive inverse of a is: - a The multiplicative inverse of a is:

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Axioms of Equality Let a, b, and c be and elements of. Reflexive Property: Symmetric Property: Transitive Property:

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1.4 – THEOREMS AND PROOF: ADDITION

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The following are basic theorems of addition. Unlike an axiom, a theorem can be proven.

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Theorem For all real numbers b and c,

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Theorem For all real numbers a, b, and c, If, then

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Theorem For all real numbers a, b, and c, if or then

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Property of the Opposite of a Sum For all real numbers a and b, That is, the opposite of a sum of real numbers is the sum of the opposites of the numbers.

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Cancellation Property of Additive Inverses For all real numbers a,

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Simplify 1. 2.

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1.5 – Properties of Products

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Multiplication properties are similar to addition properties. The following are theorems of multiplication.

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Theorem For all real numbers b and all nonzero real numbers c,

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Cancellation Property of Multiplication For all real numbers a and b and all nonzero real numbers c, if or,then

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Properties of the Reciprocal of a Product For all nonzero real numbers a and b, That is, the reciprocal of a product of nonzero real numbers is the product of the reciprocals of the numbers.

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Multiplicative Property of Zero For all real numbers a, and

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Multiplicative Property of -1 For all real numbers a, and

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Properties of Opposites of Products For all real numbers a and b,

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Explain why the statement is true. 1. A product of several nonzero real numbers of which an even number are negative is a positive number.

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Explain why the statement is true. 2. A product of several nonzero real numbers of which an odd number are negative is a negative number.

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Simplify 3.

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Simplify 8.

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Simplify the rest of the questions and then we will go over them together!

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1.6 – Properties of Differences

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Definition The difference between a and b,, is defined in terms of addition.

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Definition of Subtraction For all real numbers a and b,

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Subtraction is not commutative. Example: Subtraction is not associative. Example:

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Simplify the Expression 1.

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Simplify the expression 2.

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Your Turn! Try numbers 3 and 4 and we will check them together!

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Evaluate each expression for the value of the variable. 5.

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Evaluate each expression for the value of the variable. 6.

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