1.3

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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.