1.3 – AXIOMS FOR THE REAL NUMBERS. Goals  SWBAT apply basic properties of real numbers  SWBAT simplify algebraic expressions.

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Presentation transcript:

1.3 – AXIOMS FOR THE REAL NUMBERS

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

 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

 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.

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

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

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

1.4 – THEOREMS AND PROOF: ADDITION

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

Theorem  For all real numbers b and c,

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

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

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.

Cancellation Property of Additive Inverses  For all real numbers a,

Simplify 1. 2.

1.5 – Properties of Products

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

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

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

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.

Multiplicative Property of Zero  For all real numbers a, and

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

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

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.

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.

Simplify 3.

Simplify 8.

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

1.6 – Properties of Differences

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

Definition of Subtraction  For all real numbers a and b,

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

Simplify the Expression 1.

Simplify the expression 2.

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

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

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