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1.3 – AXIOMS FOR THE REAL NUMBERS

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1 1.3 – AXIOMS FOR THE REAL NUMBERS

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

3 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

4

5 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.

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

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

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

9 1.4 – THEOREMS AND PROOF: ADDITION

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

11 Theorem For all real numbers b and c,

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

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

14 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.

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

16 Simplify 1. 2.

17 1.5 – Properties of Products

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

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

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

21 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.

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

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

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

25 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.

26 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.

27 Simplify 3.

28 Simplify 8.

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

30 1.6 – Properties of Differences

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

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

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

34 Simplify the Expression
1.

35 Simplify the expression
2.

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

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

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


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