1.3 – AXIOMS FOR THE REAL NUMBERS

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

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

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

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

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.