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1.3 – AXIOMS FOR THE REAL NUMBERS

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.

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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. 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.

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

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

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

  8. 1.4 – THEOREMS AND PROOF: ADDITION

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

  10. Theorem For all real numbers b and c,

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

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

  13. 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.

  14. Cancellation Property of Additive Inverses For all real numbers a,

  15. Simplify 1. 2.

  16. 1.5 – Properties of Products

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

  18. Theorem • For all real numbers b and all nonzero real numbers c,

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

  20. 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.

  21. Multiplicative Property of Zero • For all real numbers a, and

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

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

  24. 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.

  25. 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.

  26. Simplify 3.

  27. Simplify 8.

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

  29. 1.6 – Properties of Differences

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

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

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

  33. Simplify the Expression 1.

  34. Simplify the expression 2.

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

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

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

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