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Table of Properties Let a, b, and c be real numbers, variables, or algebraic expressions. (These are properties you need to know.) Property 1. Example Commutative Property of Addition a+b=b+a 2+3=3+2 2. Commutative Property of Multiplication a • b = b • a 2•(3)=3•(2) 3. Associative Property of Addition a + ( b + c ) = ( a + b ) + c 4. Associative Property of Multiplication a • ( b • c ) = ( a • b ) • c 5. Distributive Property of Multiplication over Addition a • ( b + c ) = a • b + a • c 6. Additive Identity Property a + 0 = a 7. Multiplicative Identity Property a • 1 = a 8. Additive Inverse Property a + ( -a ) = 0 9. 2+(3+4)=(2+3)+4 2•(3•4)=(2•3)•4 2•(3+4)=2• 3+2•4 3+0=3 3•1 =3 3 + (-3) = 0 Multiplicative Inverse Property Note: a cannot = 0 10. Zero Product Property a •0=0 5•0=0 Above are properties you already know but another one is the property of closure. The Closure Property states that when you perform an operation (such as addition, multiplication, etc.) on any two numbers in a set, the result of the computation is another number in the same set. For example, when you multiply two integers you will always get an integer, so the integers are closed with respect to multiplication. However, when you multiply two irrational numbers, you don’t always get an irrational number ( 2 18 36 6 ), so the irrational numbers are not closed with respect to multiplication. Furthermore, when you divide two integers you don’t always get an integer (7 2 = 3.5), so the integers are not closed with respect to division. Accelerated CCGPS…Math II Name______________________ “Proofs” of Properties of Numbers Period______Date____________ To prove a mathematical statement, we must show the statement to be true in all cases, but to disprove a mathematical statement, we only need 1 counterexample. For each of the following statements, find an example that verifies the statement and an example that contradicts the statement, if possible. If no counterexample can be found, write a “proof” of the statement. If no example can be found, rewrite the statement so that it is always true. The first one is done for you. Statement The sum of two rational numbers is rational. The product of two irrational numbers is irrational. The quotient of two natural number is not natural. The area of a 30˚- 60˚- 90˚ triangle is irrational. The perimeter of a 45˚- 45˚- 90˚ triangle is rational. The area of a circle is irrational. Example Counterexample 2 7 29 3 4 12 none which is rational. “Proof” or Rationale If x a c where a, b, c, d and y b d are integers and bd ≠ 0, then a c ad bc which will x y b d bd still be rational. More Review Questions: I. Simplify completely: 1. 6 4. 4 3 3 2 2 3 2. 2 x 3k 2 7. x 10. 3 40 7 3 5 3 8 3 4 5. 8. p p p 2 8 2x 11. 6 3 2 3. 81x 6. 3 9. x x 5 n5 2 n 108 3 4 5 2 12. 2 3 2 3 27 13. Without using a calculator, approximate the following values to the nearest tenth: 80 = _____________, 2 (Check your approximations using a calculator.) 97 = _____________, 14. Use your calculator to approximate 1 17 33 = ____________. = _________ and 3 5 What do you think must be true about the above values? 3 5 = _________. 4 Can you prove it?