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PROPERTIES OF REAL NUMBERS 1 .215 -7 PI

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**Subsets of real numbers – REVIEW**

Natural numbers numbers used for counting 1, 2, 3, 4, 5, …. Whole numbers the natural numbers plus zero 0, 1, 2, 3, 4, 5, … Integers the natural numbers ( positive integers ), zero, plus the negative integers

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…,-4, -3, -2, -1, 0, 1, 2, 3, 4, … Rational numbers numbers that can be written as fractions decimal representations can either terminate or repeat Examples: fractions: 7/5 -3/ /5 Any whole number can be written as a fraction by placing it over the number 1 8 = 8/ = 100/1

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terminating decimals ¼ = /5 = .4 Repeating decimals 1/3 = /3 = .6 These will always have a bar over the repeating section. Irrational numbers Cannot be written as fractions Decimal representations do not terminate or repeat

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**if the positive rational number is not a perfect square, then its square root is irrational**

Examples: Pi - non-repeating decimal 2 - not a perfect square

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THE REAL NUMBERS Rational numbers Irrational numbers Integers Whole numbers Natural numbers

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**Graphing on a number line**

¼ Tip: Best to put them as all decimals Put the square root in the calculator and find its equivalent -1.414… ………

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Ordering numbers Use the < , >, and = symbols Compare and Here again for square roots put them in the calculator and get their equivalents .08 = = So: < or >

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**Properties of Real Numbers**

Opposite or additive inverse sum of opposites or additive inverses is 0 Examples: / /9 -400 Additive inverse of any number a is -a 4/9 - 4 1/5 . 002

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**Reciprocal or multiplicative inverse**

product of reciprocals equal 1 Examples: 4 1/ /9 1/400 Multiplicative inverse of any number a is 1/a - 9/4 5/21 - 500

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Other Properties: Addition: Closure a + b is a real number Commutative a + b = b + a 4 + 3 = 7` = 7 numbers can be moved in addition Associative (a + b) + c = a + (b + c) (1 + 2) + 3 = (2 + 3) = 6 3 + 3 = = 6 the order in which we add the numbers does not matter in addition

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Identity a + 0 = a 7 + 0 = 7 when you add nothing to a number you still only have that number Inverse a + -a = 0 = 0

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Multiplication Closure ab is a real number Commutative ab = ba 6(4) = (6) = 24 When multiplying the numbers may be switched around, will not affect product Associative (ab)c = a(bc) The order in which they are multiplied does not affect the outcome of the product

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(3*4)5 = (4*5) = 60 12(5) = (20) = 60 Identity a * 1 = a One times any number is the number itself 7 * 1 = 7 Inverse a * 1/a = 1 Product of reciprocals is one 7 * 1/7 = 7/7 = 1

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**DISTRIBUTIVE Property**

Combines addition and multiplication a(b + c) = ab + ac 2(3 + 4) = 2(3) + 2(4) 6 + 8 14

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ABSOLUTE VALUE Absolute value is its distance from zero on the number line. Absolute value is always positive because distance is always positive Examples: = = -1 * -2 = 4 2

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Assignment Page 8 – 9 Problems 34 – 60 even

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