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Properties of Real Numbers The properties of real numbers allow us to manipulate expressions and equations and find the values of a variable.

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Number Classification Natural numbers are the counting numbers. Whole numbers are natural numbers and zero. Integers are whole numbers and their opposites. Rational numbers can be written as a fraction. Irrational numbers cannot be written as a fraction. All of these numbers are real numbers.

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Number Classifications Subsets of the Real Numbers I - Irrational Integers Whole Natural Q - Rational

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There are also numbers that are NOT real. These are called imaginary or complex numbers. These numbers complete the categories of all numbers. Number Classifications

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Irrational Integers Whole Natural Rational Imaginary

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Classify each number -1 real, rational, integer real, rational, integer, whole, natural real, irrational real, rational real, rational, integer, whole real, rational 6 - 2.222 0

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Properties of Real Numbers Commutative Property Think… commuting to work. Deals with ORDER. It doesn’t matter what order you ADD or MULTIPLY. a+b = b+a 4 6 = 6 4

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Properties of Real Numbers Associative Property Think…the people you associate with, your group. Deals with grouping when you Add or Multiply. Order does not change.

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Properties of Real Numbers Associative Property (a + b) + c = a + ( b + c) (nm)p = n(mp)

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Properties of Real Numbers Additive Identity Property s + 0 = s Multiplicative Identity Property 1(b) = b

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Distributive Property a(b + c) = ab + ac (r + s)9 = 9r + 9s Properties of Real Numbers

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5 = 5 + 0 5(2x + 7) =10x + 35 8 7 = 7 8 24(2) = 2(24) (7 + 8) + 2 = 2 + (7 + 8) Additive Identity Distributive Commutative Name the Property

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7 + (8 + 2) = (7 + 8) + 2 1 v + -4 = v + -4 (6 - 3a)b = 6b - 3ab 4(a + b) = 4a + 4b Associative Mult. Identity Distributive

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Properties of Real Numbers Reflexive Property a + b = a + b The same expression is written on both sides of the equal sign.

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Properties of Real Numbers If a = b then b = a If 4 + 5 = 9 then 9 = 4 + 5 Symmetric Property

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Properties of Real Numbers Transitive Property If a = b and b = c then a = c If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5

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Properties of Real Numbers Substitution Property If a = b, then a can be replaced by b. a(3 + 2) = a(5)

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Name the property 5(4 + 6) = 20 + 30 5(4 + 6) = 5(10) 5(4 + 6) = 5(4 + 6) If 5(4 + 6) = 5(10) then 5(10) = 5(4 + 6) 5(4 + 6) = 5(6 + 4) If 5(10) = 5(4 + 6) and 5(4 + 6) = 20 + 30 then 5(10) = 20 + 30 Distributive Substitution Reflexive Symmetric Commutative Transitive

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Solving Equations To solve an equation, find replacements for the variables to make the equation true. Each of these replacements is called a solution of the equation. Equations may have {0, 1, 2 … solutions.

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Solving Equations 3(2a + 25) - 2(a - 1) = 78 6a + 75 – 2a + 2 = 78 (6a – 2a) + (75+2) = 78 4a + 77 = 78 4a = 1 a = 1/4

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Solving Equations 4(x - 7) = 2x + 12 + 2x 4x – 28 = 4x +12 4x = 4x + 40 0 = 40 This is NOT true!!! There is NO solution to this.

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Solving Literal Equations Solve: V = πr 2 h, for h The object is to get h by itself. We do this by reversing everything that is being done to h, in REVERSE of the Order of Operations rules. Right now, h is being multiplied by πr 2. We reverse this by dividing both sides by πr 2. So, h = V/ πr 2

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Solving Literal Equations Solve: de - 4f = 5g, for e Again, we need to isolate e. We do this by following the Order of Operations in REVERSE!! We will add the 4f first, then divide by d. de = 5g + 4f e = (5g + 4f) / d

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