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Transcript
1.2 Properties of Real Numbers • Objectives: To graph and order real numbers To identify properties of real numbers • Vocabulary: Opposite Additive Inverse Reciprocal Multiplicative Inverse Properties Vocabulary Terms: Properties of Real Numbers Let a, b, and c represent real numbers Property Closure Commutative Associative Identity Inverse Distributive Addition Multiplication REAL NUMBER CLASSIFICATIONS Subsets of the Real Numbers Rational Irrational Integers Whole Π, 2 Natural EXAMPLES • Use natural numbers to count - {1,2,3,4,5,6….} • The whole numbers are the natural numbers plus 0 - {0,1,2,3….} • Integers are the natural numbers and their opposites plus 0 - {...,-3,-2,-1,0,1,2,3…} • Rational numbers are all numbers that can be written as a quotient of integers. a/b, b≠0. • Rational numbers include terminating decimals…1/8 = .0125 • Rational Numbers include repeating decimals… • 1/3 = .3333333333333333333333333333333333, or 0.3 with a hat over it CLASSIFY EACH NUMBER name ALL sets to which each belongs -1 • real, rational, integer 3 • real, rational, integer, whole, natural √17 • real, irrational ⅜ • real, rational 0 • real, rational, integer, whole -5.555 • real, rational PROPERTIES OF REAL NUMBERS COMMUTATIVE • Think… commuting to school. • Deals with ORDER. It doesn’t matter what order you ADD or MULTIPLY. • a+b = b+a •4 • 6 = 6 • 4 PROPERTIES OF REAL NUMBERS ASSOCIATIVE • Think…the people you associate with; your group. Are you the member of more than 1 club? • Deals with grouping when you Add or Multiply. • Order does not change. Additive (a • + b) + c = a + ( b + c) Multiplicative • (nm)p = n(mp) PROPERTIES OF REAL NUMBERS IDENTITY Additive Identity Property •s+0=s •0 is the additive identity. Multiplicative Identity Property • 1(b) = b •1 is the multiplicative identity PROPERTIES OF REAL NUMBERS INVERSE • Multiplicative Inverse Property • Product = 1 • Additive Inverse Property • Sum = Zero • a ∙ 1/a = 1, a ≠ 0 • a + (-a) = 0 • 8(1/8) = 1 • -5(-1/5)=1 • 12 + (− 12 ) = 0 • −7 + 7 = 0 Properties of Real Numbers Distributive Distributive Property • a(b + c) = ab + ac • 9(r + s) = 9r + 9s Name the Property •5=5+0 • 5(2x + 7) =10x + 35 •8•7=7•8 • 24(2) = 2(24) • (7 + 8) + 2 =2 + (7 + 8) Name the Property •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 Commutative Commutative Name the Property • 7 + (8 + 2) = (7 + 8) + 2 • 1 • v + -4 = v + -4 • (6 - 3a)b = 6b - 3ab • 4(a + b) = 4a + 4b Name the Property • 7 + (8 + 2) = (7 + 8) + 2 • 1 • v + -4 = v + -4 • (6 - 3a)b = 6b - 3ab • 4(a + b) = 4a + 4b • Associative • Multiplicative Identity • Distributive • Distributive Vocabulary Terms: Properties of Real Numbers Let a, b, and c represent real numbers Property Addition Multiplication Closure The sum of a + b is a real number The product of ab is a real number Commutative a + b = b + a ab = ba Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Identity a + 0 = a, 0 + a = a 0 is the additive identity a●1=a, 1●a=a 1 is the multiplicative identity Inverse a + (-a) = 0 (opposite sign) Distributive a(b + c)=ab + ac reciprocal 1 a 1, a 0 a