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Numbers Foundation Concepts

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7/2/2013 Numbers 2 The Real Numbers The number line Numbers 0 1 2 3 1 2 3 4 19 12 1 2 - 2 e 3.14 = 2.718281828 2 - The set of real numbers is called R 22 7 = 3.14 = 459045… 28571428571... 15926535897… 3 - 2 Repeating group 5 Binary Relations

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7/2/2013 Numbers 3 The Real Numbers Is there always a number between a and b ? Numbers a b a + b 2 Average of a and b is midway between a and b After b, what is the next real number? Question:

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7/2/2013 Numbers 4 The Natural Numbers The Set of Counting Numbers N = { 1, 2, 3, 4, … } The Integers Positive and Negative Natural Numbers I = { … -3, -2, -1, 0, 1, 2, 3, … } Subsets of the Real Numbers … and Zero Binary Relations

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7/2/2013 Numbers 5 Rational Numbers Solutions of linear equations a x + b = 0 for integers a, b Proper fractions and integers Subsets of the Real Numbers Q = { | a, b are integers, b ≠ 0 } a b

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7/2/2013 Numbers 6 Irrational Numbers Not solutions of a x + b = 0 Algebraic numbers – roots of n th degree polynomials with rational coefficients Examples: x 2 – 2 = 0 x 3 – 5 = 0 Transcendental numbers Subsets of the Real Numbers { x | x R, x Q } x = 2 + – 3 5

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7/2/2013 Numbers 7 Irrational Numbers Not solutions of a x + b = 0 Transcendental numbers – Examples: = 3.1415 92653 58979 32384 62643 … e = 2.7182 81828 45904 52353 60287 … Subsets of the Real Numbers { x | x R, x Q } NOT roots of any polynomial with rational coefficients

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7/2/2013 Numbers 8 Scientific Notation Standard notation 38100059018442 –60850017290000.00049487173 –0.2974615490024 Scientific notation 3.8100059018442 x 10 13 –6.085001729 x 10 13 4.9487173 x 10 – 4 –2.974615490024 x 10 – 1

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7/2/2013 Numbers 9 General form For real number r, scientific notation is r = c x 10 n where n is an integer and 1 ≤ Constant c usually a single-digit integer Scientific Notation c < 10

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7/2/2013 Numbers 10 Think about it !

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7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be.

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