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# Class Note for MATH 300 at UMass(3)

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COURSE
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15
Math 300 Fall 2008 Note 5 Zhigang Han7 Urnass at Amherst 1 Rational and Real Numbers 11 Rational Numbers Def 1 De ne the equivalence relation N on the set szm mwwwezb m by 11 N 07d if and only if ad be The equivalence classes are called rational numbers7 and the equivalence class containing 11 is denoted by The set of all rational numbers is denoted by Q That is7 QM Zb0l Eg Show that the relation N above is an equivalence relation Def 2 De ne the addition and the multiplication on Q by a c adbc Eli bd a ciao b dibd39 Eg Show that these operations are well de ned That is7 the de nitions are independent of the choice of representatives of the equivalence class Prop 1 Any nonzero rational number q can be expressed in a unique way as q with b gt 0 and gcdab 1 Rmk We say the fraction is reduced to its lowest terms in this case 12 Decimal Expansions Def 3 The expression ba1a2 where b E Z and each a is a digit from 0 to 97 is called the decimal expansion of the real number r if ba1a2 an 7 10 ba1a2 an for all n E N Rmk Certain numbers have two different decimal expansions For instance7 0125000 0124999 Def 4 A decimal ba1a2 is terminating if there exists an integer n such that ai 0 for all i 2 n A decimal is called periodic if there exists positive integers p and n such that 17 a for all i 2 n Eg 1 i 025000 025 is terminating 0142857 is periodic Eg 2 Find the rational number with the decimal expansion 1234 Theorem A real number is rational if and only if its decimal expansion is periodic or terminating Hence a real number is irrational if and only if it has a nonperiodic in nite decimal expansion

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