19 - LECTURE 19: THE INTEGERS MODULO n (8.6) Appreciation...

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LECTURE 19: THE INTEGERS MODULO n ( § 8.6) Appreciation is a wonderful thing: It makes what is excellent in others belong to us as well. Voltaire (1694 - 1778) 1. Tables Recall that the relation on Z given by a b (mod n ) is an equivalence relation. The equivalence classes are [0] , [1] ,..., [ n - 1]. We will call the set of these equivalence classes the integers modulo n , written Z n . For example Z 3 = { [0] , [1] , [2] } . (Be careful [1] means different things for different n ’s.) The elements of Z 3 are sets, but we still want to think about them as integers, and be able to add them and multiply them. But what would that mean? Addition would be taking two equivalence classes and replacing them with a new equivalence class, by some rule. The rules we want to use are: [ a ] + [ b ] = [ a + b ] [ a ][ b ] = [ ab ] . These seem nice. Let’s see what happens in Z 3 . We can form the addition table and the multiplication table. (Done in class.) There is a problem here. Why does addition and multiplication work? For example, when adding
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19 - LECTURE 19: THE INTEGERS MODULO n (8.6) Appreciation...

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