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Unformatted text preview: COMPSCI 102 Discrete Mathematics for Computer Science Lecture 5 Ancient Wisdom: Unary and Binary 1 2 3 4 Prehistoric Unary Hang on a minute! Isnt unary too literal as a representation? Does it deserve to be an abstract representation? Its important to respect each representation, no matter how primitive Unary is a perfect example Consider the problem of finding a formula for the sum of the first n numbers You already used induction to verify that the answer is n(n+1) 1 + 2 3 n1 n S + + + + = 1 + 2 n1 n S + + n2 + + = n+1 + n+1 n+1 n+1 2S + + n+1 + + = n(n+1) = 2S S = n(n+1) 2 1 + 2 3 n1 n S + + + + = 1 + 2 n1 n S + + n2 + + = n(n+1) = 2S 1 2 . . . . . . . . n n . . . . . . . 2 1 There are n(n+1) dots in the grid! S = n(n+1) 2 n th Triangular Number n = 1 + 2 + 3 + . . . + n1 + n = n(n+1)/2 n th Square Number n = n 2 = n + n1 Breaking a square up in a new way Breaking a square up in a new way 1 Breaking a square up in a new way 1 + 3 Breaking a square up in a new way 1 + 3 + 5 Breaking a square up in a new way 1 + 3 + 5 + 7 Breaking a square up in a new way 1 + 3 + 5 + 7 + 9 1 + 3 + 5 + 7 + 9 = 5 2 Breaking a square up in a new way Pythagoras The sum of the first n odd numbers is n 2 Here is an alternative dot proof of the same sum. sum....
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This document was uploaded on 01/17/2012.
 Fall '09

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