IntegerFunctions

IntegerFunctions - CSE 1400 Applied Discrete Mathematics...

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CSE 1400 Applied Discrete Mathematics Integer Functions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Integer Functions 1 Floor, Ceiling, and Related Functions 1 Number Theoretic Functions 4 Problems on Integer Functions 7 Abstract Integer Functions S ome things are easy to model continuously , but only dis - crete integer values make sense : T he 2000 census reports there are 0.90 children per family in the U nited S tates of A merica ! There are several kinds of integer functions . One kind has a signature f : R Z This type of integer function maps real numbers to integers . These are useful when rounding or truncating computed values to integer approximations. Another type maps integers to integers . This kind has a signature f : Z Z and is useful in number theory and its applications. Floor, Ceiling, and Related Functions T he floor of a real number x is the integer n immediately below x : n x . The floor functions has signature bc : R Z b 2.75 c = 2 b- 2.5 c = - 3 b 18.5 c = 18 b- 12.6 c = - 13 b 5 c = 5 b- 5 c = - 5 b π c = 3 b- π c = - 4 b e c = 2 b- e c = - 3 b ϕ c = 1 b- ϕ c = - 2
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integer functions 2 x 1 2 3 0 - 1 y = b x c 1 2 3 - 1 T he ceiling lies above x . d 2.75 e = 3 d- 2.5 e = - 2 d 18.5 e = 19 d- 12.6 e = - 12 d 5 e = 5 d- 5 e = - 5 d π e = 4 d- π e = - 3 d e e = 3 d- e e = - 2 d ϕ e = 2 d- ϕ e = - 1 x 1 2 3 0 - 1 y = d x e 1 2 3 - 1 { 2.75 } = 0.75 {- 2.5 } = 0.5 { 18.5 } = 0.5 {- 12.6 } = 0.4 { 5 } = 0.0 {- 5 } = 0.0 { π } ≈ 0.1416 {- π } ≈ 0.8584 { e } ≈ 0.7183 {- e } ≈ 0.2817 { ϕ } ≈ 0.6180 {- ϕ } ≈ 0.3820 The floor and ceiling functions are useful when reasoning about the pigeonhole principle principle. Pretend you have a things to place in n places. For instance, pretend you have 13 shirts to place in 5 boxes. A natural, nearly uniform, distribution places 2 shirts in This is an application of the quotient remainder theorem : 13 = 5 · 2 + 3 and in general a = nq + r where 0 r < n . all 5 boxes and 1 shirt each of 2 boxes. Notice that b 12/5 c = 2 and d 12/5 e = 3. 1 . If all boxes contained 3 or more shirts, then there would be at least 3 · 5 = 15 shirts, but there are only 12 shirts. Therefore some box must contain 2 = b 12/5 c or less shirts. ( box )( shirts in box ≤ b 12/5 c ) It is useful to consider the function that maps shirts to boxes. Name the shirts
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This note was uploaded on 02/11/2012 for the course MTH 2051 taught by Professor Shoaff during the Fall '11 term at FIT.

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IntegerFunctions - CSE 1400 Applied Discrete Mathematics...

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