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Numbers - CSE 1400 Applied Discrete Mathematics Numbers...

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CSE 1400 Applied Discrete Mathematics Numbers Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Numbers 1 Natural Numbers 1 Integers 2 Integers Mod n 2 Rational Numbers 3 Floating Point Numbers 5 The Real Numbers 6 Complex Numbers 6 Problems on Numbers 7 Abstract Numbers are used for many purposes, chiefly to count discrete things Numbers There are many types of numbers: whole numbers, positive and negative numbers, fractional numbers, and continuous numbers. Natural Numbers The symbol N is used to refer to the set of natural numbers: zero, one, two, three, . . . . To count things, the natural numbers are used. N = { 0, 1, 2, 3, . . . } The natural numbers are unsigned, that is, no plus ( + ) or minus ( - ) sign is placed in front of a natural number. The natural numbers are closed under addition and multiplication. Addition closure: If n and m are natural numbers, then n + m is a natural number. Multiplication closure: If n and m are natural numbers, then nm is a natural number.
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cse 1400 applied discrete mathematics numbers 2 Subtraction and division can be computed on some pairs of natural numbers, but the natural numbers are not closed under these opera- tions. 8 - 9 6∈ N ; 5 ÷ 2 6∈ N . • If n is less than or equal to m , then the difference m - n is a natural number. • If m is a multiple of n , say m = nk for some natural number k , then the quotient m / n = k is a natural number. A relation < is strict if a < b , then b 6 < a . A relation < is total if for any a and b , either a < b , b < a , or a = b . The natural numbers can be placed in order. 0 < 1 < 2 < 3 < 4 < · · · Less than is a relation on the natural numbers. Less than is strict and total. Integers Integers are used to increment and decrement counts of things . The integers are the numbers in the set Z = { 0, ± 1, ± 2, ± 3, . . . } The integers are closed under addition, multiplication, and subtrac- The symbol Z is commonly used to refer to the set of integers: zero, plus or minus one, plus or minus two, plus or minus three, . . . . tion. Subtraction closure: If n and m are integers, then m - n is an integer. The integers are signed numbers: They are stored in computer mem- ory with an explicit plus (+) or minus ( - ) sign. Surprisingly, the natural numbers and the integers can be put into a one-to-one correspondence, which mathematicians understand to mean the sets have an equal number of members. A function that establishes this one-to-one correspondence maps the even natural numbers to one-half their value and the odd natural numbers to the negative of one-half times their value plus 1. n N , f ( n ) = n /2 if n is even - ( n + 1 ) /2 if n is odd For instance, 0 0 1 → - 1 2 1 3 → - 2 4 2 5 → - 3 6 3 7 → - 4 8 4 9 → - 5 . . . . . . Because of this one-to-one correspondence, the two natural numbers A set X is finite if it contains n members, where n is some natural number. For instance, B = { 0, 1 } , containing n = 2 members, is finite.
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