32 Slides--Introduction to Number Theory

32 Slides--Introduction to Number Theory - CS103A HO# 32...

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Unformatted text preview: CS103A HO# 32 Introduction to Number Theory 2/11/08 1 CS103A 2/11/08 Problem Set 7 will be distributed Wednesday, 2/13 and it will be due Friday, 2/22. Outline of topics for CS103A Basic Tools Formal Logic and Proof Techniques Number Theory and its Applications Proving Real Theorems Induction Program Proofs Recursion Combinatorics & Probability Functions Number Theory We will not try to develop number theory from the ground up, but we should acknowledge that this can be done from a small number of axioms. One example: Peano's Axioms, proposed by Giuseppe Peano (1858 1932) in 1889. Peano's Axioms There is a number 0. Every number has a successor, denoted by S(a). There is no number whose successor is 0, i.e., x (S(x) 0). Two numbers with the same successor are themselves equal, i.e., x y(S(x) = S(y) x = y) If a property is possessed by 0 and if the successor of every number possessing the property also possesses it, then it is possessed by every number, i.e., [Q(0) x(Q(x) Q(S(x))] xQ(x) What we assume . . . -3 -2 -1 0 1 2 3 . . . The existence of Z , the set of all integers. Z = { -3, -2, -1, 0, 1, 2, 3, } Commutativity: a + b = b + a a b = b a Associativity: (a + b) + c = a + (b + c) (a b) x c = a (b c) Distributivity: a (b + c) = a b + a c (a + b) c = a c + b c What we assume a+0=a a 1 = a Negative numbers: the equation a + x = 0 has the unique solution x = -a. Subtraction: x = b - a a + x = b The integers are closed under addition, multiplication, and subtraction, i.e., if a and b are integers, then so are a + b, a b, and a b. The usual algebraic manipulations such as factoring polynomials, raising to powers, etc. CS103A HO# 32 Introduction to Number Theory 2/11/08 2 Division Definition : If a and b are integers and a 0, then the statement that a divides b means that there is an integer c such that b = ac ....
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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32 Slides--Introduction to Number Theory - CS103A HO# 32...

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