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32 Slides--Introduction to Number Theory

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

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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.

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

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