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Transcript
In this lecture Number Theory ● Rational numbers ● Divisibility Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample 1 Common mistakes in proofs • Arguing from examples • Using same letter to mean two different things • Jumping to a conclusion (without adequate reasons) 2 Disproof by counterexample To disprove statement of the form “xD if P(x) then Q(x)”, find a value of x for which ● P(x) is true and ● Q(x) is false. Ex: For any prime number a, a2-1 is even integer. Counterexample: a=2. 3 Rational Numbers Definition: r is rational iff integers a and b such that r=a/b and b≠0. Examples: 5/6, -178/123, 36, 0, 0.256256256… Theorem: Every integer is a rational number. 4 Properties of Rational Numbers • Theorem: The sum of two rational numbers is rational. • Proof: Suppose r and s are rational numbers. Then r=a/b and s=c/d for some integers a,b,c,d s.t. b≠0, d≠0. (by definition) a c r s So (by substitution) b d ad bc (by basic algebra) bd Let p=ad+bc and q=bd. Then r+s=p/q where p,q Z and q≠0. Thus, r+s is rational by definition. ■ 5 Types of Mathematical Statements Theorems: Very important statements that have many and varied consequences. Propositions: Less important and consequential. Corollaries: The truth can be deduced almost immediately from other statements. Lemmas: Don’t have much intrinsic interest but help to prove other theorems. 6 Divisibility • Definition: For n,d Z and d≠0 we say that n is divisible by d iff n=d·k for some k Z . • Alternative ways to say: n is a multiple of d , d is a factor of n , d is a divisor of n , d divides n . • Notation: d | n . • Examples: 6|48, 5|5, -4|8, 7|0, 1|9 . 7 Properties of Divisibility • For xZ, 1|x . • For xZ s.t. x≠0, x|0 . • An integer x>1 is prime iff its only positive divisors are 1 and x . • For a,b,cZ, if a|b and a|c then a|(b+c) . • Transitivity: For a,b,cZ, if a|b and b|c then a|c . 8 Divisibility by a prime Theorem: Any integer n>1 is divisible by a prime number. Sketch of proof: Division into cases: ● If n is prime then we are done (since n | n). ● If n is composite then n=r1·s1 where r1,s1 Z and 1<r1<n,1<s1<n. (by definition of composite number) (Further) division into cases: ♦If r1 is prime then we are done (since r1 |n). ♦ If r1 is composite then r1=r2·s2 where r2,s2 Z and 1<r2<r1,1<s2<r1. 9 Divisibility by a prime Sketch of proof (cont.): Since r1|n and r2|r1 then r2 |n (by transitivity). Continuing the division into cases, we will get a sequence of integers r1 , r2 , r3 ,…, rk such that 1< rk< rk-1<…< r2< r1<n ; rp |n for each p=1,2,…,k ; rk is prime. Thus, r is a prime that divides n. k ■ 10 Unique Factorization Theorem • Theorem: For integer n>1, positive integer k, distinct prime numbers p1 , p2 ,, pk , positive integers e1 , e2 ,, ek ek e1 e2 s.t. n p1 p2 pk , and this factorization is unique. 6 2 3 • Example: 72,000 = 2 3 5 11