Direct Proof and Counterexample IIRational NumbersDirect ProofProof by CounterexampleSlide 5Slide 6Proof, continuedOther TheoremsAn Interesting QuestionPositive and Negative StatementsSlide 11Irrational NumbersDirect Proof and Counterexample IILecture 13Section 3.2Wed, Feb 2, 2005Rational NumbersA rational number is a number that equals the quotient of two integers.Let Q denote the set of rational numbers.An irrational number is a number that is not rational.We will assume that there exist irrational numbers.Direct ProofTheorem: The sum of two rational numbers is rational.Proof:Let r and s be rational numbers.Let r = a/b and s = c/d, where a, b, c, d are integers.Then r + s = (ad + bc)/bd, which is rational.Proof by CounterexampleDisprove: The sum of two irrationals is irrational.Counterexample:Proof by CounterexampleDisprove: The sum of two irrationals is irrational.Counterexample:Let α be irrational.Then -α is irrational. (proof?)α + (-α) = 0, which is rational.Direct ProofTheorem: Between every two distinct rationals, there is a rational.Proof:Let r, s  Q. WOLOG* WMA† r < s.Let t = (r + s)/2.Then t  Q. (proof?)We must show that r < t < s.*WOLOG = Without loss of generality†WMA = We may assumeProof, continuedGiven that r < s, it follows that 2r < r + s < 2s.Then divide by 2 to get r < (r + s)/2 < s.Therefore, r < t < s.Other TheoremsTheorem: Between every two distinct irrationals there is a rational.Proof: Difficult.Theorem: Between every two distinct irrationals there is an irrational.Proof: Difficult.An Interesting QuestionWhy are the last two theorems so hard to prove?Because they involve “negative” hypotheses and “negative” conclusions.Positive and Negative StatementsA positive statement asserts the existence of a number.A negative statement asserts the nonexistence of a number.It is much easier to use a positive hypothesis than a negative hypothesis.It is much easier to prove a positive conclusion than a negative conclusion.Positive and Negative Statements“r is rational” is a positive statement.It asserts the existence of integers a and b such that r = a/b.“α is irrational” is a negative statement.It asserts the nonexistence of integers a and b such that α = a/b.Is there a “positive” characterization of irrational numbers?Irrational NumbersTheorem: Let  be a real number  and define the two setsA = iPart({1, 2, 3, …}*( + 1))andB = iPart({1, 2, 3, …}*(-1 + 1)).Then  is irrational if and only if A  B = N and A  B =