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Chapter 3 Elementary Number Theory and Methods of Proof 3.2 Direct Proof and Counterexample 2 Rational Numbers Rational Numbers • Definition – A real number r is rational if, and only if, it can be expressed as a quotient of tow integers with a nonzero denominator. A real number that is not rational is irrational. – r is a rational ⇔∃integers a and b such that r = a/b and b ≠ 0. – (informal) quotient of integers are rational numbers. – (informal) irrational numbers are real numbers that are not a quotient of integers. Example • Is 10/3 a rational number? – Yes 10 and 3 are integers and 10/3 is a quotient of integers. • Is –(5/39) a rational number? – Yes –(5/39) = -5/39 which is a quotient of integers. • Is 0.281 rational? – Yes, 281/1000 • Is 2/0 an irrational number? – No, division by 0 is not a number of any kind. • Is 0.12121212… irrational? – No, 0.12121212… = 12/99 • If m and n are integers and neither m nore n is zero, is (m+n)/mn a rational number? – Yes, m+n is integer and mn is integer and non-zero, hence rational. Generalizing from the Generic Particular • Generalizing from the particular can be used to prove that “every integer is a rational number” 1. 2. 3. – arbitrarily select an integer x show that it is a rational number repeat until tired Example: • 7/1, -9/1, 0/1, 12345/1, -8342/1, … • Theorem 3.2.1 – Every integer is a rational number. Proving Properties of Rational Numbers • Sum of rational is rational – Prove that the sum of any two rationals is rational. – (formal)∀real numbers r and s, if r and s are rational then r + s is rational. – Starting Point: suppose r and s are rational numbers. – To Show: r + s is rational Proving Properties of Rational Numbers – r = a/b, s = c/d , for some integers a,b,c,d where b ≠ 0 and d≠0 – it follows that r + s = a/b + c/d – a/b + c/d = (ad + bc)/bd – the fraction is a ratio of integers since bd ≠ 0 – ad + bc = p (integer) and bd = q (integer) – therefore, r + s = p/q is rational by the definition. • Theorem 3.2.2 – The sum of any two rational numbers is rational. Properties of Rational Numbers • Corollary 3.2.3 – The double of a rational number is a rational number. 2r is rational. – corollary is a statement whose truth is deduced from a theorem.