Chapter 3 Elementary Number Theory and Methods of Proof

3.2 Direct Proof and Counterexample 2 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 rational? – Yes, 281/1000 Is 2/0 an irrational number? – No, division by 0 is not a number of any kind. Is … irrational? – No, … = 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.arbitrarily select an integer x 2.show that it is a rational number 3.repeat until tired – Example: 7/1, -9/1, 0/1, 12345/1, -8342/1, … Theorem – 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 – The sum of any two rational numbers is rational.

Properties of Rational Numbers Corollary – The double of a rational number is a rational number. 2r is rational. – corollary is a statement whose truth is deduced from a theorem.