Chapter 4: Elementary Number Theory and Methods of Proof 4.2 Direct Proof and Counter Example II: Rational Numbers 1 Such, then, is the whole art of convincing. It is contained in two principles; to define all notations used, and to prove everything by replacing mentally defined terms by their definitions. – Blaise Pascal,

It is important to be aware that there are a number of words that mean essentially the same thing as the word “theorem,” but which are used in slightly different ways. Theorem – In general the word “theorem” is reserved for a statement that is considered important or significant (the Pythagorean Theorem, for example). Proposition – A statement that is true but not as significant is sometimes called a proposition. 4.2 Direct Proof and Counter Example II: Rational Numbers 2

It is important to be aware that there are a number of words that mean essentially the same thing as the word “theorem,” but which are used in slightly different ways. Lemma – A lemma is a theorem whose main purpose is to help prove another theorem. Corollary – A corollary is a result that is an immediate consequence of a theorem or proposition. 4.2 Direct Proof and Counter Example II: Rational Numbers 3

Rational Number – A real number r is rational iff it can be expressed as a quotient of two integers with a nonzero denominator. More formally, if r is a real number, then r is rational   a, b  Z s.t. r = a/b and b  0. Irrational Number – A real number that is not rational is irrational. 4.2 Direct Proof and Counter Example II: Rational Numbers 4

If neither of two real numbers is zero, then their product is also not zero. 4.2 Direct Proof and Counter Example II: Rational Numbers 5

Theorem – Every integer is a rational number. Theorem – The sum of any two rational numbers is rational. Corollary – The double of a rational number is rational. 4.2 Direct Proof and Counter Example II: Rational Numbers 6

Determine whether the statements are true or false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. In case the statement is false, determine whether a small change would make it true. If so, make the change and prove the new statement. 15. The product of any two rational numbers is a rational number. 16. The quotient of any two rational numbers is a rational number. 18. If r and s are any two rational numbers, then is rational. 4.2 Direct Proof and Counter Example II: Rational Numbers 7

Use the properties of even and odd integers that are listed in example to do exercise 21. Indicate which properties you use to justify your reasoning. – True or False? If m is any even integer and n is and odd integer, then m 2 + 3n is odd. Explain. 4.2 Direct Proof and Counter Example II: Rational Numbers 8

Derive the statement below as corollaries of theorems 4.2.1, 4.2.2, and the results of exercise 15. – For any rational numbers r and s, 2r + s is rational. 4.2 Direct Proof and Counter Example II: Rational Numbers 9

Suppose a, b, c, and d are integers and a  c. Suppose also that x is a real number that satisfies the equation Must x be rational? If so, express x as the ratio of two integers. 4.2 Direct Proof and Counter Example II: Rational Numbers 10

Prove that if one solution for a quadratic equation of the form x 2 + bx + c = 0 is rational (where b and c are rational), then the other solution is also rational. Hint: Use the fact that if the solutions of the equation are r and s, then 4.2 Direct Proof and Counter Example II: Rational Numbers 11