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Chapter 3

Chapter 3. Elementary Number Theory and Methods of Proof. 3.2. Direct Proof and Counterexample 2 Rational Numbers. Rational Numbers. Definition

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Chapter 3

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  1. Chapter 3 Elementary Number Theory and Methods of Proof

  2. 3.2 Direct Proof and Counterexample 2 Rational Numbers

  3. 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.

  4. 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 mnoren is zero, is (m+n)/mn a rational number? • Yes, m+n is integer and mn is integer and non-zero, hence rational.

  5. Generalizing from the Generic Particular • Generalizing from the particular can be used to prove that “every integer is a rational number” • 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.

  6. 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

  7. 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.

  8. 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.

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