Chapter 3
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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

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Chapter 3

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Chapter 3

Chapter 3

Elementary Number Theory and Methods of Proof


Chapter 3

3.2

Direct Proof and Counterexample 2

Rational Numbers


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

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.


Generalizing from the generic particular

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.


Proving properties of rational numbers

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 numbers1

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

    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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