Rational Numbers - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Rational Numbers, Elementary Number Theory, Methods of Proof, Quotient of Tow Integers, Generalizing from Generic Particular, Properties of Rational Numbers, Ratio of Integers, Towers of Hanoi, Recursive Solution

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2012/2013

Uploaded on 04/27/2013

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Chapter 3
Elementary Number Theory and
Methods of Proof
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Chapter 3

Elementary Number Theory and

Methods of Proof

Direct Proof and Counterexample 2

Rational Numbers

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/
  • 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/
  • 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 3.2.
  • Every integer is a rational number.

Proving Properties of Rational Numbers

  • r = a/b, s = c/d , for some integers a,b,c,d where b ≠ 0 and d≠
  • 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.
  • The sum of any two rational numbers is rational.

Properties of Rational Numbers

  • Corollary 3.2.
    • The double of a rational number is a rational number. 2r is rational.
    • corollary is a statement whose truth is deduced from a theorem.