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

Download Rational Numbers - Discrete Mathematical Structures - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

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.