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# Chapter 3 - PowerPoint PPT Presentation

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

Elementary Number Theory and Methods of Proof

### 3.2

Direct Proof and Counterexample 2

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.

• 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 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.

• 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

• 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.

• 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.