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Chapter 3

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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.

- 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.