1. Direct Proof and Counterexample II
Lecture 13
Section 3.2
Wed, Feb 2, 2005

2. Rational Numbers
A rational number is a number that equals the quotient of two integers.
Let Q denote the set of rational numbers.
An irrational number is a number that is not rational.
We will assume that there exist irrational numbers.

3. Direct Proof Theorem: The sum of two rational numbers is rational.
Let r and s be rational numbers.
Let r = a/b and s = c/d, where a, b, c, d are integers.
Then r + s = (ad + bc)/bd, which is rational.

4. Proof by Counterexample
Disprove: The sum of two irrationals is irrational.
Counterexample:

5. Proof by Counterexample
Disprove: The sum of two irrationals is irrational.
Counterexample:
Let α be irrational.
Then -α is irrational. (proof?)
α + (-α) = 0, which is rational.

6. Direct Proof
Theorem: Between every two distinct rationals, there is a rational.
Proof:
Let r, s  Q. WOLOG* WMA† r < s.
Let t = (r + s)/2.
Then t  Q. (proof?)
We must show that r < t < s.
*WOLOG = Without loss of generality
†WMA = We may assume

7. Proof, continued Given that r < s, it follows that
2r < r + s < 2s.
Then divide by 2 to get
r < (r + s)/2 < s.
Therefore, r < t < s.

8. Other Theorems
Theorem: Between every two distinct irrationals there is a rational.
Proof: Difficult.
Theorem: Between every two distinct irrationals there is an irrational.

9. An Interesting Question
Why are the last two theorems so hard to prove?
Because they involve “negative” hypotheses and “negative” conclusions.

10. Positive and Negative Statements
A positive statement asserts the existence of a number.
A negative statement asserts the nonexistence of a number.
It is much easier to use a positive hypothesis than a negative hypothesis.
It is much easier to prove a positive conclusion than a negative conclusion.

11. Positive and Negative Statements
“r is rational” is a positive statement.
It asserts the existence of integers a and b such that r = a/b.
“α is irrational” is a negative statement.
It asserts the nonexistence of integers a and b such that α = a/b.
Is there a “positive” characterization of irrational numbers?

12. Irrational Numbers
Theorem: Let  be a real number  and define the two sets
A = iPart({1, 2, 3, …}*( + 1))
and
B = iPart({1, 2, 3, …}*(-1 + 1)).
Then  is irrational if and only if A  B = N and A  B = .