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Direct Proof and
Counterexample II
Lecture 12
Section 3.2
Thu, Feb 9, 2006
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.

Direct Proof
Theorem: The sum of two rational
numbers is rational.
 Proof:

Let r and s be rational numbers.
 Let r = a/b and s = c/d, where a, b, c, d are
integers, where b, d > 0.
 Then r + s = (ad + bc)/bd.

Direct Proof
We know that ad + bc is an integer.
 We know that bd is an integer.
 We also know that bd  0.
 Therefore, r + s is a rational number.

Proof by Counterexample
Disprove: The sum of two irrationals is
irrational.
 Counterexample:

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.

Direct Proof
Theorem: The sum of two odd integers is
an even integer; the product of two odd
integers is an odd integer.
 Proof:

Direct Proof
Theorem: The sum of two odd integers is
an even integer; the product of two odd
integers is an odd integer.
 Proof:

Let a and b be odd integers.
 Then a = 2s + 1 and b = 2t + 1 for some
integers s and t.

Direct Proof
Then a + b = (2s + 1) + (2t + 1)
= 2(s + t + 1).
 Therefore, a + b is an even integer.
 Finish the proof.

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?)

*WOLOG
= Without loss of generality
†WMA = We may assume
Proof, continued
We must show that r < t < s.
 Since r < s, it follows that
2r < r + s < 2s.
 Then divide by 2 to get
r < (r + s)/2 < s.
 Therefore, r < t < s.

Other Theorems
Theorem: Between every two distinct
irrationals there is a rational.
 Proof: Difficult.
 Theorem: Between every two distinct
irrationals there is an irrational.
 Proof: Difficult.

An Interesting Question
Why are the last two theorems so hard to
prove?
 Because they involve “negative”
hypotheses and “negative” conclusions.

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.

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.
Positive and Negative
Statements

Is there a “positive” characterization of
irrational numbers?
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 = .
 Try it out: Irrational.exe
