Section 3.2: Direct Proof II: Rational Numbers
1. Recall rational and irrational numbers.
2. Theorem 3.2.1: Every integer is a rational number.
3. Theorem 3.2.2: The sum of any two rational numbers is rational.
4. Theorem: The difference of any two rational numbers is rational. (Prove this.)
5. Deriving New Mathematics from Old
Proving from definitions (Section 3.1) vs. Deriving from known statements (Section 3.2).
6. Suppose that you have already proved the following:
(a) The sum, product, and difference of any two even integers are even.
(b) The sum and difference of any two even integers are even.
(c) The product of any two odd integers is odd.
(d) The product of any even integer and any odd integer is even.
(e) The sum of any odd integer and any even integer is odd.
(f) The difference of any odd integer minus any even integer is odd.
(g) The difference of any even integer minus any odd integer is odd.
7. Example #1: (p. 147, pr. 22) Use the aforementioned properties to answer the following.
True or false? If k is any even integer and m is any odd integer, then (k + 2)2 − (m − 1)2 is even.
8. Corollary - a statement whose truth can immediately be deduced from a theorem.
9. Corollary 3.2.3: 2m ∈ Q ∀m ∈ Q.
10. Example #2: (p. 147, pr. 23-24) Derive the following as results of Theorems, Examples, etc.
(a) For any rational numbers r and s, 2r + 3s is rational.
(b) If r is any rational number, then 3r2 − 24 + 4 is rational.
11. Example #3: (p. 147, pr. 32, 35, 36) Find the mistakes in the following “proofs.”
(a) Proof: Let rational numbers r = 14 and s = 21 be given. Then r + s = 14 + 21 = 34 , which is a
rational number. This shows that the sum of any two rational numbers is a rational number.
(b) Proof: Suppose r and s are rational numbers. Then r = ab and s = dc for some a, b, c, d ∈ Z,
b 6= 0, d 6= 0. But this is a sum of two fractions, which is a fraction. So r + s is a rational
number since a rational number is a fraction.
(c) Proof: Suppose r and s are rational numbers. If r + s is rational, then by definition of
rational, r + s = a/b for some a, b ∈ Z, b 6= 0. Also since r and s are rational, r = i/j and
s = m/n for i, j, m, n ∈ Z, j 6= 0, n 6= 0. It follows that r + s = i/j + m/n = a/b, which
is a quotient of two integers with nonzero denominator. Hence it is a rational number. This
shows that the sum of any two rational numbers is a rational number.
12. Example #4: (p. 147, pr. 30)
Prove that if a real number c satisfies a polynomial equation of the form
r3 x3 + r2 x2 + r1 x + r0 = 0,
where r0 , r1 , r2 , r3 ∈ Q, then c satisfies an equation of the form
n3 x3 + n2 x2 + n1 x + n0 = 0,
where n0 , n1 , n2 , n3 ∈ Z.