3.2.1
Section 2 - Rational Numbers
• Recall the definition of a Rational Number:
A real number r is rational provided there
exist integers a and b such that r = a/b and
b ≠ 0.
• Theorem: Every integer is a rational number.
Proof: Let a be an integer, then a = a/1.
Moreover, 1 is an integer and 1 ≠ 0. Therefore a
is a rational number. QED
3.2.2
Proving Properties of Rationals
• We will now look at some theorems and
corollaries (theorems that follow essentially
trivially from another theorem) about rational
numbers.
• We will rely on the Closure Properties of the
Integers under +, −, and ⋅:
If a,b are integers, then (a+b), (a−b), (b−a),
and a⋅b are also integers.
• We will also use their Zero-Product Property:
If a,b ∈ Z, with a ≠ 0 and b ≠ 0, then a⋅b ≠ 0.
3.2.3
Closure of the Rationals Under +
• Theorem: If r, s ∈ Q, then (r + s) ∈ Q.
• Proof: Let r, s ∈ Q. Thus ∃ a, b, c, d ∈ Z such
that r = a/b with b ≠ 0 and s = c/d with d ≠ 0.
Now, (r + s) = a/b + c/d = (ad + bc)/bd.
Since a, b, c, d ∈ Z, we have that (ad + bc) ∈ Z
and that bd ∈ Z. Moreover, since b ≠ 0 and d ≠
0, we conclude that bd ≠ 0. Consequently,
(r + s) is the quotient of integers with non-zero
denominator. Therefore (r + s) ∈ Q. QED
3.2.4
A Corollary
• Corollary: Double a rational is rational.
• Proof: Let r = s in the previous theorem.