Ashley Buck
To Prove:
For any rational numbers r, s
r2 - s2 ∈ Q
Proof:
By the definition of rational,
Let r = (a/b), s = (c/d), s.t. a ∈ Z, b ∈ Z, c ∈ Z, d ∈ Z and b ≠ 0 and d ≠ 0
So, r ∈ Q and s ∈ Q
Consider r2 – s2:
r2 – s2 =
=
=
=
=
=
(a/b)2 - (c/d)2
a2/b2 – c2/d2
1* (a2/b2) – 1* (c2/d2)
(d2/d2) (a2/b2) – (b2/b2) (c2/d2)
(a2d2/b2d2) – (b2 c2/b2d2)
(a2d2 – b2c2) / (b2 d2)
By substitution
By evaluating the square
As multiplying by one does not change meaning
As any number divided by itself is equal to 1
By Multiplication
By subtraction
As a ∈ Z and Z is closed under multiplication
a*a∈Z
a * a = a2
a2 ∈ Z
As d ∈ Z and Z is closed under multiplication
d*d∈Z
d * d = d2
d2 ∈ Z
As a2 ∈ Z and d2 ∈ Z and Z is closed under multiplication
a2d2 ∈ Z
As b ∈ Z and Z is closed under multiplication
b*b∈Z
b * b = b2
b2 ∈ Z
As d ∈ Z and Z is closed under multiplication
c*c∈Z
c * c = c2
c2 ∈ Z
As b ∈ Z and c ∈ Z and Z is closed under multiplication
b2c2 ∈ Z
2 2
As a d ∈ Z and b2c2 ∈ Z and Z is closed under subtraction
a2d2 – b2 c2 ∈ Z
As b2 ∈ Z and d2 ∈ Z and Z is closed under multiplication
b2d2 ∈ Z
As b ≠ 0 and d ≠ 0, b2d2 ≠ 0 by Zero Product Property
Thus, by definition of “rational”, r2 - s2 ∈ Q
▲