Proof that 2 = 1
Let a and b be equal non-zero quantities
a = b
Multiply through by a
a^2 = ab
Subtract b^2
a^2 - b^2 = ab - b^2
Factor both sides
(a - b)(a + b) = b(a - b)
Divide out (a - b)
a + b = b
Observing that a = b
b + b = b
Combine like terms on the left
2b = b
Divide by the non-zero b
2 = 1
Q.E.D.