### Flawed Proofs (some examples)

Identify the flaw in the proof
Statement: 1 = 2.
Proof: Let a=b
then $$a^2$$ = ab
then $$a^2−b^2 = ab−b^2$$
then (a+b)(a−b) = b(a−b)
then a+b = b
then 2b = b
hence 2 = 1

In this case, (a-b) can be 0, and you cannot divide both sides by 0. The mistake happens on the third line.

Statement: 4 = 5
$$4^2 −9*4=5^2 −9*5\\ 4^2−2*4*(\frac{9}{2})^2+ 2 = 5^2−2*5*\frac{9}{2}+\frac{9}{2}\\ (4−\frac{9}{2})^2 = (5-\frac{9}{2})^2\\ 4 - \frac{9}{2} = 5 - \frac{9}{2}\\ 4 = 5$$