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
Proof Follow the steps
−20 = −20
16 − 36 = 25 − 35
\(4^2 −9*4=5^2 −9*5\\ 4^2−2*4*(\frac{9}{2})^2+ 2 = 5^2−2*5*\frac{9}{2}+\frac{9}{2}\\ (4−\frac{9}{2})^2 = (5-\frac{9}{2})^2\\ 4 - \frac{9}{2} = 5 - \frac{9}{2}\\ 4 = 5\)

In this case you cannot get rid of the exponents on both sides. The mistake happens on the fifth line.