What's wrong with this proof of 1 = 2?

Obviously 1 does not equal 2, but this ``proof'' is discussed from time to time:

$$a = b$$

$$ab = b^2 b$$

$$ab-a^2 = b^2-a^2 a^2$$

$$a(b-a) = (b+a)(b-a)$$ Factor

$$a = b+a b-a$$

$$a = 2a b+a=2a$$

$$1 = 2 a$$

The erroneous step was dividing both sides by $b-a$ , which was dividing by 0 since $a=b\Leftrightarrow b-a=0$ . There are other (equally invalid) ways to prove $1=2$ .