You know, multiplying by zero makes no sense at all too.

Agreed!

I don't think, that the following 'proof'

Given that a = b (e.g. 1/2 = 2/4);

1) a^2 = ab
2) a^2 - b^2 = ab - b^2
3) (a + b)(a - b) = b(a - b)
4) (a + b) = b
5) (b + b) = b
6) 2b = b
7) 2 = 1
8) QED?

is meant as a series of equivalence transformations (I hope this expression exists in english...), but more as a series of conclusions: If equation '1' is correct, then equation '2' will be correct, too.
An equivalence transformation is an operation done in order to find a solution of equation, which does not affect the regular solutions of this equation. From this point of view step 'a=b' to 'a^2=ab' is no equivalence transformation, because 'a=b' has the solution a=b, but 'a^2=ab' has two solutions: 1. a=b, 2. a=0.
Nevertheless you can conlude: If a=b is valid, then a^2=ab is also valid.

Equivalence transformations are additions and subtractions of TERMS and multiplications and divisions of NUMBERS unequally to zero.

Examples:

Adding 'a' to both sides of equation a=b does not change the solutions. But multiplication of 'a' does. Here we have made the mistake of multiplying a TERM instead of NUMBER.

If we multiply a=b on both sides with zero we get:

a x 0 = b x 0

But a=b is no longer THE solution of this equation. So, multiplying by zero is no equivalence transformation.

But nevertheless you can conlude: If a=b is valid, then a x 0 = b x 0 is also valid.

Now the interesting question. What are we doing if we try to conlude:

If

(a + b)(a - b) = b(a - b)

is valid, then

(a + b) = b

is also valid?

We did not multiply both sides by zero. It's the result of an equivalence transformation. So, we havn't done anything wrong. The series of conclusions up to the line (a + b)(a - b) = b(a - b) is correct. But if we now want to conlude (a+b) = b, then this would be the result of an equivalence transformation, namley dividing both sides by (a-b). But as (a-b) is zero, this would be an unallowed equivalence transformation.

But can we nevertheless at least conclude that if (a + b)(a - b) = b(a - b) is valid, then also (a + b) = b must be valid?

NO! Because our equation looks like:

(a+b) x 0 = b x 0

And this equation is solved even if (a+b) <> b. So, we cannot conlude (a+b) = b

Kai