1.- Suppose we have the following mathematical equation:
a = b
where to It can be any positive number.
2.- If we multiply both members of equality by to, the validity of the equation is maintained:
to2 = ab
3.- The same happens if we add (a² - 2ab) to both sides. In this case, we will have the following equation:
to2 + (to2 - 2ab) = ab + (a2 - 2ab)
4.- Simplifying we have to:
2nd2 - 2ab = a2 - ab
5.- Now we get a common factor and we have the following:
2a (a - b) = a (a - b)
6.- If we simplify (a - b) we have to:
2nd = a
7.- And if we now divide both terms by the result is:
2 = 1
Which certainly seems a false statement ... Where have we been wrong?
The error is subtly hidden in the problem statement. Initially we are told that a = b, so the term we simplify in step 6 (a - b) will be zero. It is clear that we cannot divide by zero to simplify the equation, so this would be the wrong step of our deduction.