This falls down on line 5, divide by (a-b). If a=b then (a-b)=0. This leads on to a situation where 0 is divided by 0. (a-b)/(a-b). The common but, incorrect assumption is then made that 0/0=1. This is what the trick relies on. At the stage where you divide through the equation by (a-b) to leave a+b = b, the trickster has lead you to believe that (a-b)/(a-b) = 1, otherwise the equation would collapse to 0 on both sides. It is generally accepted that 0/0 can't be determined, or is an indeterminate number. Whatever its value, it certainly isnt equal to 1, so the whole equation is invalid and a nonsense.let a =b
therefore a*a = b*a....................................multiply by a
therefore a*a-b*b= b*a-b*b.........................subtract b*b
therefore (a+b)(a-b) = b(a-b)......................factorise
Therefore a+b = b......................................divide by (a-b)
therefore 2b =b..........................................because a=b
therefore 2=1...................................divide by b
QED
It reminds me of this old chestnut from my days of studying mathematics. (slightly modified)
You are on a game show and you stand to win the electric bicycle of your choice. There are three inverted cups numbered 1,2 and 3 standing on a table. Under one of the cups are the keys to the bike of your dreams. All you have to do is lift the cup which covers the keys. You have two chances to select the correct cup.
You lift the first cup, number 2...............there is nothing beneath it. So now it is a choice between the two remaining cups, Cup 1 and Cup 3. You put your hand on Cup 3 and are just about to lift it. The host shouts stop! He offers you the chance to change your mind and pick the other cup.
The question is, if you swap, are your chances of success increased?