Tricky proofs in Mathematics:

You would have come across this proof many times. Probably you might have also known the error in one of the steps that had lead to the wrong result.

The error is in the step number 7 which divides both sides by (a – b) and it is obvious that we are dividing by 0 (as a = b )

Ok. We do accept the weird result is due to this error in dividing by a 0.

Now look at this proof:

And the proof continues further to get many good results (which are out of the scope of this article)

Here, we had divided the equation by z -1 which assumes  z  not equal to 1 and then we substitute z =1!! Very clever isn’t it? This is mathematics. You can actually cheat – as long as your steps are logical. Also, you would have noticed how cleverly the step number (2) had been rewritten in a convenient form to suit the substitution. If this conversion of 

 was not done, making z = 1 would have lead to  form which is unacceptable in mathematical proofs. But still we have used the value of z =1 in the same expression by writing it in a different format.

We can see such approaches in various other places. For example, while using First Principle method to find the derivative,

 we simplify the numerators so that the ‘h’ in the denominator first gets cancelled and make it safe to substitute the limit as h = 0. L’Hospital’s rule basically changes the given function to a form which makes it convenient for the application of the specified limits.

There are number of proofs like this in mathematics. You just need to think out of the box and see how you can proceed without losing the logic.

As a very simple example, consider the quadratic formula. We wouldn't have got the quadratic formula if someone hadn’t thought of adding and subtracting the extra termto make it a perfect square. Can we do something similar to cubic? Can we make a normal cubic equation to a perfect cube, for example, and invent a formula for finding the roots? It is possible though so far no one could get the knack of it yet! May be someone in future might come up with a special trick like we saw above.

Hence, all you need is to have a good visualisation of the question and what you have in hand that could be applied in the process of solving it. Keep looking for such approaches while you learn new concepts and you will definitely be able to appreciate the way mathematicians think. Soon, you might also be able to think out of the box and could come out with some wonderful proof.