Vijay For Victory

False Proof is one of the concepts in Mathematics. This is a quite interesting area. One of my Mathematics professor started his first class with the following question.

Can you prove 1 = -1?

We all said nope.

Then he went to the blackboard and wrote the following proof.

Version 1:

Amazing. Isn’t? This is called False Proof.

There is one more way to prove 1 = -1.

Version 2 :

Now let us see, how is this possible?

Let us take the first proof (False Proof)

The rule  is generally valid only if at least one of the two numbers x or y is positive, which is not the case here. Alternatively, one can view the square root as a 2-valued function over the complex numbers; in this case both sides of the above equation evaluate to {1, -1}.

How about the second version? Is that true? The answer is false again. Below is the explanation.

The equation a ^bc = (a ^b )^c, when b and/or c are fractions, is generally valid only when a is positive, which is not the case here, leading to an invalid proof.