Could you catch it?

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In my previous article I proved that “1=2”.

Actually, that’s not a proof. That’s a pseudo proof rather. 🙂
So, there must be something wrong. But, that logical error cannot be observed at a glance. We have to have a careful and closer look on it. Consider the following step in my proof.

e2πi = e0

What’s wrong with this implication . Is it because the base is well known ghostly constant “e” ? No. That’s not the case !!. It’s true that “e” has strange special properties. Nevertheless in this case it is merely a number whose magnitude is approximately 2.718281828459 .

The thing is that, here I’ve equated the exponents , as the bases are same.
Mmmm…… You may ask , “So, what?”
Are you sure that you can always equate the exponents when bases are equal?
You may answer “why not.”
So, how do you know? Because , your high school maths teacher taught so? Ha? 😛
That’s going to be true only in the real number domain.
In general, “if the bases are equal, then the real parts of the exponents are equal.” More precisely,

ea = eb –> Re(a) = Re(b)

We proved previously that,

e2πi = 1

So, more in detail,

ea = eb –> a = b + 2πni

Where n is any integer. (Think about it.)

Observe that a=b is merely a special case of the above implication where n=0 !!! So, see how unfair is to equate the exponents blindly. So far your blind luck may have worked. But that’s only until you encounter such a contradiction as 1=2.

To be continued…….

Image : http://www.telegraph.co.uk/