Look at this thread .
Look at this thread .
Thanks for the reply.
From the thread that you directed me to I understand how to derive at .
My question, I feel, is not sufficiently answered. I would like to know whether there are any areas of mathematics that "break down" if this equation does not hold?
An analogy would be: What difficulties would mathematics suffer if .
I had a discussion with a person and it got me thinking.
the laws of exponents wouldn't work. You might as well ask what would happen if addition didn't work.
You would multiply things by 1 and not end up with what you started with. Everything would either blow up or implode depending on whether was greater or less than 1.
None of the laws of physics would work if the laws of exponents didn't work.
Actually for a whole school of modern analyst it is true that .
We think that is defined if and only if .
We believe that it far better to go with the ideas from modern non-standard analysis of a enlargement.
Add one new number, , to the real number system.
We define the number as being a root of the equation
There is a group centered is St Louis calling for a more geometric approach to complex numbers.
But I say that using the enlargement idea is just as natural.
Look at this. If then and it is a root of !
Hence it seems as if .