[Guille]

Originally Posted by AntonioLao the math behind Euler's identity is the imaginary exponent of real is real. I have the book that explains a little bit about this. I try to look into it.

Some new questions, thus, arise to me:

If the imaginary exponent of real, is real, then, what are the:

Imaginary exponent of imaginary?
Real exponent of real?
Real exponent of imaginary?
Imaginary root of real?
Imaginary root of imaginary?
Real root of real?
Real root of imaginary?
Complex root of imaginary?
Complex root of real?
Complex exponent of real?
Complex exponent of imaginary?

I know these are many questions, but you can answer each with a capital I, R, or C.

[AntonioLao]

Originally Posted by GUILLE then, what are the:

These are good thesis topics for you to work on. By the time you answer all of these you would be holding in your hand a degree of Doctor of Philosophy in mathematics, a higher degree than an Honours or Masters degree, involving at least two and a half years of supervised research resulting in a thesis. PhD graduates may call themselves “Dr” If I do the same, I will be in the cemetary.

[Guille]

Originally Posted by AntonioLao These are good thesis topics for you to work on. By the time you answer all of these you would be holding in your hand a degree of Doctor of Philosophy in mathematics, a higher degree than an Honours or Masters degree, involving at least two and a half years of supervised research resulting in a thesis. PhD graduates may call themselves “Dr” If I do the same, I will be in the cemetary.

ahahahha........

Aren't these questions answered by mathematicians yet?!

[AntonioLao]

Originally Posted by GUILLE Aren't these questions answered by mathematicians yet?!

I am not a mathematician by trade. So, I will be the last person to know the answer to your question. My guesses are

Imaginary exponent of imaginary? not defined
Real exponent of real? trivial
Real exponent of imaginary? cyclical between real and imaginary
Imaginary root of real? Euler did them
Imaginary root of imaginary? not defined
Real root of real? trivial
Real root of imaginary? not defined completely
Complex root of imaginary? not defined
Complex root of real? not defined
Complex exponent of real? Euler did them
Complex exponent of imaginary? not defined

[Guille]

Originally Posted by AntonioLao I am not a mathematician by trade. So, I will be the last person to know the answer to your question. My guesses are Imaginary exponent of imaginary? not defined Real exponent of real? trivial Real exponent of imaginary? cyclical between real and imaginary Imaginary root of real? Euler did them Imaginary root of imaginary? not defined Real root of real? trivial Real root of imaginary? not defined completely Complex root of imaginary? not defined Complex root of real? not defined Complex exponent of real? Euler did them Complex exponent of imaginary? not defined

Thanks. Although it doesn't solve everything, this post gives me start point for my investigations on lattice and the complex plane. I've now noticed that I've left out a few more possibilities, tell me if they are defined or not, and where to investigate them:

real exponent of complex?
real root of complex?
imaginary exponent of complex?
imaginary root of complex?

[AntonioLao]

According to Keith Devlin in his book 'The language of Mathematics: Making the Invisible Visible' page 134, the imaginary exponent of imaginary number is real.

[Guille]

Originally Posted by AntonioLao According to Keith Devlin in his book 'The language of Mathematics: Making the Invisible Visible' page 134, the imaginary exponent of imaginary number is real.

Is this widelly accepted or not?

What about my propositions in my previous post? Are the answrs all unknown?

[AntonioLao]

Originally Posted by GUILLE Is this widelly accepted or not?

Keith Devlin didn't say.

Originally Posted by GUILLE Are the answrs all unknown?

I don't know. A resident mathematician at the university would most like be able to give all the answers.
Exponentiation and extracting roots are both classified as operators of functional analysis and in that sense they can operate only on certain type of numbers or functions for all that matters. Similar to transformation theory, the answers must be within the same mathematical groups otherwise it is meaningless.

[Guille]

Originally Posted by AntonioLao Keith Devlin didn't say. I don't know. A resident mathematician at the university would most like be able to give all the answers. Exponentiation and extracting roots are both classified as operators of functional analysis and in that sense they can operate only on certain type of numbers or functions for all that matters. Similar to transformation theory, the answers must be within the same mathematical groups otherwise it is meaningless.

Do you think If I contact a proper mathematician s/he will be willing to give me the answers?

About the other operators, this is, +, -, x and /, are they all defined?

[AntonioLao]

Originally Posted by GUILLE About the other operators, this is, +, -, x and /, are they all defined?

I know for sure that division is not defined in vector analysis and there are more than one definition for vector multiplications.

Originally Posted by GUILLE Do you think If I contact a proper mathematician s/he will be willing to give me the answers?

Most likely he/she would ask you to enroll in his/hers classes or pay him/her money for the efforts. The answers you get from them would be the most authoritative answers but you still have to find the most authoritative mathematicians alive as of today.