Euler Identity

[Guille]

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

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.

Yes, but there is a problem: proportionality. Basically:

The more authoritative the mathematician is, the more likelly it is that the mathematician asks for more lucratives (enrolling clases, paying, etz...).

So I'll develop the answers myself, better.

Why do you talk about vectors here? If we are actually talking about numbers, scalars?

[AntonioLao]

Why do you talk about vectors here? If we are actually talking about numbers, scalars?

Vectors are hypercomplex numbers. They belong to a larger set than complex numbers and complex numbers is in a bigger set than real numbers. To become a number system some of the following properties must be defined for binary operations of addition and multiplication.
1. commutativity
2. associativity
3. identity
4. reciprocity
5. distributativity
maybe more that I'm not aware of...

[Guille]

Vectors are hypercomplex numbers. They belong to a larger set than complex numbers and complex numbers is in a bigger set than real numbers. To become a number system some of the following properties must be defined for binary operations of addition and multiplication.
1. commutativity
2. associativity
3. identity
4. reciprocity
5. distributativity
maybe more that I'm not aware of...

And are tensors and spinors hypercomplex numbers?

[AntonioLao]

And are tensors and spinors hypercomplex numbers?

I think they are. But the site says otherwise http://en.wikipedia.org/wiki/Hypercomplex_number

[bsaucer]

Some new questions, thus, arise to me:

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

Imaginary exponent of imaginary?

Actually, the imaginary exponent of a real number can be real, imaginary, or complex. Same with imaginary exponent of an imaginary.

e^(pi*i/2) = i
e^(pi*i) = -1
e^(3*pi*i/2) = -i
e^(2*pi*i) = 1.

Also, i^i = e^(-pi/2). This one is real, but if you change the exponent a little bit, you can get non-real values.

I've even played around with quaternions, but certain problems crop up... What is i^j? It's either k or -k, depending on which order you multiply exponents when you work it out. I think Hamilton chose -k as the result.

Actually, with quaternions, exponents "operate" on the base either "from the left" or "from the right". I believe Hamilton used the "left exponent", even though the exponent is written to the right of the base.

I'll research that some more...

[AntonioLao]

For now, I have to take your words since I need some time to work it out.