01/22/2017, 10:29 PM

A few days ago, I post a question about the derivatives of these units. But to do it, I need to know everything about base units. I think this forum is optimal to collect our knowledges of it.

So a multidimensional number is from a linear combination of these units,

We multiply them like this way: \( i_x i_y = \eps _{ijk} i_{x\^k} \)

where i!=j, \( \eps \) is the levi-civita tensor which gives the sign of the multiplication, and ^ is a logic binary operator: xor.

And of course, we can represent them as matrices, too ... But just from reals to quaternions, because of the non-associativity of the octonions, the multiplication of the matrices of the representation of these octonions does not work ... I do not know why. (?)

I have many questions:

Why cannot we represent the octonions, sedenions ... etc.?

What is the derivative of \( i_x \)?

What is \( i_x^{-1} \) (inverse)?

Which logic binary operators could we substitute instead of \( \eps _{ijk} \)?

What is the taylor series of the xor and the levi-civita ops?

How can we calculate with i[x] where x is not integer, so it is real or complex?

So a multidimensional number is from a linear combination of these units,

We multiply them like this way: \( i_x i_y = \eps _{ijk} i_{x\^k} \)

where i!=j, \( \eps \) is the levi-civita tensor which gives the sign of the multiplication, and ^ is a logic binary operator: xor.

And of course, we can represent them as matrices, too ... But just from reals to quaternions, because of the non-associativity of the octonions, the multiplication of the matrices of the representation of these octonions does not work ... I do not know why. (?)

I have many questions:

Why cannot we represent the octonions, sedenions ... etc.?

What is the derivative of \( i_x \)?

What is \( i_x^{-1} \) (inverse)?

Which logic binary operators could we substitute instead of \( \eps _{ijk} \)?

What is the taylor series of the xor and the levi-civita ops?

How can we calculate with i[x] where x is not integer, so it is real or complex?

Xorter Unizo