1. ## Complex numbers

I have a question, asking it from curiosity, When I was reading the book Linear Algebra Done right, it talked about higher dimension in complex numbers, it didn't say much other than that the it's hard for the human brain to provide geometric model of higher dimension of complex numbers , higher than 2,

Firstly my question is, Isn't a complex number from 2 dimensional space, I mean we define complex numbers as C={a+ib|a,b are real numbers}
What would be a complex number from 1 dimensional space? (In that case, C=R I guess.) And if this the case would C be an ordered field? (Because on higher dimension doesn't have the order ''<'' like R and Q)

Secondly, what would be a complex number from higher dimensions ( higher that 2), I mean we thing of the axis x as the real number axis and y as the imaginary number axis, which takes i in front of it. So let's say C3 , what would the z axis be? And would it be just a point in 3 dimensional space, or more than that?

2. ## Re: Complex numbers

Originally Posted by davidciprut
I have a question, asking it from curiosity, When I was reading the book Linear Algebra Done right, it talked about higher dimension in complex numbers, it didn't say much other than that the it's hard for the human brain to provide geometric model of higher dimension of complex numbers , higher than 2,

Firstly my question is, Isn't a complex number from 2 dimensional space, I mean we define complex numbers as C={a+ib|a,b are real numbers}
What would be a complex number from 1 dimensional space? (In that case, C=R I guess.) And if this the case would C be an ordered field? (Because on higher dimension doesn't have the order ''<'' like R and Q)

Secondly, what would be a complex number from higher dimensions ( higher that 2), I mean we thing of the axis x as the real number axis and y as the imaginary number axis, which takes i in front of it. So let's say C3 , what would the z axis be? And would it be just a point in 3 dimensional space, or more than that?