# Thread: Complex numbers

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?

Thirdly, where can I learn more about this subject? Thank you.

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?

Thirdly, where can I learn more about this subject? Thank you.
There is no such thing as a one dimensional complex number. You clearly need at least two degrees of freedom in order for the concept of a complex to number to make any sense. I suppose you could have purely imaginary numbers. This would be a 1D space and I guess you couldn't call them real numbers even though they are just the Reals multiplied by i.

I've seen lecturers refer to "complex dimension" where a 2D complex number has a complex dimension of 1. I think Penrose uses this idea when talking about spinors.

The phrase complex numbers of a higher dimension is a bit of misnomer. When you try to extend fields to higher dimensions it generally doesn't work and even when it does, i.e. quaternions, the field elements are not complex numbers, they are quaternions, and they don't obey all the same laws as complex numbers do. Multiplication for example isn't commutative over the quaternions I believe.

One of the real math experts here can expand on this I'm sure.

In the meantime you can read this.