# Imaginary number dimensions

• B
Complex numbers ##a+bi## can be thought of as a second dimension extension of the real number line.

Is there a third dimension version of this? Are there even more complex numbers that not only extend into the y axis but also the z axis?

tex

PeroK
Homework Helper
Gold Member
2020 Award
Complex numbers ##a+bi## can be thought of as a second dimension extension of the real number line.

Is there a third dimension version of this? Are there even more complex numbers that not only extend into the y axis but also the z axis?

tex
Not 3, but 4 dimensions:

https://en.wikipedia.org/wiki/Quaternion

Ssnow
Gold Member
yes, You can obtain it considering ##\mathbb{C}\times \mathbb{R}## as structure space and giving rules for the addition and multiplication, from the history point of view is not used so much this extension ...
More interesting is the ##4## dimensional extension, this is the set of quaternions ...

Last edited:
Mark44
Mentor
Fixed that for you...

• SammyS and Ssnow
So is it correct to say that the single dimension real numbers can be considered a subset of the two dimensional complex number set where the real numbers are complex numbers with a 0 imaginary component?

tex

PeroK