Can Complex Numbers Extend Beyond Two Dimensions?

[thetexan]

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]

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]

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 ...

 

[Mark44]

Fixed that for you...

giving rules for the addiction addition and multiplication

 

[thetexan]

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]

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

Yes!