Are Hypercomplex Extensions Like g²=i Practically Useful in Mathematics?

[Isaac0427]

Hi,

I was just wondering if an extension of hypercomplex numbers like this have any use or if it would be pointless:

The number g is defined by $g^2=i$. Then, the powers of g from 0 to 8 (where the cycle restarts) would be 1, g, i, ig, -1, -g, -i, -ig, 1. There's a lot of interesting things I found that you can do with this. I was surprised, however, that I couldn't find a hypercomplex system like this that was established, making me wonder if this system is pointless; it does seem like something someone would come up with easily. Any thoughts?

 

[mfb]

There is a complex number with that property. Can you find it?

More general: all polynomial expressions like yours have a solution. This is one of the most important properties of the complex numbers: The fundamental theorem of algebra.

 

[Isaac0427]

There is a complex number with that property. Can you find it?

More general: all polynomial expressions like yours have a solution. This is one of the most important properties of the complex numbers: The fundamental theorem of algebra.

I know that $\frac{1+i}{\sqrt{2}}$ fits that definition. But, could this also be a useful hypercomplex system as well?

 

[fresh_42]

How do you define hypercomplex numbers? I've read a definition which says it's a division algebra over the reals. With this definition $\mathbb{R}$ and $\mathbb{C}$ are also hypercomplex and therefore $\mathbb{C}[g]$ as well, but $\mathbb{C}=\mathbb{C}[g]$, so how would this help? If you read the corresponding Wikipedia entry you will find interesting objects (number systems like dual numbers, octonions, sedenions, bicomplex numbers, biquaternion numbers) and historical explanations.
https://en.wikipedia.org/wiki/Hypercomplex_number