Very recently I learned about the number $j^2 =1, j\neq 1$ which I found a fascinating idea and in hindsight quite an obvious one also.
The though of this number existing and meaningfully describing minkowsi space got me wondering: is it meaningful to formulate a number $k^2=0, k\neq0$, and does it have any useful properties?
Now, because I had already done some work myself on how such a number would most likely behave I have decided to add that here as well in the hopes that someone might recognize the behaviour of such a number to say describe some obscure variant on minkowski or euclidian space.
How I theorised such a number could work is as follows.
We define a number $$z = x + yk$$
Addition of two such numbers should give
$$z_1+z_2 = (x_1+x_2) + (y_1+y_2)k$$
And multiplication should give
$$z_1\cdot z_2 = x_1\cdot x_2 + (x_1\cdot y_2+x_2\cdot y_1)k + y_1\cdot y_2 k^2 = x_1\cdot x_2 + (x_1\cdot y_2+x_2\cdot y_1)k$$
As seen above, all terms where the power of $k$ is greater than one cancel as they are multiplied by zero. This means that we get a definition for positive integer powers that is surprisingly easy to expand to negtaive, fractional or even complex powers
$$z^n = x^n+\binom{n}{1}x^{n-1}yk = x^n+nx^{n-1}yk$$
Without any concern for what this would mean I have extended that definition from $n \in \mathbb{N}$ to $n \in \mathbb{R}$ to create the plot below which shows lines connecting powers of the same number.
I have yet to formulate a definition for $z_1^{z_2}$ and if there is no existing body of work describing this type of numbers I would love to hear everyone's thoughts on how a neat definition for said operation could be formulated.