Why this my upvoted answer has been deleted by moderator with no explanation?

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This my answer seems to be absolutely relevant to the question:

https://math.stackexchange.com/a/4436362/2513

This looks like some personal attack.

The answer is below. If it is undeleted I would like to expand it considerably to include other examples.

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Example of 3D numbers.

Take $\mathbb{R}^3$ with Hadamard product. In other words, triplets of numbers with element-wise multiplication.

Now assign $(1,1,1)=1,(-1,1,1)=j, (1,1,-1)=k$.

A number would be written in the form $a+bj+ck$. Algebraically it will be a commutative ring with zero divisors (hence, not a field, but that's OK). For instance $(j-1)(k-1)=0$.

Here is a Mathematica code to experiment with:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
$Pre = (# /. {j -> {-1, 1, 1}, k -> {1, 1, -1}}) /. {x_, y_, z_} -> 
     x/2 + z/2 + (j (y - x))/2 + (k (y - z))/2 &;

Using this code you can see that

$j^2=k^2=1$

$jk=j+k-1$

$j^j=j^k=j$

$k^k=k^j=k$

$\sqrt{j+k}=\frac{j}{\sqrt{2}}+\frac{k}{\sqrt{2}}$

$\log (j+k+1)=\frac{1}{2} j \log (3)+\frac{1}{2} k \log (3)$

$0^{j+k}=1-\frac{j}{2}-\frac{k}{2}$

A division formula would be: $\frac{a_1+b_1 j+c_1 k}{a_2+b_2 j+c_2 k}=\frac{j}{2} \left(\frac{a_1+b_1+c_1}{a_2+b_2+c_2}-\frac{a_1-b_1+c_1}{a_2-b_2+c_2}\right)+\frac{k}{2} \left(\frac{a_1+b_1+c_1}{a_2+b_2+c_2}-\frac{a_1+b_1-c_1}{a_2+b_2-c_2}\right)+\frac{a_1+b_1-c_1}{2 \left(a_2+b_2-c_2\right)}+\frac{a_1-b_1+c_1}{2 \left(a_2-b_2+c_2\right)}$

If you add a complex unity $i$, you will get a 6-dimensional number system.

Particularly, you will see that

$i^j=ij$

$j^i=\frac{1}{2} \left(1-(-1)^i\right) j+\frac{(-1)^i}{2}+\frac{1}{2}$

$i^{j+k}=1-j-k$

$\log (j k)=i \pi-\frac{i \pi j}{2}-\frac{i \pi k}{2}$

and

$\sqrt{j}=\left(\frac{1}{2}-\frac{i}{2}\right) j+\left(\frac{1}{2}+\frac{i}{2}\right)$

Answer 1

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