# Were 3-dimensional split-complex numbers ever described in literature?

Basically, if you add two complex dimensions to reals, say $$i$$ and $$j$$, you automatically get a fourth dimension $$ij$$ because this number cannot be expressed using only the three dimensions. The system you get then is called bicomplex numbers and 4-dimensional.

On the other hand, if you add two split-complex dimensions to reals, say $$j$$ and $$k$$, you do not get a fourth dimension automatically because we can define $$jk=j+k-1$$, which can be expressed in the already existing 3 dimensions. Thus, you get a 3D algebra.

It seems that each of the two added split-complex dimensions are isomorphic to the classic split-complex axis.

Construction and properties

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 one can see that

$$j^2=k^2=1$$

$$jk=j+k-1$$

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

$$j^j=j^k=j$$

$$k^k=k^j=k$$

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

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

The 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 we add a complex unity $$i$$, we will get a 6-dimensional number system.

Particularly, we will see that

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

and

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

So, I wonder whether anyone ever described such a system separately from just $$\mathbb{R}^3$$ (to which it is isomorphic similarly to how usual split-complex numbers are isomorphic to $$\mathbb{R}^2$$)? Maybe in the 19th century?

• This does not really look like a historical question to me. Apr 30 at 16:47