# Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I’m programming a maths environment. I’m computer science, so please excuse any probable ignorance and lack of precision in my question.

It seems $i$ and complex numbers were “invented” out of necessity to solve equations like

$$x^2 = -1$$

As far as I can tell, the imaginary units present in the split-complex numbers, quaternions and other number systems don’t have that property. They weren’t invented out of necessity to solve a previously expressible equation. They were truly “invented” for such conveniences as being able to express multiple dimensions in one number and easy rotations.

Is this correct?

If I take the sequence of hyperoperations – $\{a+b,ab,a^b,\dots\}$ – and their inverses – $\{x-b,{x\over b},\dots\}$ – and form equations with them and the complex numbers what, if any, non-complex numbers can be generated?

In my aforementioned maths environment, I plan to implement every number as a pair of real numbers $(a, b)$ representing $a+bi$ – i.e. every number is a point on the complex plane. Besides issues with representing each of $a$ and $b$ with sufficient numerical precision and accuracy, is this sufficient to represent a solution to every equation composed of the hyperoperators, their inverses and complex numbers, that is known to have at least one solution? Is it sufficient to represent every solution to every equation composed of the same, that is known to have at least one solution?

I allow $n$-tuples and simple user defined functions in this environment. Are these, in combination with complex numbers or not, sufficient to implement all of split-complex numbers, quaternions, split-biquaternions, coquaternions, octonions, spacetime algebra, etc?

Let me know if there’s anything that requires clarification.

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Let me try to give at least a partial answer since there hasn’t been an answer since 3 years.

If I take the sequence of hyperoperations (…) and form equations with them and the complex numbers what, if any, non-complex numbers can be generated?

In a previous question I came to the conclusion that very likely the opposite hasn’t been proven. So it seems we call call that an open research question.

To construct the complex numbers from hyperoperations, I think, you will either need to add some concept of convergence to your code or simply be contented with less than perfect precision.