Intereting Posts

Using the unit circle to prove the double angle formulas for sine and cosine?
Book recommendations for self-study at the level of 3rd-4th year undergraduate
Discrete probability problem: what is the probability that you will win the game?
In what ways has physics spurred the invention of new mathematical tools?
Lipschitz in $\mathbb R^1$ implies Lipschitz along any line in $\mathbb R^k$ (for convex functions)
Math behind rotation in MS Paint
Volume form on $(n-1)$-sphere $S^{n-1}$
Why isn't this a valid argument to the “proof” of the Axiom of Countable Choice?
Book suggestions: Introduction to Measure Theory for non-mathematicians
Are proofs by induction inferior to other proofs?
Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?
Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric
Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds
Solutions to exp(x) + x = 2
Statistics: Joint Moment Generating Function Question

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

$$ x^2 = -1$$

- Complex numbers system of equations problem with 5 variables
- How to prove that every number to the power of any other numbers can be express in a+bi?
- Is this a sound demonstration of Euler's identity?
- If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
- Type of singularity of $\log z$ at $z=0$
- How many roots have modulus less than $1$?

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.

- Given complex $|z_{1}| = 2\;\;,|z_{2}| = 3\;\;, |z_{3}| = 4\;\;$ : when and what is $\max$ of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$
- What is the most useful/intuitive way to define $\Bbb{C}$, the complex set?
- Does the limit of $e^{-1/z^4}$ as $z\to 0$ exist?
- Describe the set of all complex numbers $z$ such that $|z-a |+| z-b |=c$
- $(-1)^{\sqrt{2}} = ? $
- Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant
- Etymology of “topological sorting”
- How can one intuitively think about quaternions?
- A contradiction involving exponents
- Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

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.

- Convergence of integrals in $L^p$
- Do these inequalities regarding the gamma function and factorials work?
- Understanding second derivatives
- Finding the limit $\lim_{x\rightarrow \infty} \sqrt{x+1}-\sqrt{x}$
- Continuous injective map is strictly monotonic
- N circles in the plane
- Contradictory; Homotopy equivalence and deformation retract problem
- Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$
- History of Lagrange Multipliers
- Solving $x\sqrt1+x^2\sqrt2+x^3\sqrt3+…+x^n\sqrt{n}+\dots=1$ with $x\in \mathbb{R}$ and $n\in \mathbb{N}$
- primegaps w.r.t. the m first primes / jacobsthal's function
- Does the limit of a descending sequence of connected sets still connected?
- When each prime ideal is maximal
- Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
- Open Sets Boundary question