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I was thinking about other complex-like systems the other day, and I decided to define a number $o$ such that $o^2 = 1, o \ne \pm 1$. I wondered if there was a formula like Euler's formula for this number system, where an exponential expression can be converted to a trignometric expression. As it turns out, there is: $$ \begin{align} e^{o\theta} & = \sum_{n = 0}^\infty \frac{(o\theta)^n}{n!} \\ & = \sum_{n = 0}^\infty \frac{(o\theta)^{2n}}{(2n)!} + \sum_{n = 0}^\infty \frac{(o\theta)^{2n + 1}}{(2n + 1)!} \\ & = \sum_{n = 0}^\infty \frac{\theta^{2n}}{(2n)!} + o\sum_{n = 0}^\infty \frac{\theta^{2n + 1}}{(2n + 1)!} \\ & = \cosh \theta + o \sinh \theta. \end{align} $$ It seems that where Euler's formula for complex numbers is related to a unit circle, Euler's formula for these numbers is related to a unit hyperbola. So, there is a sense in which $a + bi, i^2 = -1$ are circular numbers and $a + bo, o^2 = 1, o \ne \pm 1$ are hyperbolic numbers.

I tried this again with a number system built on the definition $k^2 = 0, k \ne 0$. Euler's formula for this system is: $$ \begin{align} e^{k\theta} & = \sum_{n = 0}^\infty \frac{(k\theta)^n}{n!} \\ & = \sum_{n = 0}^\infty \frac{(k\theta)^{2n}}{(2n)!} + \sum_{n = 0}^\infty \frac{(k\theta)^{2n + 1}}{(2n + 1)!} \\ & = 1 + 0\sum_{n = 1}^\infty \frac{\theta}{(2n)!} + k\theta + 0\sum_{n = 1}^\infty \frac{k\theta}{(2n + 1)!} \\ & = 1 + k\theta. \end{align} $$ This is analagous to the parametric equation of a "unit line", $(1, t)$.

As it turns out, these two number systems are already well-known: the hyperbolic numbers are called split-complex numbers (with $j$ as $o$) and the constant numbers are called dual numbers (with $\varepsilon$ as $k$).

It seems intuitively true that if one can express a function in terms of parametric equations for a set of axes (parameters) and make Taylor series for those equations and show that the sum of those series can be represented in terms of the Taylor series representation of the exponentional function given an arbitrary complex-like number system, one can have a set of "hypercomplex" numbers which in some sense describes a geometry (as defined by the function). So, in a sense, reverse engineer what was done above.

For example, assume complex numbers are not defined and one wants to make a number system that describes a circle. They could use the fact that the parametric equation for a circle is $(\cos t, \sin t)$,

My question is whether or not this reasoning is true or useful and if something like this has been done before.

Thank you for your answers.

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    $\begingroup$ What do you mean when you say "hypercomplex number"? If you mean that you add on a new number that satisfies a quadratic equation over the reals, then you've basically covered all the types of cases and someone could write an answer explaining why (and linking Michael Penn's video on it), if this isn't a duplicate. If you meant something more general, do you have anything concrete in mind? $\endgroup$
    – Mark S.
    May 27, 2022 at 16:35
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    $\begingroup$ Check John Stillwell's Naive Lie theory, the exponential is a really deep thing actually $\endgroup$ May 30, 2022 at 13:32
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    $\begingroup$ Ahh. Maybe there is something more to it. mathoverflow.net/questions/423657/… $\endgroup$
    – Anixx
    May 31, 2022 at 17:08
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    $\begingroup$ @MarkS. mathoverflow.net/questions/423657/… $\endgroup$
    – Anixx
    May 31, 2022 at 17:08
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    $\begingroup$ If you use $a+bi$ you get a spiral. If you use $a+bj$ you get the equation ${\left(x+(a+b)y\right)}^{a-b}={\left(x+(a-b)y\right)}^{a+b}$. Use $a+b\epsilon$ get $x+ay=1$. $\endgroup$ Nov 22, 2023 at 9:29

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No. If we take 2D case, those shapes characterize the sets of points that have magnitude $1$.

Since real numbers $1$ and $-1$ as we know have magnitude $1$, our shape should cross these points. Other investigations show that it should cross them at right angles so that the multiplication to work, no line from origin should cross the shape twice, in order that $(-1)j=-j$, $-1$ should have twice the argument (that is area between the unit curve and origin) of $j$, which means the curve should be symmetric both against real and $j$ axis, etc.

Basically, in 2 dimensions there are only 3 shapes that fit: unit circle (circle with radius 1 and centered at origin), hyperbola with apexes at $1$ and $-1$ and diagonal asymptotes, and two vertical lines perpendicular to the real line via $1$ and $-1$. This third shape defines dual numbers.

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