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So, $i$ and complex expressions are used as a sort of stepping stone to bypass domain issues when solving expressions and equations algebraically; it is very convenient to be able to factor out or solve the square root of a negative number. But is it just $i$? Are there any constants with no direct connection to the real world, that are not involved with the square root of $-1$?

For example, $\frac{1}{0}$ would be useful for sidestepping restricted domain in expressions like $\frac{3x+2}{x}$ or something... except that $\frac{1}{0}$ doesn't play well with the regular rules of math.

So, my question is, is there any comparable style of imaginary numbers, that I don't know about? Or is $i$ just special?

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  • $\begingroup$ en.wikipedia.org/wiki/Hypercomplex_number This isn't quite what you are asking for because although they add extra functionality as field extensions, they don't add to the algebraic completeness -> like $\mathbb{C}$ guarantees, or the analytic completeness like $\mathbb{R}$ guarantees. For the $\frac{1}{0}$ case specifically, you can sort of deal with a wheel algebra, although this actually costs some of the properties that were trivial before: en.wikipedia.org/wiki/Wheel_theory. $\endgroup$
    – Neptune
    Sep 25, 2021 at 4:15
  • $\begingroup$ Double numbers (a.k.a. split-complex numbers) have $j\neq1$ with $j^2=1$, they are related to the Minkowski plane. Dual numbers have $\varepsilon$ with $\varepsilon^2=0$, and are related to the Galilean plane. Up to isomorphism, those and complex numbers are the only 2D possibilities over $\mathbb{R}$. $\endgroup$
    – Conifold
    Sep 25, 2021 at 6:20
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    $\begingroup$ It is part of the ethos of abstract algebra that any equation you wish to hold can be made to hold if we allow ourselves to extend the domain of discourse. For example, going from the integers to the rational is an extension that lets us solve equations like $2x = 1$. Point being that the style in which $i$ is created is common. I can try writing a more detailed answer along these lines with more examples, but I'd like to ask if this sort of thing is what you had in mind. Also, would you happen to know any ring theory? $\endgroup$ Sep 25, 2021 at 6:43
  • $\begingroup$ @Neptune I do not think $\mathbb{R}$ is analytically complete, because there are sequences of real numbers (even monotonic) that do not have limit in $\mathbb{R}$. In other words, you can easily add divergent sequences (series, integrals) to $\mathbb{R}$. And you still would have a field! $\endgroup$
    – Anixx
    Sep 25, 2021 at 7:04
  • $\begingroup$ @Anixx analytic completeness is typically the DEFINING property of the real numbers, from which its other properties come. It means that the supremum property holds (any nonempty set that has an upper bound has a least upper bound). $\endgroup$
    – Neptune
    Sep 25, 2021 at 7:09

2 Answers 2

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There are many sorts of hypecomplex numbers.

There are hyperbolic numbers, also known as split-complex numbers, where another constant, $j$, which plays a role, similar to $i$.

If you combine complex numbers and hyperbolic numbers, you get tessarines, also known as bicomplex numbers.

There are also dual numbers with a similar but distinct constant $\varepsilon$.

You also can treat divergent integrals or growth rates as some kind of numbers (Hardy fields is a useful link here).

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After learning the complex numbers, it is natural to look for other extensions of the real numbers. There are many and others have given some examples but the other extensions have many more limitations.

With the complex numbers, most of the properties of the real numbers are retained and you can manipulate them in almost the same way. The main property that is lost is a definition of order, < and >, that works nicely with arithmetic. So, positive and negative cannot be usefully defined. The has important consequences on $\sqrt x$. On the other hand, calculus extends very nicely to the complex numbers.

Since adding $i$ as a solution to $x^2 + 1 = 0$ works so well, it is tempting to do the same for $\frac{1}{0}$. You can, you won't be arrested by the mathematics police, but so many things fail then few people judge it to be useful.

Maybe the next most nicely behaved extension of the reals is the quaternions. A lot of algebra still works (much more than when you add $\frac{1}{0}$) but you lose commutativity of multiplication: $xy$ is not necessarily equal to $yx$.

Square matrices are another extension but as well as losing commutativity, you also cannot be sure of division. You can have $xy = 0$ even if neither $x$ nor $y$ is $0$.

Maybe the most exotic extension is the surreal numbers.

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