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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?

Palbitt
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  • https://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: https://en.wikipedia.org/wiki/Wheel_theory. – Neptune Sep 25 '21 at 04:15
  • 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 $\mathbb{R}$. – Conifold Sep 25 '21 at 06:20
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    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? – paul blart math cop Sep 25 '21 at 06:43
  • @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! – Anixx Sep 25 '21 at 07:04
  • @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). – Neptune Sep 25 '21 at 07:09
  • @Neptune okay, but this is too strict. – Anixx Sep 25 '21 at 07:11
  • @Anixx in what sense do you mean too strict? – Neptune Sep 25 '21 at 07:12
  • @Neptune There are sets of reals that do not have the upper bound in $\mathbb{R}$. If you reformulate the phrasing so that any set of reals should have the least upper bound, you get a wider domain than reals. – Anixx Sep 25 '21 at 07:15
  • @Neptune and, there are multiple approaches at compactification of reals. From trivial ones like $\overline{\mathbb{R}}$ to highly non-trivial. It depends on how you define the equivalence classes of sequences. – Anixx Sep 25 '21 at 07:19
  • @Anixx that's true but when people refer to analytic (or Dedekind) completeness, it means that the bound is within $\Bbb{R}$. As far as I can tell, the issue with making it the projective line, for example, is that it allows for nonmonotonic sequences to have your infinite limit as its supremum to converge to, which is why people are OK with making analytic completeness require the upper bound to be within $\Bbb{R}$. Regardless of the approach at the compactification, the LUB will still hold, though. – Neptune Sep 25 '21 at 07:20
  • @Neptune my point was that by choosing compactification and defining equivalence classes of divergent sequences one can extend reals quite easy. And after that one can speak about field closure of the newly-added numbers so to add even more. For instance, we can add a divergent integral $\int_0^\infty dx$ via compactification, and then something exotic like $(-1)^{\int_0^\infty dx}$ via field operations. The resulting "number" would not be even infinite: its modulus (exponent of real part) is still $1$ but its regularized value is $\pi/2$! – Anixx Sep 25 '21 at 07:27
  • Let us continue this discussion in chat. – Neptune Sep 25 '21 at 07:29
  • If you want something "more complete" than real numbers take a look at Conway's surreal numbers, they have both infinities and infinitesimals, form an ordered field, and every ordered field is their subfield. In particular, both $ω$ (countable infinity) and $1/ω$ are surreal numbers. – Conifold Sep 25 '21 at 08:39
  • @Conifold this "surreal numbers" nonsense always puzzled me, and when people started to explain it to me, it started to look even more nonsense. For instance, "countable infinity" is not a single entity. For instance, [math]\sum_{k=0}^\infty 1[/math] and [math]\sum_{k=1}^\infty[/math] are both "countable infinities" but the first one is greater by 1. The [math]\sum_{k=-\infty}^\infty1[/math] and [math]\sum_{k=-\infty}^\infty \frac{(-1)^k+1}2[/math] are both "countable infinities", but the first one is twice greater. Yet, you somehow have one "countable infinity" in surreal numbers. – Anixx Sep 29 '21 at 13:41

2 Answers2

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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).

Anixx
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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.

badjohn
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