# Extending the complex numbers by the solution of $|x| = -1$

I don't think I've ever encountered a situation where I've wanted to solve equations of the form $|x| = -1$, but you often hear that mathematics should be explored for the sake of mathematics. I'm wondering if this venue has been explored and what came up, and if it hasn't been explored, why not?

I guess whatever number satisfies $|x| = -1$ would be annoying to work with since a lot of nice stuff like the triangle inequality would fail, but is there any intrinsic reason that it would not make sense as a definition of a new number?

• A "number" doesn't exist on its own. You have to be able to add it to, subtract it from, multiply and divide it by, other numbers. What would you get for $|x+1|$? for $|x^2|$? If you can make sense out of these, you may be onto something. Otherwise, not so much. – Gerry Myerson Aug 11 '15 at 7:27
• Regarding your guess: you're forgetting about the fact that, for example, complex exponentiation doesn't work exactly like real exponentiation. Would you say complex numbers are annoying due to this? Indeed, continuing to extend the complexes, we get the octonions and finally the sedenions, losing however both commutativity and associativity. That is arguably annoying, but sedenions are still interesting and useful. – Vincenzo Oliva Aug 11 '15 at 7:32
• Somewhat related: math.stackexchange.com/questions/259584/… – Hans Lundmark Aug 11 '15 at 9:12
• A related question. – Lucian Aug 11 '15 at 9:37
• @Lucian Very nice. So, this extension would lack natural way of defining distances. I'm still wondering if that is a dealbreaker for the study of the field. Maybe I'm mistaken, but a field without a distance metric can still be a field, right? – Benjamin Lindqvist Aug 11 '15 at 10:52

The way we originally defined $|~|$, it was a Norm which means in particular that $|x|>0$ for every $x$ except $x=0$ in which case $|0|=0$. If there were to be some solution for $|x|=-1$, it would no longer be a norm, but instead just an arbitrary function with no other required properties.
Certainly, you can define a function on a space $\mathbb{X}=\{a+bi+cj~:~a,b,c\in\mathbb{R}\}$, isomorphic to $\mathbb{R}^3$ as $f(a+bi+cj)=|a+bi|+c$ where $j$ is our new "imaginary unit." This function $f$ winds up agreeing with the norm when the input is a strictly complex number (of the form $a+bi+0j$) but can take any value in general, in particular $-1$.
• Yeah, we can extend the definition of $|x|$ such that it's positive definite for $x \in \mathbb{C}$. That's what I mean by 'annoying', we'd lose a bunch of norm related stuff in our new field. But I'm wondering if there's a reason why this intrinsically makes the concept not worthy of study! – Benjamin Lindqvist Aug 11 '15 at 8:32
• Well, as mentioned, our version of $|~~|$ won't even satisfy the properties required of being a distance function, so this won't be a metric space and there would be no concept of convergence. Of course, we could define a different norm, but that would seemingly defeat the purpose since you seem to have wanted to move to a space where the "(not)norm" could have a negative output. – JMoravitz Aug 11 '15 at 15:55