What distinguishes $i$ as an 'imaginary number' worth study versus other numerical ideas/numbers that should not/do not exist?

Question

In an earlier post, I entertained the idea that a number (would it be a number? Well, it would be something) $Φ$ existed such that the interval $[0,Φ]$ would include only $0$ and $Φ$.

In other words, $Φ$ such that $0<Φ< ℝ+$.

This pretty clearly does not make sense. But today I was thinking about this and how it compares to $i$, another number that does not seem sensible at first glance.

My question is: how exactly does a term like $i$ find general approval and integration into standard mathematics while other 'imaginary' concepts such as the $Φ$ I speculated on above not?

(just a thought experiment past this point, I'm making this up as I go)

Just like $i$ can find practical usage in mathematics and physics, can't ideas/numbers like the $Φ$ I described be able to find usage as well; for instance, could a number such as $Φ$ be used a useful shorthand in epsilon-delta problems?

Concepts of that kind do get used, if they find applications. Random unmotivated systems fall by the wayside. — rschwieb, Sep 30 '18 at 02:16
What logic of the real numbers do you want to keep in this new system that includes $\Phi$. Do you want to keep a notion of $+$, $\times$, $\leq$? Depending on which you say this may or may not be possible. — AHusain, Sep 30 '18 at 02:19
Related (duplicate?): Why don't we define "imaginary” numbers for every “impossibility”?. — Blue, Sep 30 '18 at 02:26
The square root of $-1$ wasn’t a solution in search of a problem, as your $\Phi$ appears to be. It arose organically in the course of solving problems that were interesting and important in their own right. — amd, Sep 30 '18 at 02:37
@fleablood No, it's not: in the hyperreals we still always have $\epsilon\over 2$ between $0$ and $\epsilon$ for any positive $\epsilon$. (The OP wants more than just infinitesimality.) — Noah Schweber, Sep 30 '18 at 03:13

score 1 · Accepted Answer · answered Sep 30 '18 at 02:33

One difference is, unlike the number system in this thought experiment, complex numbers are well-defined. Using properties to characterise something can be an intuitive way of talking about something, but there has to be an actual definition behind it. The idea of "everything stays the same except..." is not clear or mathematically rigorous.

In the case of the complex numbers, we can define them a few equivalent ways. One way is to define the complex numbers to be the points of $\mathbb{R}^2$, and equip it with the usual $\mathbb{R}^2$ addition, and multiplication defined as so: $$(a, b)(c, d) = (ac - bd, ad + bc).$$ It "contains" the reals, in the sense that the points $(x, 0)$ can be identified with the real numbers $x$, and addition and multiplication are respected. Also, it has a square root of $(-1, 0) \sim -1$, specifically $(0, -1)$ (or $(0, 1)$). It has the properties that we asked for, when we asked for a square root of $-1$.

But, we do lose some stuff. There's no longer a sensible order. In particular, surely since $i \neq 0$, we must have $i > 0$ or $i < 0$. But in either case, surely $-1 = i^2 > 0 > -1$?

We also no longer have nice rules with radicals, such as $\sqrt{ab} = \sqrt{a}\sqrt{b}$ (e.g. $a = b = -1$), see my answer here for more on this.

Then, of course, we have the quaternions which lose commutativity of the multiplication, and octonions which lose associativity and commutativity of the multiplication. Basically, changing the number system may introduce some nice structure, but you're probably going to lose some structure too.

Which structure you keep and which you lose often depends on the specific definition.

What distinguishes $i$ as an 'imaginary number' worth study versus other numerical ideas/numbers that should not/do not exist?

1 Answers1