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

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?

• That's actually the hyper-reals and it (sort of) works. – fleablood Sep 30 '18 at 2:15
• Concepts of that kind do get used, if they find applications. Random unmotivated systems fall by the wayside. – rschwieb Sep 30 '18 at 2:16
• 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 2:19
• Related (duplicate?): Why don't we define "imaginary” numbers for every “impossibility”?. – Blue Sep 30 '18 at 2:26
• 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 2:37

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.