Can $i$ be conceptualized as a free variable?

Question

Stuart Hollingdale's book Makers of Mathematics states the following:

In 1833 Hamilton read a paper to the Royal Irish Academy in which he pointed out that the plus sign in $a + ib$ was a misnomer, as $a$ and $ib$ cannot be added arithmetically. Following Gauss, he proposed that a complex number should be regarded as an ordered pair of real numbers $(a, b)$ which obey certain operational rules (etc.)

With a computer science hat on, as I understand this, the problem is that multiples of $i$ are not the same type as multiples of $1$, so it simply not possible to perform the addition operation. The expression as a whole doesn't type-check, because addition (implicitly, of reals) has type $\mathbb{R} * \mathbb{R} \rightarrow \mathbb{R}$. Hamilton recognizes the need to keep the real and imaginary components apart, because they are incompatible.

I have two vague doubts about this objection. The first is just that we could imagine an addition operator that has type $\mathbb{C} * \mathbb{C} \rightarrow \mathbb{C}$, with which $a + ib$ would be equivalent to $(a + i0) + (0 + ib) = a + ib$. This is equivalent to addition in Hamilton's Theory of Couplets, I guess.

The second thought is more about the nature of $i$. Can we think of $i$ as a free variable? What I mean is, since $i$ is a placeholder for an imaginary number, can we treat it as a free variable of type $\mathbb{R}$ whose value, we know, will never become available? This would make $a + ib$ into an acceptable (but irreducible) expression, from the point of view of typing.

I know this is a bit philosophical, but I suppose what I'm trying to understand is the relationship between the way $i$ behaves (as a placeholder) and how variables behave. If it's possible to think of $i$ as a variable that will never get a value, that would seem straightforward, at least from a syntactic point of view.

The apparently identical notation for field extensions and polynomial rings ($\mathbb{C} = \mathbb{R}[i]$ and $K[X]$) seems to support this interpretation, but I'd like to hear about any wrinkles.

Answer 1

One cannot consider $\mathbf i$ to be a free variable, since the relation $\mathbf i^2+1=0$ is written in stone, and a truly free variable would not respect that relation. However, one can treat a formal symbol $X$ as an indeterminate (which is not the same as a free variable since it does not have the ambition the hide an unchanging but unknown number) and form the polynomial ring $\Bbb R[X]$, and in which it is perfectly valid to form expressions like $(X-3)*(X^3-7)$ even though $X$ is a different kind on entity than numbers (which should soften your apprehension about invalid types a bit). Now you can form the quotient of the ring $\Bbb R[X]$ by its ideal generated by $X^2+1$, and call the image of $X$ in the quotient $\mathbf i$, which by construction satisfies $\mathbf i^2+1=0$. This is the usual construction of the ring (and indeed field)$~\mathbb C$.

Answer 2

It’s natural to think of complex numbers as an extension of the real numbers, not just as pairs of them. The notation $a+bi$ makes dealing with complex numbers much easier than using $(a,b)$ e.g. Multiplying $(a,b)$ and $(c,d)$ is nonobvious, but $(a+bi)(c+di)$ clearly just requires using the distributive law.

This kind of redefining of old notation to match our intuitions in new contexts is very common throughout math. In CS, this kind of simplification is often known as syntactic sugar.

If you want to be rigorous about it, you could define $\mathbb C$ as $\mathbb R[X]/(1+X^2)$, meaning that you just consider $\mathbb C$ as being any combination of $i$ and reals numbers under the assumption that $i^2=-1$, though it requires some work to show that division is well-defined I.e. that this is a field. This is a particularly nice approach since it’s clear that the real numbers are a subfield.