Imaginary Numbers Aren't Imaginary

I actually wrote this over a year ago but this seems like the right platform so please enjoy an alternative motivation of the extension from the real numbers to the complex numbers.

Firstly, we're all happy with real numbers (denoted R because I don't have blackboard bold font): the number line along with its usual addition and multiplication. And we're also hopefully happy with the Euclidean plane, R^2 and the addition we do with vectors in R^2:

And this obeys all the things we like about addition of real numbers, namely the order of addition doesn’t matter, adding zero [the zero vector] changes nothing and adding something to its negative gives us zero [the vector]. The natural next question is how can we define multiplication for vectors in R^2? The naïve approach would be:

Here I’ve used the star so I can reserve the multiplication symbol for later. Again, this does all the nice things we like about multiplication of real numbers but there’s a subtle problem. When we multiply vectors in R^2 our answer shouldn’t care about how we orient the coordinate axes: it shouldn't matter if we rotate our vectors and then multiply or if we multiply and then rotate i.e. if rho is a rotation then

But as a concrete example taking (0,1) and (1,0) their 'product' is (0,0) but rotating anti-clockwise 45 degrees and multiplying gives (−1/2, 1/2) which is not the rotation of (0, 0) However adding them gives (1, 1) which rotates to (0, √ 2) and rotating them gives (√ 2/2, √ 2/2) and (− √ 2/2, √ 2/2)  which add up to (0, √ 2). We say that rotation "respects addition". Can we cook up a different definition of multiplication of vectors such that rotations "respect multiplication"?  Unfortunately I have no sort of nice way to motivate the following definition but I hope you can tolerate having it be handed down on high and seeing that it in fact does what we want. 

For this to be any kind of recognisable definition we want it to obey all the familiar nice properties of multiplication. If z and w are vectors in R^2 then we want:

[[These properties stipulate that × is commutative, distributive and associative.]] As this is maths you can go check these properties for yourself using the definitions or you can trust me but I won’t bore you with the details (I’ve worked through (3) by myself and it’s awful). One thing you might spot is that we have a ’copy’ of R inside our multiplication just by taking b and d to be zero:

So we know the first entry can just be viewed as a familiar real number. But this second entry clearly isn’t so independent, for example:

Which looks harmless, right? but if z = (0, 1) then z^2 = (−1, 0) and (−1, 0) added and multiplied just like -1. So we've found some sort of vector square root of -1? It feels wrong to call this the beginning of a new branch of mathematics: that doesn’t do justice to how important this is. 2-dimensional vectors are interesting in their own right but the richness that comes from imbuing R^2 with this multiplication is phenomenal. Conventionally we never write in terms of the vectors as we have: we write instead

This is usually the first definition in a complex numbers course but I hope this route of motivation has been insightful or at the least refreshing. This may feel artificial but the important fact this is the only commutative, associative, distributive multiplication on R^2 which indicates that maybe we discovered this operation and the necessity of defining i dropped out organically rather than making a u-turn and saying that -1 now has a square root and seeing what follows.