Over at YouTube, there is a bit of history of Math, on study of cubic functions — and yes there is as usual, some less than exemplary detail — that led to the “invention” of imaginary numbers:
Now of course, I contend that this was discovery not invention (I often don’t buy Veritas Sum’s narrative, but here is a way to see the story).
In News’ thread on i, I commented at 33:
your definition [– Eugene at 8: “there exists such a pair of real numbers (0, 1) that (0, 1) * (0, 1) = -1, where “*” is the specific multiplication rule defined for these types of pairs” –] is tantamount to describing the role of sqrt – 1 in a vector system of numbers, extensible to the ijk vectors and onward to quaternions etc. I often just say take the j* operator as rot 0x 90 degrees anticlockwise. do a double and we see j*j* x –> – x, i.e, the sqrt has a natural, rotation linked sense as j^2 = – 1. Where, oscillations and waves including transients [think here Laplace and Fourier transforms] are closely tied to rotations. Where of course waves are pervasive in quantum mechanics, the Schrodinger expression is about waves and of course energy with standing waves under constraints naturally being quantised as we know from school physics. Further to such, I showed that from {} –> 0 thence N,Z,Q,R,C,R* etc, once distinct identity of possible worlds obtains, core math is embedded in any possible world which gives it universal power as Wigner marvelled at.
So, we are back to Wigner’s wonder. END