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What about things like quaternions, which suddenly became relevant when we needed to interpolate 3D transforms and do rotations without Gimbal lock? The current best process for calibrating cameras is to use dual quats, which also means needing dual number theory. Were those areas originally expected to be useful for engineering? My understanding is no, but I'm not a mathematician.
That's a good question. I'm not sure of the exact reason quaternions were invented - you can indeed stumble on them just by trying to extend the complex numbers in an abstract way - but the Wikipedia article suggests they were already being used for 3D mechanics within a couple of years of invention. (BTW, "number theory" involves integers, primes, that kind of thing, not quaternions. Complex numbers do show up though.)
You could ask the same question about complex numbers too, but they originally arose from the search for an algorithm to solve cubic equations, which is a fairly practical question. That they later turned out to be essential for electronics and quantum mechanics is a case of some new applications of an already useful math concept.
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