2021-03-06, 12:37 PM (ISO 8601)
Barbarian in the Playground
Re: Why does ijk=-1
Which isn't a surprise as basically he wanted vectors and cross products 40 years too early.
Originally Posted by Rockphed
I looked it up, and I was right historically i*j*k=-1 is axiomatic and written on Brougham bridge. Although 3B1B's presentation of complex numbers is based on a more modern thinking of what they mean, so he'd probably start by constructing the rotational analogue and then derive the historical axioms as the interesting parallel rather than the other way round. A third option would be to start with the multiplication timeline.
Again as Rockphed implies, many of the other options have the wrong sort of consequences.
If i,j,k aren't symettric then that's a bit messy (though you do have Tessarines, in this case you have values such that a*b=0).
If the orthoganal 'vectors' were unaffected by the rotation then the resultant 'direction' is basically dominated by the last two numbers (I think), etc... (which I'm sure you could do something with but isn't a good 4D complex number)