4 ms·
I don't agree. Complex numbers are the algebraic closure of the reals. Or the quotient of the real polynomial ring by (x^2+1=0). Or whatever other construction.
by nh23423fefe 2y ago
I don't agree. Complex numbers are the algebraic closure of the reals. Or the quotient of the real polynomial ring by (x^2+1=0). Or whatever other construction. The multiplication rule is the essence of C.
Orthogonality is captured linear algebra over R^2, but R^2 isn't a field or an algebra.
- markisus 2y agoI think there is still a geometric viewpoint you can bring to the multiplicative structure of C. For example there is the extremely natural homeomorphism between unit C and SO(2). And C minus origin to (R+, SO(2)). It’s completely intuitive for mathematicians to say that 1 and i are separated by 90 degrees.
- btilly 2y agoIndeed, there is a polar coordinates representation of the complex numbers with the nice property that when you multiply you multiply lengths and add angles.
- quantadev 2y agoSniffy nailed it (above) when he mentioned Cayley-Dickson. That's exactly what I was getting at, and had never researched.