search for: complex_number

...over the reals, Yes, but they _also_ are field (and as others have argued mathematically only have one infinity point), and I think here we are fighting with which definition should take precedence here. The English Wikipedia page is even more extensive and precise, https://en.wikipedia.org/wiki/Complex_number (line breaking by me): " The complex numbers form a rich structure that is simultaneously - an algebraically closed field, - a commutative algebra over the reals, and - a Euclidean vector space of dimension two." our problem "of course" is that we additionally add...

...t they _also_ are field (and as others have argued mathematically only > have one infinity point), > and I think here we are fighting with which definition should > take precedence here. > The English Wikipedia page is even more extensive and precise, > https://en.wikipedia.org/wiki/Complex_number (line breaking by me): > > " The complex numbers form a rich structure that is simultaneously > - an algebraically closed field, > - a commutative algebra over the reals, and > - a Euclidean vector space of dimension two." > > our problem "of course&qu...

The thing is that real*complex, complex*real, and complex/real are not "complex arithmetic" in the requisite sense. The complex numbers are a vector space over the reals, and complex*real and real*complex are vector*scalar and scalar*vector. For example, in the Ada programming language, we have function "*" (Left, Right : Complex) return Complex; function "*" (Left :