(in)exactness of complex numbers

[Michael Abbott]

Michael Ackerman <ack at nethere.com> wrote in 
news:mailman.996787686.29273.python-list at python.org:

> 
> 
> "David C. Ullrich" wrote:
> 
>> Exactly what definition of
>> "complex number" do you have in mind here?
>> The _standard_ definition _is_ "pair of real numbers".)
> 
> If you define the complex numbers C as pairs of reals, you have given at
> most its structure of real vector space. Then you must define
> multiplication and prove that what you have is actually a ring. But if
> you define C as the quotient ring R[X]/(X^2+1), then you needn't do
> anything more. This definition is often used, e.g. in MacLane &
> Birhkoff's "Algebra", the most categorically oriented basic algebra
> text.  
> 
> -- Michael Ackerman


Actually, this is kinda amusing: I thought I'd try writing a Complex class 
using the representation above (ie, the value of a complex number is a 
polynomial, and equality is modulo X^2+1); in this case the only real 
complexity (sic) is implementing equality and division by quotienting by 
X^2+1 -- but then I realised that I might as well quotient my 
representation by X^2+1 throughout.
    Then, surprise, surprise, it turns out that we recover the standard 
(real,imaginary) presentation of reals.  Of course this isn't all that 
surprising, given that X is there precisely to represent the root of 
X^2+1=0.