[Haskell-cafe] Num instances for 2-dimensional types

Miguel Mitrofanov miguelimo38 at yandex.ru
Mon Oct 5 10:55:11 EDT 2009

No, they aren't. They are polynomials in one variable "i" modulo i^2+1.

Seriously, if you say complex numbers are just pairs of real numbers - 
you have to agree that double numbers (sorry, don't know the exact 
English term), defined by

(a,b)+(c,d) = (a+c,b+d)
(a,b)(c,d) = (ac, ad+bc)

are just pairs of real numbers too. After that, you have two choices: a) 
admit that complex numbers and double numbers are the same - and most 
mathematicians would agree they aren't - or b) admit that the relation 
"be the same" is not transitive - which is simply bizarre.

Lennart Augustsson wrote:
> But complex numbers are just pairs of numbers.  So pairs of numbers
> can obviously be numbers then.
> On Mon, Oct 5, 2009 at 4:40 PM, Miguel Mitrofanov <miguelimo38 at yandex.ru> wrote:
>> Lennart Augustsson wrote:
>>> And what is a number?
>> Can't say. You know, it's kinda funny to ask a biologist what it means to be
>> alive.
>>> Are complex numbers numbers?
>> Beyond any reasonable doubt. Just like you and me are most certainly alive.
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