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

[Lennart Augustsson]

Complex numbers are just pairs of numbers, and then the various
operations on them are defined in a specific way.
There may be other ways to define the operations on pairs of numbers
that makes sense too.

You can also view complex numbers as polynomials if you wish.  Or two
element lists of numbers.

My real point is that you shouldn't tell others what they should
regard as numbers and what not.
Being a number is in the eye of the beholder. :)

On Mon, Oct 5, 2009 at 4:55 PM, Miguel Mitrofanov <miguelimo38 at yandex.ru> wrote:
> 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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>