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

[Miguel Mitrofanov]

On 5 Oct 2009, at 19:17, Lennart Augustsson wrote:

> Complex numbers are just pairs of numbers,

What about dual numbers? (Yes, I've remembered the term) Aren't they  
also just pairs of numbers?

> There may be other ways to define the operations on pairs of numbers
> that makes sense too.

But these aren't the ways to define pairs of numbers, right?

> My real point is that you shouldn't tell others what they should
> regard as numbers and what not.

Suppose somebody asks why his Haskell program doesn't work, and after  
some interrogation he
reveals that he just renamed Program.hs to Program.exe. Shouldn't we  
tell him NOT to do this?

> Being a number is in the eye of the beholder. :)

Yes, and some points of view aren't as good as other. That's why we  
have psychiatric hospitals.

>
> 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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