[Haskell-cafe] Re: 0/0 > 1 == False

[Achim Schneider]

Jonathan Cast <jonathanccast at fastmail.fm> wrote:

> On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote:
> 
> > David Roundy <droundy at darcs.net> wrote:
> >
> >> Prelude> let x=1e-300/1e300
> >> Prelude> x
> >> 0.0
> >> Prelude> x/x
> >> NaN
> >>
> >> The "true" answer here is that x/x == 1.0 (not 0 or +Infinity), but
> >> there's no way for the computer to know this, so it's NaN.
> 
> Didn't catch this the first time around, but: only to a physicist.   
> (I mean no disrespect to the author of darcs, but nevertheless the  
> point stands).  Back in the real world, 0 / 0 may be defined  
> arbitrarily, or left undefined.  (Defining it breaks the wonderful  
> property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim  
> (xn / yn) = z.  This is considered a Bad Thing by real  
> mathematicians).  In fact, in integration theory 0 * inf = 0 for  
> certain 'multiplications', which gives the lie to 0 / 0.

whereas lim( 0 ) * lim( inf ) is anything you want, or, more precisely,
the area of the thing you started with. It's like taking seven balls,
assembling them into a hexagon and claiming it's a circle.

Such things just "happen" if you are working with pure abstractions.

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