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<p class="MsoNormal">[Attempting to resend]</p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Hi - I think your logic is:<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><i>I can define these two:<o:p></o:p></i></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fFrac :: Fractional a => a -> a<o:p></o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fFrac n = n * (n+1) / 2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fInt :: Integral a => a -> a<o:p></o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fInt n = n * (n+1) `div` 2<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><i>so that, e.g.<o:p></o:p></i></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fFrac (5.0 :: Float) == 15.0 :: Float<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">fInt (5 :: Integer) == 15 :: Integer<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><i>And all number types are either Integral or Fractional, so surely I should be able to define a single function of type:<o:p></o:p></i></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">f :: Num a => a -> a<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">This would seem reasonable, but I think there’s a problem with the last assumption. It is indeed possible for other types to be instances of Num, but not of Integral or Fractional.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">For example, I could define:<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New"">instance Num Bool where<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> fromInteger 0 = False<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> fromInteger _ = True<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> (+) = (&&)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> (*) = (||)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> abs = id<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> signum _ = True<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-family:"Courier New""> negate = not<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoNormal">Now this would probably be pretty dumb (and probably doesn’t comply with
<a href="https://hackage.haskell.org/package/base-4.14.0.0/docs/Prelude.html#t:Num">
expectations</a>), but is possible. (And also pretty dumb to define it without also define an
<span style="font-family:"Courier New"">instance Integral Bool where ...</span>, but still possible).<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">So I don’t think<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span style="font-family:"Courier New"">f :: Num a => a -> a<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">could be possible, since Num by itself (& the dumb Bool instance) has no way to do the division. (At least that I can think of, but would be very interested to hear if there is).<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Regards, David.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><img border="0" width="702" height="2" style="width:7.3125in;height:.0208in" id="Horizontal_x0020_Line_x0020_1" src="cid:image001.png@01D6CFDC.18287E30"><o:p></o:p></p>
<div id="divRplyFwdMsg">
<p class="MsoNormal"><b><span style="color:black">From:</span></b><span style="color:black"> Beginners <beginners-bounces@haskell.org> on behalf of M Douglas McIlroy <m.douglas.mcilroy@dartmouth.edu><br>
<b>Sent:</b> Friday, December 11, 2020 1:03:03 PM<br>
<b>To:</b> beginners@haskell.org <beginners@haskell.org><br>
<b>Subject:</b> [Haskell-beginners] unwanted Fractional constraint</span> <o:p></o:p></p>
<div>
<p class="MsoNormal"> <o:p></o:p></p>
</div>
</div>
<p class="MsoNormal">For rational functions that take on integer values at integer<br>
arguments, for example n*(n+1)/2, is there a way to doctor the<br>
corresponding Haskell definition<br>
<br>
f n = n*(n+1)/2<br>
<br>
so that the type signature becomes<br>
<br>
f :: Num a => a -> a<br>
<br>
rather than<br>
<br>
f :: Fractional a => a -> a<br>
<br>
Doug McIlroy<br>
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