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diff --git a/miralib/ex/rational.m b/miralib/ex/rational.m
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+||rational numbers as an abstract data type, say
+||	%include <ex/rat>
+||to include this in one of your own scripts.  Quoting the  filename  in
+||this form makes the %include directive work from any directory.
+
+abstype rational
+with ratio :: num -> num -> rational
+     mkrat :: num->rational
+     rplus, rminus, rtimes, rdiv :: rational -> rational -> rational
+     rpow :: num -> rational -> rational
+     numerator, denominator :: rational -> num
+     rparts :: rational -> (num,num)
+     showrational :: rational -> [char]
+
+rational == (num,num)
+
+||a rational is represented as a pair of integers
+||note that rationals are kept in their lowest terms, with positive
+||denominator, and (0,1) is the unique representation of zero
+
+ratio p q = error "illegal ratio", if  q=0\/~integer p\/~integer q
+          = ratio (-p) (-q), if q<0
+          = (0,1), if  p=0
+          = (p div h,q div h), otherwise
+            where
+            h = hcf (abs p) q
+            hcf a b = hcf b a, if  a>b
+	            = b, if a=0
+	            = hcf (b mod a) a, otherwise
+
+mkrat n = ratio n 1, if integer n
+	= error ("mkrat "++shownum n), otherwise
+
+(a,b) $rplus (c,d) = ratio (a*d+c*b) (b*d)
+(a,b) $rminus (c,d) = ratio (a*d-c*b) (b*d)
+(a,b) $rtimes (c,d) = ratio (a*c) (b*d)
+(a,b) $rdiv (c,d) = ratio (a*d) (b*c)
+
+rpow 0 x = (1,1)
+rpow n x = thing, if n mod 2 = 0
+         = x $rtimes thing, otherwise
+           where
+	   thing = rpow (n div 2) (x $rtimes x)
+
+numerator = fst
+denominator = snd
+rparts = id
+
+showrational (a,b) = "(ratio "++shownum1 a++" "++shownum1 b++")"
+shownum1 n = "("++shownum n++")", if n<0
+           = shownum n, otherwise
+
+||Attempts to print a rational will automatically pick up  the  function
+||called showrational - see manual section on abstract data types.  Note
+||that we have chosen to print rationals in such a way that Miranda  can
+||read them back in again at the same type.