diff --git a/CHANGELOG.markdown b/CHANGELOG.markdown
--- a/CHANGELOG.markdown
+++ b/CHANGELOG.markdown
@@ -1,3 +1,7 @@
+0.6
+---
+* Added `Numeric.Interval.Exception`. For consistency, we tend to throw exceptions now instead of rely on `NaN` when working with empty intervals.
+
 0.5.1.1
 -------
 * Misc `doctest` fixes.
diff --git a/intervals.cabal b/intervals.cabal
--- a/intervals.cabal
+++ b/intervals.cabal
@@ -1,5 +1,5 @@
 name:              intervals
-version:           0.5.1.1
+version:           0.6
 synopsis:          Interval Arithmetic
 description:
   A 'Numeric.Interval.Interval' is a closed, convex set of floating point values.
@@ -40,6 +40,7 @@
 
   exposed-modules:
     Numeric.Interval
+    Numeric.Interval.Exception
     Numeric.Interval.Internal
     Numeric.Interval.Kaucher
     Numeric.Interval.NonEmpty
diff --git a/src/Numeric/Interval/Exception.hs b/src/Numeric/Interval/Exception.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Interval/Exception.hs
@@ -0,0 +1,24 @@
+{-# LANGUAGE DeriveDataTypeable #-}
+module Numeric.Interval.Exception
+  ( EmptyInterval(..)
+  , AmbiguousComparison(..)
+  ) where
+
+import Control.Exception
+import Data.Data
+
+data EmptyInterval = EmptyInterval
+  deriving (Eq,Ord,Typeable,Data)
+
+instance Show EmptyInterval where
+  show EmptyInterval = "empty interval"
+
+instance Exception EmptyInterval
+
+data AmbiguousComparison = AmbiguousComparison
+  deriving (Eq,Ord,Typeable,Data)
+
+instance Show AmbiguousComparison where
+  show AmbiguousComparison = "ambiguous comparison"
+
+instance Exception AmbiguousComparison
diff --git a/src/Numeric/Interval/Internal.hs b/src/Numeric/Interval/Internal.hs
--- a/src/Numeric/Interval/Internal.hs
+++ b/src/Numeric/Interval/Internal.hs
@@ -46,6 +46,7 @@
   , ifloat
   ) where
 
+import Control.Exception as Exception
 import Data.Data
 import Data.Foldable hiding (minimum, maximum, elem, notElem)
 import Data.Function (on)
@@ -53,6 +54,7 @@
 #if defined(__GLASGOW_HASKELL) && __GLASGOW_HASKELL__ >= 704
 import GHC.Generics
 #endif
+import Numeric.Interval.Exception
 import Prelude hiding (null, elem, notElem)
 
 -- $setup
@@ -89,9 +91,6 @@
   | otherwise = Nothing
 {-# INLINE interval #-}
 
-nan :: Fractional a => a
-nan = 0/0
-
 fmod :: RealFrac a => a -> a -> a
 fmod a b = a - q*b where
   q = realToFrac (truncate $ a / b :: Integer)
@@ -119,7 +118,7 @@
 empty = Empty
 {-# INLINE empty #-}
 
--- | negation handles NaN properly
+-- | Check if an interval is empty
 --
 -- >>> null (1 ... 5)
 -- False
@@ -142,22 +141,28 @@
 singleton a = I a a
 {-# INLINE singleton #-}
 
--- | The infinumum (lower bound) of an interval
+-- | The infimum (lower bound) of an interval
 --
 -- >>> inf (1.0 ... 20.0)
 -- 1.0
-inf :: Fractional a => Interval a -> a
+--
+-- >>> inf empty
+-- *** Exception: empty interval
+inf :: Interval a -> a
 inf (I a _) = a
-inf Empty = nan
+inf Empty = Exception.throw EmptyInterval
 {-# INLINE inf #-}
 
 -- | The supremum (upper bound) of an interval
 --
 -- >>> sup (1.0 ... 20.0)
 -- 20.0
-sup :: Fractional a => Interval a -> a
+--
+-- >>> sup empty
+-- *** Exception: empty interval
+sup :: Interval a -> a
 sup (I _ b) = b
-sup Empty = nan
+sup Empty = Exception.throw EmptyInterval
 {-# INLINE sup #-}
 
 -- | Is the interval a singleton point?
@@ -212,11 +217,12 @@
 -- >>> magnitude (singleton 5)
 -- 5
 --
+-- throws 'EmptyInterval' if the interval is empty.
+--
 -- >>> magnitude empty
--- 0
+-- *** Exception: empty interval
 magnitude :: (Num a, Ord a) => Interval a -> a
-magnitude (I a b) = on max abs a b
-magnitude Empty = 0
+magnitude = sup . abs
 {-# INLINE magnitude #-}
 
 -- | \"mignitude\"
@@ -225,16 +231,17 @@
 -- 1
 --
 -- >>> mignitude (-20 ... 10)
--- 10
+-- 0
 --
 -- >>> mignitude (singleton 5)
 -- 5
 --
+-- throws 'EmptyInterval' if the interval is empty.
+--
 -- >>> mignitude empty
--- 0
+-- *** Exception: empty interval
 mignitude :: (Num a, Ord a) => Interval a -> a
-mignitude (I a b) = on min abs a b
-mignitude Empty = 0
+mignitude = inf . abs
 {-# INLINE mignitude #-}
 
 instance (Num a, Ord a) => Num (Interval a) where
@@ -294,10 +301,10 @@
 -- 5.0
 --
 -- >>> midpoint empty
--- NaN
+-- *** Exception: empty interval
 midpoint :: Fractional a => Interval a -> a
 midpoint (I a b) = a + (b - a) / 2
-midpoint Empty = nan
+midpoint Empty = Exception.throw EmptyInterval
 {-# INLINE midpoint #-}
 
 -- | Determine if a point is in the interval.
@@ -340,7 +347,7 @@
 
 -- | 'realToFrac' will use the midpoint
 instance Real a => Real (Interval a) where
-  toRational Empty = nan
+  toRational Empty = Exception.throw EmptyInterval
   toRational (I ra rb) = a + (b - a) / 2 where
     a = toRational ra
     b = toRational rb
@@ -354,7 +361,7 @@
     | bx < ay = LT
     | ax > by = GT
     | bx == ay && ax == by = EQ
-    | otherwise = error "Numeric.Interval.compare: ambiguous comparison"
+    | otherwise = Exception.throw AmbiguousComparison
   {-# INLINE compare #-}
 
   max (I a b) (I a' b') = max a a' ... max b b'
@@ -409,7 +416,7 @@
   _ / Empty = Empty
   x / y@(I a b)
     | 0 `notElem` y = divNonZero x y
-    | iz && sz  = empty -- division by 0
+    | iz && sz  = Exception.throw DivideByZero
     | iz        = divPositive x a
     |       sz  = divNegative x b
     | otherwise = divZero x
diff --git a/src/Numeric/Interval/Kaucher.hs b/src/Numeric/Interval/Kaucher.hs
--- a/src/Numeric/Interval/Kaucher.hs
+++ b/src/Numeric/Interval/Kaucher.hs
@@ -47,6 +47,7 @@
   ) where
 
 import Control.Applicative hiding (empty)
+import Control.Exception as Exception
 import Data.Data
 import Data.Distributive
 import Data.Foldable hiding (minimum, maximum, elem, notElem)
@@ -56,6 +57,7 @@
 #if defined(__GLASGOW_HASKELL) && __GLASGOW_HASKELL__ >= 704
 import GHC.Generics
 #endif
+import Numeric.Interval.Exception
 import Prelude hiding (null, elem, notElem)
 
 -- $setup
@@ -341,7 +343,7 @@
 -- | 'realToFrac' will use the midpoint
 instance Real a => Real (Interval a) where
   toRational x
-    | null x   = nan
+    | null x    = Exception.throw EmptyInterval
     | otherwise = a + (b - a) / 2
     where
       a = toRational (inf x)
@@ -353,7 +355,7 @@
     | sup x < inf y = LT
     | inf x > sup y = GT
     | sup x == inf y && inf x == sup y = EQ
-    | otherwise = error "Numeric.Interval.compare: ambiguous comparison"
+    | otherwise = Exception.throw AmbiguousComparison
   {-# INLINE compare #-}
 
   max (I a b) (I a' b') = max a a' ... max b b'
diff --git a/src/Numeric/Interval/NonEmpty/Internal.hs b/src/Numeric/Interval/NonEmpty/Internal.hs
--- a/src/Numeric/Interval/NonEmpty/Internal.hs
+++ b/src/Numeric/Interval/NonEmpty/Internal.hs
@@ -44,6 +44,7 @@
   , ifloat
   ) where
 
+import Control.Exception as Exception
 import Data.Data
 import Data.Foldable hiding (minimum, maximum, elem, notElem)
 import Data.Function (on)
@@ -51,6 +52,7 @@
 #if defined(__GLASGOW_HASKELL) && __GLASGOW_HASKELL__ >= 704
 import GHC.Generics
 #endif
+import Numeric.Interval.Exception
 import Prelude hiding (null, elem, notElem)
 
 -- $setup
@@ -177,7 +179,7 @@
 -- >>> magnitude (singleton 5)
 -- 5
 magnitude :: (Num a, Ord a) => Interval a -> a
-magnitude (I a b) = on max abs a b
+magnitude = sup . abs
 {-# INLINE magnitude #-}
 
 -- | \"mignitude\"
@@ -186,12 +188,12 @@
 -- 1
 --
 -- >>> mignitude (-20 ... 10)
--- 10
+-- 0
 --
 -- >>> mignitude (singleton 5)
 -- 5
 mignitude :: (Num a, Ord a) => Interval a -> a
-mignitude (I a b) = on min abs a b
+mignitude = inf . abs
 {-# INLINE mignitude #-}
 
 instance (Num a, Ord a) => Num (Interval a) where
@@ -285,7 +287,7 @@
     | bx < ay = LT
     | ax > by = GT
     | bx == ay && ax == by = EQ
-    | otherwise = error "Numeric.Interval.compare: ambiguous comparison"
+    | otherwise = Exception.throw AmbiguousComparison
   {-# INLINE compare #-}
 
   max (I a b) (I a' b') = max a a' ... max b b'
@@ -331,7 +333,7 @@
   -- TODO: check isNegativeZero properly
   x / y@(I a b)
     | 0 `notElem` y = divNonZero x y
-    | iz && sz  = error "division by zero"
+    | iz && sz  = Exception.throw DivideByZero
     | iz        = divPositive x a
     |       sz  = divNegative x b
     | otherwise = divZero x