"All Haskell Number Types"

In some languages there is a notion of a "numeric tower." The reasoning behind this is: there is an obvious way to inject less precise numbers into a more precise number type. While this makes sense mathematically, this doesn't even strictly work for the humble Int and Double types[1]!

Haskell has more of a "numeric salad bowl" of number types. Each kind of number has different strengths and drawbacks. Which one you want to use depends on factors like

• convenience: The standard library uses Int in many places.
• performance: Is numeric code your program's bottleneck? Double and Int compute faster due to hardware support. Also they can take less memory and be unboxed.
• faithfulness: Double can't represent an arbitrary rational number. But Rational can.
• security: Allowing users to specify arbitrary Integers or Rationals can be a denial of service vulnerability.
• safety: Natural will crash on some subtractions while Integer won't.

This is a list of the most common number types and what to import to get them.

Type	Import	Ex.	Notes
Int		3	Signed machine integer (at least 30 bits wide)
Integer		3	Arbitrary integer
Float		3.125	Single precision float
Double		3.125	Double precision float
Rational	Data.Ratio	5 % 16	Rational
Complex t	Data.Complex	3.0 :+ 2.0	Complex number
Word8 Word16 Word32 Word64	Data.Word	3	Fixed-size unsigned int
Word	Data.Word	3	Unsigned machine int
Int8 Int16 Int32 Int64	Data.Int	3	Fixed-size signed int
Deci Milli Micro Nano Pico Fixed n	Data.Fixed	3.00	Decimal fixed-point
Scientific	Data.Scientific	9.109e-31	Decimal scientific notation (scientific package)
Natural	Numeric.Natural	3	Arbitrary non-negative
CReal	Data.Number.CReal	sqrt 2	Computable real (numbers package)
CReal n	Data.CReal	sqrt 2 :: CReal 100	Computable real (exact-real package)

Converting between number types

There are a lot of number types, but generally there's only a few functions necessary to do conversions.

• fromIntegral = fromInteger . toInteger converts any integral type to any other number type.
• realToFrac = fromRational . toRational converts between fractional types. This isn't strictly safe if the source type can't be mapped to the rationals.
• floor, ceiling, round, or truncate will convert any fractional number to any integral type.

All of the above conversions might lose information depending on the source and target types.

[1]: let n = 2^62+1 :: Int in n == floor (fromIntegral n :: Double)