# Converting numbers

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Conversion between numerical types in Haskell must be done explicitly. This is unlike many traditional languages (such as C or Java) that automatically coerce between numerical types.

## Converting from and between integral types (integer-like types)

`Integral` types contain only whole numbers and not fractions. The most commonly used integral types are:

• `Integer`, which are arbitrary-precision integers, often called "bignum" or "big-integers" in other languages, and
• `Int`, which fixed-width machine-specific integers with a minimum guaranteed range of `−229` to `229 − 1`. In practice, its range can be much larger: on the x86-64 version of Glasgow Haskell Compiler, it can store any signed 64-bit integer.

The workhorse for converting from integral types is `fromIntegral`, which will convert from any `Integral` type into any `Num`eric type (which includes `Int`, `Integer`, `Rational`, and `Double`):

```fromIntegral :: (Num b, Integral a) => a -> b
```

For example, given an `Int` value `n`, one does not simply take its square root by typing `sqrt n`, since `sqrt` can only be applied to `Floating`-point numbers. Instead, one must write `sqrt (fromIntegral n)` to explicitly convert `n` to a floating-point number.

There are special cases for converting from `Integer`s:

```fromInteger :: Num a => Integer -> a
```

as well as for converting to `Integer`s:

```toInteger:: Integral a => a -> Integer
```

## Converting from real and between real-fractional types (rational-like types)

`RealFrac`tional types can contain either whole numbers or fractions. The most commonly used real-fractional types are:

`Real` types include both `Integral` and `RealFrac`tional types. The name "real" indicates that it excludes `Complex` numbers.

The workhorse for converting from real types is `realToFrac`, which will convert from any `Real` type into any `Fractional` type (which includes `Rational` and `Double`):

```realToFrac:: (Real a, Fractional b) => a -> b
```

It can also be used to convert between real-fractional types. (Warning: Avoid using `realToFrac` to convert between floating-point types; see below.)

There are special cases for converting from `Rational`s:

```fromRational :: Fractional a => Rational -> a
```

as well as for converting to `Rational`s:

```toRational :: Real a => a -> Rational
```

## Converting from real-fractional numbers to integral numbers

This is an inherently lossy transformation since integral types cannot express non-whole numbers. Depending on how you wish to convert, you may choose any of the following:

```ceiling  :: (RealFrac a, Integral b) => a -> b
floor    :: (RealFrac a, Integral b) => a -> b
truncate :: (RealFrac a, Integral b) => a -> b
round    :: (RealFrac a, Integral b) => a -> b
```

## Converting between different floating-point precisions

Conversion between `Float` and `Double` can be done using the GHC-specific functions in the GHC.Float module:

```float2Double :: Float -> Double
double2Float :: Double -> Float
```

Avoid using `realToFrac` to convert between floating-point types as the intermediate type `Rational` is unable to represent exceptional values like infinity or NaN. See GHC ticket #3676.

## Automatic conversion

Repeatedly people ask for automatic conversion between numbers. This is usually not a good idea; for more information, refer to the thoughts about a Generic number type.

## Example

Hi, I am trying to write some functions that convert between two coordinate systems. The first coordinate system, which ill call coord1, starts in the upper left at (0, 0) and ends in the lower right at (500, 500). Coordinates in coord1 have type (Int, Int). The second coord system, which I'll call coord2, starts in the lower left at (0.0, 0.0) and ends in the upper right at (1.0, 1.0). Coords in coord2 have type (Float, Float). I was hoping someone could help me figure out how I can rewrite the two functions below so that the type checker will accept them.

``` coord1ToCoord2 :: (Int, Int) -> (Float, Float)
coord1ToCoord2 (x, y) = (x/500, (500-y)/500)

coord2ToCoord1 :: (Float, Float) -> (Int, Int)
coord2ToCoord1 (x, y) = (500/(1/x), 500 - 500/(1/y))
```

examples of what i want. i think i have the logic right :)

``` coord1ToCoord2 (0, 0) -> (0.0, 1.0)
coord1ToCoord2 (250, 250) -> (0.5, 0.5)
coord1ToCoord2 (350, 350) -> (0.7, 0.3)
coord1ToCoord2 (500, 500) -> (1.0, 0.0)

coord2ToCoord1 (0.0, 0.0) -> (0, 500)
coord2ToCoord1 (0.5, 0.5) -> (250, 250)
coord2ToCoord1 (0.7, 0.7) -> (350, 150)
coord2ToCoord1 (1.0, 1.0) -> (500, 0)
```

One of the thing that confused me was that I expected 500 to be an Int, but in fact the literals are automatically converted to a correct Num instance.

The solution here was to use fromIntegral and round :

```coord1ToCoord2 :: (Int, Int) -> (Float, Float)
coord1ToCoord2 (x, y) = (fromIntegral x/500, (500 - fromIntegral y)/500)

coord2ToCoord1 :: (Float, Float) -> (Int, Int)
coord2ToCoord1 (x, y) = (round (500 * x), round (500 - 500 * y))
```