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This library allows you to use Rational numbers in Elixir, to enable exact calculations with all numbers big and small.

Ratio follows the Numeric behaviour from Numbers, and can therefore be used in combination with any data type that uses Numbers (such as Tensor and ComplexNum).


Using Ratio

Ratio defines arithmetic and comparison operations to work with rational numbers.

Usually, you probably want to add the line import Ratio, only: [<|>: 2] to your code.


Shorthand operator

Rational numbers can be written using the operator <|> (as in: 1 <|> 2), which is also how Ratio structs are pretty-printed when inspecting. a <|> b is a shorthand for, b).


Basic functionality

Rational numbers can be manipulated using the functions in the Ratio module.

iex> Ratio.mult( 1 <|> 3, 1 <|> 2)
1 <|> 6
iex> Ratio.div(2 <|> 3, 8 <|> 5)
5 <|> 12
iex> Ratio.pow(, 4)
16 <|> 1

The ratio module also contains:

  • a guard-safe is_rational/1 check.
  • a compare/2 function for use with e.g. Enum.sort.
  • to_float to (lossly) convert a rational into a float.


Inline Math Operators and Casting

Ratio interopts with the Numbers library: If you want to overload Elixir's builtin math operators, you can add use Numbers, overload_operators: true to your module.

This also allows you to pass in a rational number as one argument and an integer, float or Decimal (if you have installed the Decimal library), which are then cast to rational numbers whenever necessary.



The package can be installed from hex, by adding :ratio to your list of dependencies in mix.exs:

    def deps do
        {:ratio, "~> 3.0"}



  • 3.0.2 -
    • Fixes: A bug with <|> when the numerator was a rational and the denuminator an integer. (c.f. #104) Thank you, @varsill!
  • 3.0.1 -
    • Fixes:
      • Problem where Ratio.ceil/1 would be off-by-one (c.f. #89). Thank you, @Hajto!
      • Problem where Ratio.pow/2 would return an integer rather than a new Ratio.(c.f. #100). Thank you, @speeddragon!
  • 3.0.0 -
    • All operators except <|> are removed from Ratio. Instead, the operators defined by Numbers (which Ratio depends on) can be used, by adding use Numbers, overload_operators: true to your modules. (c.f. #34)
    • All math-based functions expect and return Ratio structs (rather than also working on integers and returning integers sometimes if the output turned out to be a whole number). (c.f. #43) This makes the code more efficient and more clear for users.
      • Ratio structs representing whole numbers are no longer implicitly converted 'back' to integers, as this behaviour was confusing. (c.f. #28)
      • If conversion to/from other number-like types is really desired, use the automatic conversions provided by, <|> or (a bit slower but more general) the math functions exposed by Numbers. Ratio ships with implementations of Coerce.defcoercion for Integer -> Ratio, Float -> Ratio and Decimal -> Ratio.
    • is_rational?/1 is replaced with the guard-safe is_rational/1 (only exported on Erlang versions where :erlang.map_get/2 is available, i.e. >= OTP 21.0.) (c.f. #37)
    • Float.ratio/1 is now used to convert floats into Ratio structs, rather than maintaining a hand-written version of this logic. (c.f #46) Thank you, @marcinwasowicz !
    • A lot of property-based tests have been added to get some level of confidence of the correctness of the library's operations.
  • 2.4.2 Uses extra_applications in mix.exs to silence warnings in Elixir 1.11 and onwards.
  • 2.4.1 Fixes a bug in the decimal conversion implementation where certain decimals were not converted properly. Thank you, @iterateNZ!
  • 2.4.0 Adds optional support for automatic conversion from Decimals. Thank you, @kipcole !
  • 2.3.1 Removes spurious printing statement in Rational.FloatConversion that would output a line of text at compile-time. Fixes support for Numbers v5+ which was broken.
  • 2.3.0 Adds trunc and to_floor_error functions.
  • 2.1.1 Fixes implementation of floor and ceil which was counter-intuitive for negative numbers (it now correctly rounds towards negative infinity).
    • Drops support for Elixir versions older than 1.4, because of use of Integer.floor_div.
    • First version to support new Erlang versions (20 and onward) that have native floor and ceil functions.
  • 2.1.0 Adds optional overloaded comparison operators.
  • 2.0.0 Breaking change: Brought in line with Elixir's comparison function guideline, to return :lt | :eq | :gt. (This used to be -1 | 0 | 1).

  • 1.2.9 Improved documentation. (Thanks, @morontt!)
  • 1.2.8 Adding :numbers to the applications: list, to ensure that no warnings are thrown when building releases on Elixir < 1.4.0.
  • 1.2.6, 1.2.7 Improving documentation.
  • 1.2.5 added ceil/1 and floor/1.
  • 1.2.4 Fixes Elixir 1.4 warnings in the mix.exs file.
  • 1.2.3 Upgraded version of the Numbers dependency to 2.0.
  • 1.2.2 Added default argument to, to follow the Numeric behaviour fully, and added Ratio.minus/1 as alias for Ratio.negate/1 for the same reason.
  • 1.2.0 Changed name of Ratio.mul/2 to Ratio.mult/2, to avoid ambiguety, and to allow incorporation with Numbers. Deprecation Warning was added to using Ratio.mul/2.
  • 1.1.1 Negative floats are now converted correctly.
  • 1.1.0 Elixir 1.3 compliance (Statefree if/else/catch clauses, etc.)
  • 1.0.0 Proper __using__ macro, with more readable option names. Stable release.
  • 0.6.0 First public release
  • 0.0.1 First features


Difference with the 'rational' library

Observant readers might notice that there also is a 'rational' library in The design idea between that library vs. this one is a bit different: Ratio hides the internal data representation as much as possible, and numbers are therefore created using Rational.<|>/2 or This has as mayor advantage that the internal representation is always correct and simplified.

The Ratio library also (optionally) overrides the built-in math operations +, -, *, /, div, abs so they work with combinations of integers, floats and rationals.

Finally, Ratio follows the Numeric behaviour, which means that it can be used with any data types that follow Numbers.