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Rational number library for Elixir.

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# Ratio

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 `Ratio.new(a, 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(Ratio.new(2), 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.

## Installation

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

``````    def deps do
[
{:ratio, "~> 3.0"}
]
end
``````

## Changelog

• 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 `Ratio.new`, `<|>` 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 `Ratio.compare/2` 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 `Ratio.new/2`, 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 Hex.pm. 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 `Ratio.new/2`. 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.

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Rational number library for Elixir.

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