/Nim

# Module rationals

This module implements rational numbers, consisting of a numerator num and a denominator den, both of type int. The denominator can not be 0.

math, hashes

## Types

```Rational[T] = object
num*, den*: T```
a rational number, consisting of a numerator and denominator

## Procs

`proc initRational[T: SomeInteger](num, den: T): Rational[T]`
Create a new rational number.
`proc `//`[T](num, den: T): Rational[T]`
A friendlier version of initRational. Example usage:
`var x = 1//3 + 1//5`
`proc `\$`[T](x: Rational[T]): string`
Turn a rational number into a string.
`proc toRational[T: SomeInteger](x: T): Rational[T]`
Convert some integer x to a rational number.
`proc toRational(x: float; n: int = high(int) shr 32): Rational[int] {...}{.raises: [], tags: [].}`

Calculates the best rational numerator and denominator that approximates to x, where the denominator is smaller than n (default is the largest possible int to give maximum resolution).

The algorithm is based on the theory of continued fractions.

```import math, rationals
for i in 1..10:
let t = (10 ^ (i+3)).int
let x = toRational(PI, t)
let newPI = x.num / x.den
echo x, " ", newPI, " error: ", PI - newPI, "  ", t```
`proc toFloat[T](x: Rational[T]): float`
Convert a rational number x to a float.
`proc toInt[T](x: Rational[T]): int`
Convert a rational number x to an int. Conversion rounds towards 0 if x does not contain an integer value.
`proc reduce[T: SomeInteger](x: var Rational[T])`
Reduce rational x.
`proc `+`[T](x, y: Rational[T]): Rational[T]`
Add two rational numbers.
`proc `+`[T](x: Rational[T]; y: T): Rational[T]`
Add rational x to int y.
`proc `+`[T](x: T; y: Rational[T]): Rational[T]`
Add int x to rational y.
`proc `+=`[T](x: var Rational[T]; y: Rational[T])`
Add rational y to rational x.
`proc `+=`[T](x: var Rational[T]; y: T)`
Add int y to rational x.
`proc `-`[T](x: Rational[T]): Rational[T]`
Unary minus for rational numbers.
`proc `-`[T](x, y: Rational[T]): Rational[T]`
Subtract two rational numbers.
`proc `-`[T](x: Rational[T]; y: T): Rational[T]`
Subtract int y from rational x.
`proc `-`[T](x: T; y: Rational[T]): Rational[T]`
Subtract rational y from int x.
`proc `-=`[T](x: var Rational[T]; y: Rational[T])`
Subtract rational y from rational x.
`proc `-=`[T](x: var Rational[T]; y: T)`
Subtract int y from rational x.
`proc `*`[T](x, y: Rational[T]): Rational[T]`
Multiply two rational numbers.
`proc `*`[T](x: Rational[T]; y: T): Rational[T]`
Multiply rational x with int y.
`proc `*`[T](x: T; y: Rational[T]): Rational[T]`
Multiply int x with rational y.
`proc `*=`[T](x: var Rational[T]; y: Rational[T])`
Multiply rationals y to x.
`proc `*=`[T](x: var Rational[T]; y: T)`
Multiply int y to rational x.
`proc reciprocal[T](x: Rational[T]): Rational[T]`
Calculate the reciprocal of x. (1/x)
`proc `/`[T](x, y: Rational[T]): Rational[T]`
Divide rationals x by y.
`proc `/`[T](x: Rational[T]; y: T): Rational[T]`
Divide rational x by int y.
`proc `/`[T](x: T; y: Rational[T]): Rational[T]`
Divide int x by Rational y.
`proc `/=`[T](x: var Rational[T]; y: Rational[T])`
Divide rationals x by y in place.
`proc `/=`[T](x: var Rational[T]; y: T)`
Divide rational x by int y in place.
`proc cmp(x, y: Rational): int {...}{.procvar.}`
Compares two rationals.
`proc `<`(x, y: Rational): bool`
`proc `<=`(x, y: Rational): bool`
`proc `==`(x, y: Rational): bool`
`proc abs[T](x: Rational[T]): Rational[T]`
`proc `div`[T: SomeInteger](x, y: Rational[T]): T`
Computes the rational truncated division.
`proc `mod`[T: SomeInteger](x, y: Rational[T]): Rational[T]`
Computes the rational modulo by truncated division (remainder). This is same as `x - (x div y) * y`.
`proc floorDiv[T: SomeInteger](x, y: Rational[T]): T`

Computes the rational floor division.

Floor division is conceptually defined as `floor(x / y)`. This is different from the `div` operator, which is defined as `trunc(x / y)`. That is, `div` rounds towards `0` and `floorDiv` rounds down.

`proc floorMod[T: SomeInteger](x, y: Rational[T]): Rational[T]`

Computes the rational modulo by floor division (modulo).

This is same as `x - floorDiv(x, y) * y`. This proc behaves the same as the `%` operator in python.

`proc hash[T](x: Rational[T]): Hash`
Computes hash for rational x

© 2006–2018 Andreas Rumpf
Licensed under the MIT License.
https://nim-lang.org/docs/rationals.html