std/rationals

This module implements rational numbers, consisting of a numerator and a denominator. The denominator can not be 0.

Example:

import std/rationals
let
  r1 = 1 // 2
  r2 = -3 // 4

doAssert r1 + r2 == -1 // 4
doAssert r1 - r2 ==  5 // 4
doAssert r1 * r2 == -3 // 8
doAssert r1 / r2 == -2 // 3

Imports

math, hashes

Types

Rational[T] = object num*, den*: T: A rational number, consisting of a numerator num and a denominator den. Source Edit

Procs

func `$`[T](x: Rational[T]): string: Turns a rational number into a string.
Example:

doAssert $(1 // 2) == "1/2"
Source Edit
func `*`[T](x, y: Rational[T]): Rational[T]: Multiplies two rational numbers. Source Edit
func `*`[T](x: Rational[T]; y: T): Rational[T]: Multiplies the rational x with the int y. Source Edit
func `*`[T](x: T; y: Rational[T]): Rational[T]: Multiplies the int x with the rational y. Source Edit
func `*=`[T](x: var Rational[T]; y: Rational[T]): Multiplies the rational x by y in-place. Source Edit
func `*=`[T](x: var Rational[T]; y: T): Multiplies the rational x by the int y in-place. Source Edit
func `+`[T](x, y: Rational[T]): Rational[T]: Adds two rational numbers. Source Edit
func `+`[T](x: Rational[T]; y: T): Rational[T]: Adds the rational x to the int y. Source Edit
func `+`[T](x: T; y: Rational[T]): Rational[T]: Adds the int x to the rational y. Source Edit
func `+=`[T](x: var Rational[T]; y: Rational[T]): Adds the rational y to the rational x in-place. Source Edit
func `+=`[T](x: var Rational[T]; y: T): Adds the int y to the rational x in-place. Source Edit
func `-`[T](x, y: Rational[T]): Rational[T]: Subtracts two rational numbers. Source Edit
func `-`[T](x: Rational[T]): Rational[T]: Unary minus for rational numbers. Source Edit
func `-`[T](x: Rational[T]; y: T): Rational[T]: Subtracts the int y from the rational x. Source Edit
func `-`[T](x: T; y: Rational[T]): Rational[T]: Subtracts the rational y from the int x. Source Edit
func `-=`[T](x: var Rational[T]; y: Rational[T]): Subtracts the rational y from the rational x in-place. Source Edit
func `-=`[T](x: var Rational[T]; y: T): Subtracts the int y from the rational x in-place. Source Edit
func `/`[T](x, y: Rational[T]): Rational[T]: Divides the rational x by the rational y. Source Edit
func `/`[T](x: Rational[T]; y: T): Rational[T]: Divides the rational x by the int y. Source Edit
func `/`[T](x: T; y: Rational[T]): Rational[T]: Divides the int x by the rational y. Source Edit
func `//`[T](num, den: T): Rational[T]: A friendlier version of initRational.
Example:

let x = 1 // 3 + 1 // 5 doAssert x == 8 // 15
Source Edit
func `/=`[T](x: var Rational[T]; y: Rational[T]): Divides the rational x by the rational y in-place. Source Edit
func `/=`[T](x: var Rational[T]; y: T): Divides the rational x by the int y in-place. Source Edit
func `<`(x, y: Rational): bool: Returns true if x is less than y. Source Edit
func `<=`(x, y: Rational): bool: Returns tue if x is less than or equal to y. Source Edit
func `==`(x, y: Rational): bool: Compares two rationals for equality. Source Edit
func `^`[T: SomeInteger](x: Rational[T]; y: T): Rational[T]: Computes x to the power of y.

The exponent y must be an integer. Negative exponents are supported but floating point exponents are not.

Example:

doAssert (-3 // 5) ^ 0 == (1 // 1) doAssert (-3 // 5) ^ 1 == (-3 // 5) doAssert (-3 // 5) ^ 2 == (9 // 25) doAssert (-3 // 5) ^ -2 == (25 // 9)
Source Edit
func abs[T](x: Rational[T]): Rational[T]: Returns the absolute value of x.
Example:

doAssert abs(1 // 2) == 1 // 2 doAssert abs(-1 // 2) == 1 // 2
Source Edit
func cmp(x, y: Rational): int: Compares two rationals. Returns

a value less than zero, if x < y

a value greater than zero, if x > y

zero, if x == y

Source Edit
func `div`[T: SomeInteger](x, y: Rational[T]): T: Computes the rational truncated division. Source Edit
func 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.
Source Edit
func 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 func behaves the same as the % operator in Python.
Source Edit
func hash[T](x: Rational[T]): Hash: Computes the hash for the rational x. Source Edit
func initRational[T: SomeInteger](num, den: T): Rational[T]: Creates a new rational number with numerator num and denominator den. den must not be 0.

Note: den != 0 is not checked when assertions are turned off.
Source Edit
func `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. Source Edit
func reciprocal[T](x: Rational[T]): Rational[T]: Calculates the reciprocal of x (1/x). If x is 0, raises DivByZeroDefect. Source Edit
func reduce[T: SomeInteger](x: var Rational[T]): Reduces the rational number x, so that the numerator and denominator have no common divisors other than 1 (and -1). If x is 0, raises DivByZeroDefect.

Note: This is called automatically by the various operations on rationals.

Example:

var r = Rational[int](num: 2, den: 4) # 1/2 reduce(r) doAssert r.num == 1 doAssert r.den == 2
Source Edit
func toFloat[T](x: Rational[T]): float: Converts a rational number x to a float. Source Edit
func toInt[T](x: Rational[T]): int: Converts a rational number x to an int. Conversion rounds towards 0 if x does not contain an integer value. Source Edit
func toRational(x: float; n: int = high(int) shr 32): Rational[int] {. ...raises: [], tags: [], forbids: [].}: Calculates the best rational approximation of x, where the denominator is smaller than n (default is the largest possible int for maximal resolution).

The algorithm is based on the theory of continued fractions.

Example:

let x = 1.2 doAssert x.toRational.toFloat == x
Source Edit
func toRational[T: SomeInteger](x: T): Rational[T]: Converts some integer x to a rational number.
Example:

doAssert toRational(42) == 42 // 1
Source Edit