# 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   Source Edit

# Procs

`proc initRational[T: SomeInteger](num, den: T): Rational[T]`
Create a new rational number.   Source Edit
`proc `//`[T](num, den: T): Rational[T]`
A friendlier version of initRational. Example usage:
`var x = 1//3 + 1//5`
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`proc `\$`[T](x: Rational[T]): string`
Turn a rational number into a string.   Source Edit
`proc toRational[T: SomeInteger](x: T): Rational[T]`
Convert some integer x to a rational number.   Source Edit
```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```
Source Edit
`proc toFloat[T](x: Rational[T]): float`
Convert a rational number x to a float.   Source Edit
`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.   Source Edit
`proc reduce[T: SomeInteger](x: var Rational[T])`
Reduce rational x.   Source Edit
`proc `+`[T](x, y: Rational[T]): Rational[T]`
Add two rational numbers.   Source Edit
`proc `+`[T](x: Rational[T]; y: T): Rational[T]`
Add rational x to int y.   Source Edit
`proc `+`[T](x: T; y: Rational[T]): Rational[T]`
Add int x to rational y.   Source Edit
`proc `+=`[T](x: var Rational[T]; y: Rational[T])`
Add rational y to rational x.   Source Edit
`proc `+=`[T](x: var Rational[T]; y: T)`
Add int y to rational x.   Source Edit
`proc `-`[T](x: Rational[T]): Rational[T]`
Unary minus for rational numbers.   Source Edit
`proc `-`[T](x, y: Rational[T]): Rational[T]`
Subtract two rational numbers.   Source Edit
`proc `-`[T](x: Rational[T]; y: T): Rational[T]`
Subtract int y from rational x.   Source Edit
`proc `-`[T](x: T; y: Rational[T]): Rational[T]`
Subtract rational y from int x.   Source Edit
`proc `-=`[T](x: var Rational[T]; y: Rational[T])`
Subtract rational y from rational x.   Source Edit
`proc `-=`[T](x: var Rational[T]; y: T)`
Subtract int y from rational x.   Source Edit
`proc `*`[T](x, y: Rational[T]): Rational[T]`
Multiply two rational numbers.   Source Edit
`proc `*`[T](x: Rational[T]; y: T): Rational[T]`
Multiply rational x with int y.   Source Edit
`proc `*`[T](x: T; y: Rational[T]): Rational[T]`
Multiply int x with rational y.   Source Edit
`proc `*=`[T](x: var Rational[T]; y: Rational[T])`
Multiply rationals y to x.   Source Edit
`proc `*=`[T](x: var Rational[T]; y: T)`
Multiply int y to rational x.   Source Edit
`proc reciprocal[T](x: Rational[T]): Rational[T]`
Calculate the reciprocal of x. (1/x)   Source Edit
`proc `/`[T](x, y: Rational[T]): Rational[T]`
Divide rationals x by y.   Source Edit
`proc `/`[T](x: Rational[T]; y: T): Rational[T]`
Divide rational x by int y.   Source Edit
`proc `/`[T](x: T; y: Rational[T]): Rational[T]`
Divide int x by Rational y.   Source Edit
`proc `/=`[T](x: var Rational[T]; y: Rational[T])`
Divide rationals x by y in place.   Source Edit
`proc `/=`[T](x: var Rational[T]; y: T)`
Divide rational x by int y in place.   Source Edit
`proc cmp(x, y: Rational): int`
Compares two rationals.   Source Edit
`proc `<`(x, y: Rational): bool`
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`proc `<=`(x, y: Rational): bool`
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`proc `==`(x, y: Rational): bool`
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`proc abs[T](x: Rational[T]): Rational[T]`
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`proc `div`[T: SomeInteger](x, y: Rational[T]): T`
Computes the rational truncated division.   Source Edit
`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.   Source Edit
`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.

Source Edit
`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.

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