62 lines
3.1 KiB
Markdown
62 lines
3.1 KiB
Markdown
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# Rational Numbers
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Welcome to Rational Numbers on Exercism's Elixir Track.
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If you need help running the tests or submitting your code, check out `HELP.md`.
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## Instructions
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A rational number is defined as the quotient of two integers `a` and `b`, called the numerator and denominator, respectively, where `b != 0`.
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~~~~exercism/note
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Note that mathematically, the denominator can't be zero.
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However in many implementations of rational numbers, you will find that the denominator is allowed to be zero with behaviour similar to positive or negative infinity in floating point numbers.
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In those cases, the denominator and numerator generally still can't both be zero at once.
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~~~~
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The absolute value `|r|` of the rational number `r = a/b` is equal to `|a|/|b|`.
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The sum of two rational numbers `r₁ = a₁/b₁` and `r₂ = a₂/b₂` is `r₁ + r₂ = a₁/b₁ + a₂/b₂ = (a₁ * b₂ + a₂ * b₁) / (b₁ * b₂)`.
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The difference of two rational numbers `r₁ = a₁/b₁` and `r₂ = a₂/b₂` is `r₁ - r₂ = a₁/b₁ - a₂/b₂ = (a₁ * b₂ - a₂ * b₁) / (b₁ * b₂)`.
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The product (multiplication) of two rational numbers `r₁ = a₁/b₁` and `r₂ = a₂/b₂` is `r₁ * r₂ = (a₁ * a₂) / (b₁ * b₂)`.
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Dividing a rational number `r₁ = a₁/b₁` by another `r₂ = a₂/b₂` is `r₁ / r₂ = (a₁ * b₂) / (a₂ * b₁)` if `a₂` is not zero.
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Exponentiation of a rational number `r = a/b` to a non-negative integer power `n` is `r^n = (a^n)/(b^n)`.
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Exponentiation of a rational number `r = a/b` to a negative integer power `n` is `r^n = (b^m)/(a^m)`, where `m = |n|`.
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Exponentiation of a rational number `r = a/b` to a real (floating-point) number `x` is the quotient `(a^x)/(b^x)`, which is a real number.
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Exponentiation of a real number `x` to a rational number `r = a/b` is `x^(a/b) = root(x^a, b)`, where `root(p, q)` is the `q`th root of `p`.
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Implement the following operations:
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- addition, subtraction, multiplication and division of two rational numbers,
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- absolute value, exponentiation of a given rational number to an integer power, exponentiation of a given rational number to a real (floating-point) power, exponentiation of a real number to a rational number.
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Your implementation of rational numbers should always be reduced to lowest terms.
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For example, `4/4` should reduce to `1/1`, `30/60` should reduce to `1/2`, `12/8` should reduce to `3/2`, etc.
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To reduce a rational number `r = a/b`, divide `a` and `b` by the greatest common divisor (gcd) of `a` and `b`.
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So, for example, `gcd(12, 8) = 4`, so `r = 12/8` can be reduced to `(12/4)/(8/4) = 3/2`.
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The reduced form of a rational number should be in "standard form" (the denominator should always be a positive integer).
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If a denominator with a negative integer is present, multiply both numerator and denominator by `-1` to ensure standard form is reached.
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For example, `3/-4` should be reduced to `-3/4`
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Assume that the programming language you are using does not have an implementation of rational numbers.
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## Source
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### Created by
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- @jiegillet
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### Contributed to by
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- @angelikatyborska
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- @kszambelanczyk
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### Based on
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Wikipedia - https://en.wikipedia.org/wiki/Rational_number
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