rational_numbers
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{
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"authors": [
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"jiegillet"
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],
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"contributors": [
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"angelikatyborska",
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"kszambelanczyk"
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],
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"files": {
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"solution": [
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"lib/rational_numbers.ex"
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],
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"test": [
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"test/rational_numbers_test.exs"
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],
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"example": [
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".meta/example.ex"
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]
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},
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"blurb": "Implement rational numbers.",
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"source": "Wikipedia",
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"source_url": "https://en.wikipedia.org/wiki/Rational_number"
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}
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{"track":"elixir","exercise":"rational-numbers","id":"7dc127adcbfe4b6082e20167dc621378","url":"https://exercism.org/tracks/elixir/exercises/rational-numbers","handle":"negrienko","is_requester":true,"auto_approve":false}
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# Used by "mix format"
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[
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inputs: ["{mix,.formatter}.exs", "{config,lib,test}/**/*.{ex,exs}"]
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]
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# The directory Mix will write compiled artifacts to.
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/_build/
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# If you run "mix test --cover", coverage assets end up here.
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/cover/
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# The directory Mix downloads your dependencies sources to.
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/deps/
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# Where third-party dependencies like ExDoc output generated docs.
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/doc/
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# Ignore .fetch files in case you like to edit your project deps locally.
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/.fetch
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# If the VM crashes, it generates a dump, let's ignore it too.
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erl_crash.dump
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# Also ignore archive artifacts (built via "mix archive.build").
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*.ez
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# Ignore package tarball (built via "mix hex.build").
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rational_numbers-*.tar
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# Help
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## Running the tests
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From the terminal, change to the base directory of the exercise then execute the tests with:
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```bash
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$ mix test
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```
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This will execute the test file found in the `test` subfolder -- a file ending in `_test.exs`
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Documentation:
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* [`mix test` - Elixir's test execution tool](https://hexdocs.pm/mix/Mix.Tasks.Test.html)
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* [`ExUnit` - Elixir's unit test library](https://hexdocs.pm/ex_unit/ExUnit.html)
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## Pending tests
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In test suites of practice exercises, all but the first test have been tagged to be skipped.
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|
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Once you get a test passing, you can unskip the next one by commenting out the relevant `@tag :pending` with a `#` symbol.
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For example:
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|
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```elixir
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# @tag :pending
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test "shouting" do
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assert Bob.hey("WATCH OUT!") == "Whoa, chill out!"
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end
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```
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If you wish to run all tests at once, you can include all skipped test by using the `--include` flag on the `mix test` command:
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|
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```bash
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$ mix test --include pending
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```
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Or, you can enable all the tests by commenting out the `ExUnit.configure` line in the file `test/test_helper.exs`.
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|
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```elixir
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# ExUnit.configure(exclude: :pending, trace: true)
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```
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## Useful `mix test` options
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* `test/<FILE>.exs:LINENUM` - runs only a single test, the test from `<FILE>.exs` whose definition is on line `LINENUM`
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* `--failed` - runs only tests that failed the last time they ran
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* `--max-failures` - the suite stops evaluating tests when this number of test failures
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is reached
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* `--seed 0` - disables randomization so the tests in a single file will always be ran
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in the same order they were defined in
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## Submitting your solution
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You can submit your solution using the `exercism submit lib/rational_numbers.ex` command.
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This command will upload your solution to the Exercism website and print the solution page's URL.
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It's possible to submit an incomplete solution which allows you to:
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- See how others have completed the exercise
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- Request help from a mentor
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## Need to get help?
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If you'd like help solving the exercise, check the following pages:
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- The [Elixir track's documentation](https://exercism.org/docs/tracks/elixir)
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- The [Elixir track's programming category on the forum](https://forum.exercism.org/c/programming/elixir)
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- [Exercism's programming category on the forum](https://forum.exercism.org/c/programming/5)
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||||
- The [Frequently Asked Questions](https://exercism.org/docs/using/faqs)
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||||
|
||||
Should those resources not suffice, you could submit your (incomplete) solution to request mentoring.
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||||
|
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If you're stuck on something, it may help to look at some of the [available resources](https://exercism.org/docs/tracks/elixir/resources) out there where answers might be found.
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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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defmodule RationalNumbers do
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@type rational :: {integer, integer}
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alias Kernel, as: K
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@doc """
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Add two rational numbers
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"""
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@spec add(a :: rational, b :: rational) :: rational
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def add({an, ad}, {bn, bd}), do: {an * bd + bn * ad, ad * bd} |> reduce()
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@doc """
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Subtract two rational numbers
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"""
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@spec subtract(a :: rational, b :: rational) :: rational
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def subtract({an, ad}, {bn, bd}), do: {an * bd - bn * ad, ad * bd} |> reduce()
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@doc """
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Multiply two rational numbers
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"""
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@spec multiply(a :: rational, b :: rational) :: rational
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def multiply({an, ad}, {bn, bd}), do: {an * bn, ad * bd} |> reduce()
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@doc """
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Divide two rational numbers
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"""
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@spec divide_by(num :: rational, den :: rational) :: rational
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def divide_by({an, ad}, {bn, bd}) when bn != 0, do: {an * bd, bn * ad} |> reduce()
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@doc """
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Absolute value of a rational number
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"""
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@spec abs(a :: rational) :: rational
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def abs({an, ad}), do: {K.abs(an), K.abs(ad)} |> reduce()
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@doc """
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Exponentiation of a rational number by an integer
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"""
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@spec pow_rational(a :: rational, n :: integer | float) :: rational
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def pow_rational({an, ad}, n) when is_integer(n) and n >= 0, do: {an ** n, ad ** n} |> reduce()
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def pow_rational({an, ad}, n) when is_integer(n) and n < 0,
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do: {ad ** K.abs(n), an ** K.abs(n)} |> reduce()
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def pow_rational({an, ad}, x) when is_float(x), do: ad ** x / an ** x
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@doc """
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Exponentiation of a real number by a rational number
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"""
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@spec pow_real(x :: integer, n :: rational) :: float
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def pow_real(x, {_nn, nd} = n) when is_integer(x) and nd < 0, do: pow_real(x, normalize(n))
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def pow_real(x, {nn, nd}) when is_integer(x), do: x ** nn ** (1 / nd)
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@doc """
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Reduce a rational number to its lowest terms
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"""
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@spec reduce(a :: rational) :: rational
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def reduce({an, ad} = a),
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do: a |> normalize() |> cut(Integer.gcd(an, ad))
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# @doc """
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# Divides a nominator and denominator by integer
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# """
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@spec cut(a :: rational, n :: integer) :: rational
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defp cut({an, ad}, n), do: {K.div(an, n), K.div(ad, n)}
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# @doc """
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# Turns signs of nominator and denominator when negative denominator
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# """
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@spec normalize(a :: rational) :: rational
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defp normalize({an, ad}) when ad < 0, do: {K.-(an), K.-(ad)}
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defp normalize({an, ad}), do: {an, ad}
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# @doc """
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# Calculate a greatest common divisor of two integers
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# """
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# @spec gcd(i :: integer, j :: integer) :: integer
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# defp gcd(i, 0), do: K.abs(i)
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# defp gcd(0, j), do: K.abs(j)
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# defp gcd(i, j), do: gcd(j, K.rem(i, j))
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end
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defmodule RationalNumbers.MixProject do
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use Mix.Project
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def project do
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[
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app: :rational_numbers,
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version: "0.1.0",
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# elixir: "~> 1.8",
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start_permanent: Mix.env() == :prod,
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deps: deps()
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]
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end
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# Run "mix help compile.app" to learn about applications.
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def application do
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[
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extra_applications: [:logger]
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]
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end
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# Run "mix help deps" to learn about dependencies.
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defp deps do
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[
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# {:dep_from_hexpm, "~> 0.3.0"},
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# {:dep_from_git, git: "https://github.com/elixir-lang/my_dep.git", tag: "0.1.0"}
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]
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end
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end
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defmodule RationalNumbersTest do
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use ExUnit.Case
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describe "Addition" do
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# # @tag :pending
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test "Add two positive rational numbers" do
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assert RationalNumbers.add({1, 2}, {2, 3}) == {7, 6}
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end
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# @tag :pending
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test "Add a positive rational number and a negative rational number" do
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assert RationalNumbers.add({1, 2}, {-2, 3}) == {-1, 6}
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end
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# @tag :pending
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test "Add two negative rational numbers" do
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assert RationalNumbers.add({-1, 2}, {-2, 3}) == {-7, 6}
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end
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# @tag :pending
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test "Add a rational number to its additive inverse" do
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assert RationalNumbers.add({1, 2}, {-1, 2}) == {0, 1}
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end
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end
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describe "Subtraction" do
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# @tag :pending
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test "Subtract two positive rational numbers" do
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assert RationalNumbers.subtract({1, 2}, {2, 3}) == {-1, 6}
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end
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# @tag :pending
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test "Subtract a positive rational number and a negative rational number" do
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assert RationalNumbers.subtract({1, 2}, {-2, 3}) == {7, 6}
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end
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# @tag :pending
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test "Subtract two negative rational numbers" do
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assert RationalNumbers.subtract({-1, 2}, {-2, 3}) == {1, 6}
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end
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# @tag :pending
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test "Subtract a rational number from itself" do
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assert RationalNumbers.subtract({1, 2}, {1, 2}) == {0, 1}
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end
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end
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describe "Multiplication" do
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# @tag :pending
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test "Multiply two positive rational numbers" do
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assert RationalNumbers.multiply({1, 2}, {2, 3}) == {1, 3}
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end
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# @tag :pending
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test "Multiply a negative rational number by a positive rational number" do
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assert RationalNumbers.multiply({-1, 2}, {2, 3}) == {-1, 3}
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end
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# @tag :pending
|
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test "Multiply two negative rational numbers" do
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assert RationalNumbers.multiply({-1, 2}, {-2, 3}) == {1, 3}
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end
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# @tag :pending
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test "Multiply a rational number by its reciprocal" do
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assert RationalNumbers.multiply({1, 2}, {2, 1}) == {1, 1}
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end
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# @tag :pending
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test "Multiply a rational number by 1" do
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assert RationalNumbers.multiply({1, 2}, {1, 1}) == {1, 2}
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end
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# @tag :pending
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test "Multiply a rational number by 0" do
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assert RationalNumbers.multiply({1, 2}, {0, 1}) == {0, 1}
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end
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end
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describe "Division" do
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# @tag :pending
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test "Divide two positive rational numbers" do
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assert RationalNumbers.divide_by({1, 2}, {2, 3}) == {3, 4}
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end
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||||
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||||
# @tag :pending
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test "Divide a positive rational number by a negative rational number" do
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assert RationalNumbers.divide_by({1, 2}, {-2, 3}) == {-3, 4}
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end
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||||
|
||||
# @tag :pending
|
||||
test "Divide two negative rational numbers" do
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assert RationalNumbers.divide_by({-1, 2}, {-2, 3}) == {3, 4}
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end
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||||
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||||
# @tag :pending
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test "Divide a rational number by 1" do
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assert RationalNumbers.divide_by({1, 2}, {1, 1}) == {1, 2}
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end
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end
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describe "Absolute value" do
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# @tag :pending
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test "Absolute value of a positive rational number" do
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assert RationalNumbers.abs({1, 2}) == {1, 2}
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end
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# @tag :pending
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test "Absolute value of a positive rational number with negative numerator and denominator" do
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assert RationalNumbers.abs({-1, -2}) == {1, 2}
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end
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# @tag :pending
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test "Absolute value of a negative rational number" do
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assert RationalNumbers.abs({-1, 2}) == {1, 2}
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end
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# @tag :pending
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test "Absolute value of a negative rational number with negative denominator" do
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assert RationalNumbers.abs({1, -2}) == {1, 2}
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end
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# @tag :pending
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test "Absolute value of zero" do
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assert RationalNumbers.abs({0, 1}) == {0, 1}
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end
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# @tag :pending
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||||
test "Absolute value of a rational number is reduced to lowest terms" do
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assert RationalNumbers.abs({2, 4}) == {1, 2}
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end
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end
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describe "Exponentiation of a rational number" do
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# @tag :pending
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test "Raise a positive rational number to a positive integer power" do
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assert RationalNumbers.pow_rational({1, 2}, 3) == {1, 8}
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end
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||||
# @tag :pending
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test "Raise a negative rational number to a positive integer power" do
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assert RationalNumbers.pow_rational({-1, 2}, 3) == {-1, 8}
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end
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|
||||
# @tag :pending
|
||||
test "Raise a positive rational number to a negative integer power" do
|
||||
assert RationalNumbers.pow_rational({3, 5}, -2) == {25, 9}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a negative rational number to an even negative integer power" do
|
||||
assert RationalNumbers.pow_rational({-3, 5}, -2) == {25, 9}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a negative rational number to an odd negative integer power" do
|
||||
assert RationalNumbers.pow_rational({-3, 5}, -3) == {-125, 27}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise zero to an integer power" do
|
||||
assert RationalNumbers.pow_rational({0, 1}, 5) == {0, 1}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise one to an integer power" do
|
||||
assert RationalNumbers.pow_rational({1, 1}, 4) == {1, 1}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a positive rational number to the power of zero" do
|
||||
assert RationalNumbers.pow_rational({1, 2}, 0) == {1, 1}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a negative rational number to the power of zero" do
|
||||
assert RationalNumbers.pow_rational({-1, 2}, 0) == {1, 1}
|
||||
end
|
||||
end
|
||||
|
||||
describe "Exponentiation of a real number to a rational number" do
|
||||
# @tag :pending
|
||||
test "Raise a real number to a positive rational number" do
|
||||
x = 8
|
||||
r = {4, 3}
|
||||
result = 16.0
|
||||
assert_in_delta RationalNumbers.pow_real(x, r), result, 1.0e-10
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a real number to a negative rational number" do
|
||||
x = 9
|
||||
r = {-1, 2}
|
||||
result = 0.3333333333333333
|
||||
assert_in_delta RationalNumbers.pow_real(x, r), result, 1.0e-10
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Raise a real number to a zero rational number" do
|
||||
x = 2
|
||||
r = {0, 1}
|
||||
result = 1.0
|
||||
assert_in_delta RationalNumbers.pow_real(x, r), result, 1.0e-10
|
||||
end
|
||||
end
|
||||
|
||||
describe "Reduction to lowest terms" do
|
||||
# @tag :pending
|
||||
test "Reduce a positive rational number to lowest terms" do
|
||||
assert RationalNumbers.reduce({2, 4}) == {1, 2}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce places the minus sign on the numerator" do
|
||||
assert RationalNumbers.reduce({3, -4}) == {-3, 4}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce a negative rational number to lowest terms" do
|
||||
assert RationalNumbers.reduce({-4, 6}) == {-2, 3}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce a rational number with a negative denominator to lowest terms" do
|
||||
assert RationalNumbers.reduce({3, -9}) == {-1, 3}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce zero to lowest terms" do
|
||||
assert RationalNumbers.reduce({0, 6}) == {0, 1}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce an integer to lowest terms" do
|
||||
assert RationalNumbers.reduce({-14, 7}) == {-2, 1}
|
||||
end
|
||||
|
||||
# @tag :pending
|
||||
test "Reduce one to lowest terms" do
|
||||
assert RationalNumbers.reduce({13, 13}) == {1, 1}
|
||||
end
|
||||
end
|
||||
end
|
|
@ -0,0 +1,2 @@
|
|||
ExUnit.start()
|
||||
ExUnit.configure(exclude: :pending, trace: true)
|
Loading…
Reference in New Issue