[#765] New practice exercise complex-numbers (#769)

Author: jiegillet

config.json

@@ -2430,6 +2430,22 @@
         ],
         "difficulty": 4
       },
+      {
+        "slug": "complex-numbers",
+        "name": "Complex Numbers",
+        "uuid": "0cc86a92-7118-4cb9-8417-db4557baf364",
+        "prerequisites": [
+          "pattern-matching",
+          "erlang-libraries",
+          "floating-point-numbers",
+          "tuples"
+        ],
+        "practices": [
+          "erlang-libraries",
+          "floating-point-numbers"
+        ],
+        "difficulty": 4
+      },
       {
         "slug": "yacht",
         "name": "Yacht",

exercises/practice/complex-numbers/.docs/instructions.append.md

@@ -0,0 +1,3 @@
+In this exercise, complex numbers are represented as a tuple-pair containing the real and imaginary parts.
+
+For example, the real number `1` is `{1, 0}`, the imaginary number `i` is `{0, 1}` and the complex number `4+3i` is `{4, 3}'.

exercises/practice/complex-numbers/.docs/instructions.md

@@ -0,0 +1,29 @@
+# Description
+
+A complex number is a number in the form `a + b * i` where `a` and `b` are real and `i` satisfies `i^2 = -1`.
+
+`a` is called the real part and `b` is called the imaginary part of `z`.
+The conjugate of the number `a + b * i` is the number `a - b * i`.
+The absolute value of a complex number `z = a + b * i` is a real number `|z| = sqrt(a^2 + b^2)`. The square of the absolute value `|z|^2` is the result of multiplication of `z` by its complex conjugate.
+
+The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
+`(a + i * b) + (c + i * d) = (a + c) + (b + d) * i`,
+`(a + i * b) - (c + i * d) = (a - c) + (b - d) * i`.
+
+Multiplication result is by definition
+`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`.
+
+The reciprocal of a non-zero complex number is
+`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`.
+
+Dividing a complex number `a + i * b` by another `c + i * d` gives:
+`(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i`.
+
+Raising e to a complex exponent can be expressed as `e^(a + i * b) = e^a * e^(i * b)`, the last term of which is given by Euler's formula `e^(i * b) = cos(b) + i * sin(b)`.
+
+Implement the following operations:
+ - addition, subtraction, multiplication and division of two complex numbers,
+ - conjugate, absolute value, exponent of a given complex number.
+
+
+Assume the programming language you are using does not have an implementation of complex numbers.

exercises/practice/complex-numbers/.formatter.exs

@@ -0,0 +1,4 @@
+# Used by "mix format"
+[
+  inputs: ["{mix,.formatter}.exs", "{config,lib,test}/**/*.{ex,exs}"]
+]

exercises/practice/complex-numbers/.meta/config.json

@@ -0,0 +1,21 @@
+{
+  "blurb": "Implement complex numbers.",
+  "authors": [
+    "jiegillet"
+  ],
+  "contributors": [
+  ],
+  "files": {
+    "solution": [
+      "lib/complex_numbers.ex"
+    ],
+    "test": [
+      "test/complex_numbers_test.exs"
+    ],
+    "example": [
+      ".meta/example.ex"
+    ]
+  },
+  "source": "Wikipedia",
+  "source_url": "https://en.wikipedia.org/wiki/Complex_number"
+}

exercises/practice/complex-numbers/.meta/example.ex

@@ -0,0 +1,64 @@
+defmodule ComplexNumbers do
+  @typedoc """
+  In this module, complex numbers are represented as a tuple-pair containing the real and
+  imaginary parts.
+  For example, the real number `1` is `{1, 0}`, the imaginary number `i` is `{0, 1}` and
+  the complex number `4+3i` is `{4, 3}'.
+  """
+  @type complex :: {float, float}
+
+  @doc """
+  Return the real part of a complex number
+  """
+  @spec real(a :: complex) :: float
+  def real({real, _im}), do: real
+
+  @doc """
+  Return the imaginary part of a complex number
+  """
+  @spec imaginary(a :: complex) :: float
+  def imaginary({_real, im}), do: im
+
+  @doc """
+  Multiply two complex numbers
+  """
+  @spec mul(a :: complex, b :: complex) :: complex
+  def mul({a, b}, {c, d}), do: {a * c - b * d, b * c + a * d}
+
+  @doc """
+  Add two complex numbers
+  """
+  @spec add(a :: complex, b :: complex) :: complex
+  def add({a, b}, {c, d}), do: {a + c, b + d}
+
+  @doc """
+  Subtract two complex numbers
+  """
+  @spec sub(a :: complex, b :: complex) :: complex
+  def sub({a, b}, {c, d}), do: {a - c, b - d}
+
+  @doc """
+  Divide two complex numbers
+  """
+  @spec div(a :: complex, b :: complex) :: complex
+  def div({a, b}, {c, d}),
+    do: {(a * c + b * d) / (c * c + d * d), (b * c - a * d) / (c * c + d * d)}
+
+  @doc """
+  Absolute value of a complex number
+  """
+  @spec abs(a :: complex) :: float
+  def abs({a, b}), do: :math.sqrt(a * a + b * b)
+
+  @doc """
+  Conjugate of a complex number
+  """
+  @spec conjugate(a :: complex) :: complex
+  def conjugate({a, b}), do: {a, -b}
+
+  @doc """
+  Exponential of a complex number
+  """
+  @spec exp(a :: complex) :: complex
+  def exp({a, b}), do: {:math.exp(a) * :math.cos(b), -:math.exp(a) * :math.sin(b)}
+end

exercises/practice/complex-numbers/.meta/tests.toml

@@ -0,0 +1,96 @@
+# This is an auto-generated file. Regular comments will be removed when this
+# file is regenerated. Regenerating will not touch any manually added keys,
+# so comments can be added in a "comment" key.
+
+[9f98e133-eb7f-45b0-9676-cce001cd6f7a]
+description = "Real part of a purely real number"
+
+[07988e20-f287-4bb7-90cf-b32c4bffe0f3]
+description = "Real part of a purely imaginary number"
+
+[4a370e86-939e-43de-a895-a00ca32da60a]
+description = "Real part of a number with real and imaginary part"
+
+[9b3fddef-4c12-4a99-b8f8-e3a42c7ccef6]
+description = "Imaginary part of a purely real number"
+
+[a8dafedd-535a-4ed3-8a39-fda103a2b01e]
+description = "Imaginary part of a purely imaginary number"
+
+[0f998f19-69ee-4c64-80ef-01b086feab80]
+description = "Imaginary part of a number with real and imaginary part"
+
+[a39b7fd6-6527-492f-8c34-609d2c913879]
+description = "Imaginary unit"
+
+[9a2c8de9-f068-4f6f-b41c-82232cc6c33e]
+description = "Add purely real numbers"
+
+[657c55e1-b14b-4ba7-bd5c-19db22b7d659]
+description = "Add purely imaginary numbers"
+
+[4e1395f5-572b-4ce8-bfa9-9a63056888da]
+description = "Add numbers with real and imaginary part"
+
+[1155dc45-e4f7-44b8-af34-a91aa431475d]
+description = "Subtract purely real numbers"
+
+[f95e9da8-acd5-4da4-ac7c-c861b02f774b]
+description = "Subtract purely imaginary numbers"
+
+[f876feb1-f9d1-4d34-b067-b599a8746400]
+description = "Subtract numbers with real and imaginary part"
+
+[8a0366c0-9e16-431f-9fd7-40ac46ff4ec4]
+description = "Multiply purely real numbers"
+
+[e560ed2b-0b80-4b4f-90f2-63cefc911aaf]
+description = "Multiply purely imaginary numbers"
+
+[4d1d10f0-f8d4-48a0-b1d0-f284ada567e6]
+description = "Multiply numbers with real and imaginary part"
+
+[b0571ddb-9045-412b-9c15-cd1d816d36c1]
+description = "Divide purely real numbers"
+
+[5bb4c7e4-9934-4237-93cc-5780764fdbdd]
+description = "Divide purely imaginary numbers"
+
+[c4e7fef5-64ac-4537-91c2-c6529707701f]
+description = "Divide numbers with real and imaginary part"
+
+[c56a7332-aad2-4437-83a0-b3580ecee843]
+description = "Absolute value of a positive purely real number"
+
+[cf88d7d3-ee74-4f4e-8a88-a1b0090ecb0c]
+description = "Absolute value of a negative purely real number"
+
+[bbe26568-86c1-4bb4-ba7a-da5697e2b994]
+description = "Absolute value of a purely imaginary number with positive imaginary part"
+
+[3b48233d-468e-4276-9f59-70f4ca1f26f3]
+description = "Absolute value of a purely imaginary number with negative imaginary part"
+
+[fe400a9f-aa22-4b49-af92-51e0f5a2a6d3]
+description = "Absolute value of a number with real and imaginary part"
+
+[fb2d0792-e55a-4484-9443-df1eddfc84a2]
+description = "Conjugate a purely real number"
+
+[e37fe7ac-a968-4694-a460-66cb605f8691]
+description = "Conjugate a purely imaginary number"
+
+[f7704498-d0be-4192-aaf5-a1f3a7f43e68]
+description = "Conjugate a number with real and imaginary part"
+
+[6d96d4c6-2edb-445b-94a2-7de6d4caaf60]
+description = "Euler's identity/formula"
+
+[2d2c05a0-4038-4427-a24d-72f6624aa45f]
+description = "Exponential of 0"
+
+[ed87f1bd-b187-45d6-8ece-7e331232c809]
+description = "Exponential of a purely real number"
+
+[08eedacc-5a95-44fc-8789-1547b27a8702]
+description = "Exponential of a number with real and imaginary part"

exercises/practice/complex-numbers/lib/complex_numbers.ex

@@ -0,0 +1,72 @@
+defmodule ComplexNumbers do
+  @typedoc """
+  In this module, complex numbers are represented as a tuple-pair containing the real and
+  imaginary parts.
+  For example, the real number `1` is `{1, 0}`, the imaginary number `i` is `{0, 1}` and
+  the complex number `4+3i` is `{4, 3}'.
+  """
+  @type complex :: {float, float}
+
+  @doc """
+  Return the real part of a complex number
+  """
+  @spec real(a :: complex) :: float
+  def real(a) do
+  end
+
+  @doc """
+  Return the imaginary part of a complex number
+  """
+  @spec imaginary(a :: complex) :: float
+  def imaginary(a) do
+  end
+
+  @doc """
+  Multiply two complex numbers
+  """
+  @spec mul(a :: complex, b :: complex) :: complex
+  def mul(a, b) do
+  end
+
+  @doc """
+  Add two complex numbers
+  """
+  @spec add(a :: complex, b :: complex) :: complex
+  def add(a, b) do
+  end
+
+  @doc """
+  Subtract two complex numbers
+  """
+  @spec sub(a :: complex, b :: complex) :: complex
+  def sub(a, b) do
+  end
+
+  @doc """
+  Divide two complex numbers
+  """
+  @spec div(a :: complex, b :: complex) :: complex
+  def div(a, b) do
+  end
+
+  @doc """
+  Absolute value of a complex number
+  """
+  @spec abs(a :: complex) :: float
+  def abs(a) do
+  end
+
+  @doc """
+  Conjugate of a complex number
+  """
+  @spec conjugate(a :: complex) :: complex
+  def conjugate(a) do
+  end
+
+  @doc """
+  Exponential of a complex number
+  """
+  @spec exp(a :: complex) :: complex
+  def exp(a) do
+  end
+end

exercises/practice/complex-numbers/mix.exs

@@ -0,0 +1,28 @@
+defmodule ComplexNumbers.MixProject do
+  use Mix.Project
+
+  def project do
+    [
+      app: :complex_numbers,
+      version: "0.1.0",
+      # elixir: "~> 1.8",
+      start_permanent: Mix.env() == :prod,
+      deps: deps()
+    ]
+  end
+
+  # Run "mix help compile.app" to learn about applications.
+  def application do
+    [
+      extra_applications: [:logger]
+    ]
+  end
+
+  # Run "mix help deps" to learn about dependencies.
+  defp deps do
+    [
+      # {:dep_from_hexpm, "~> 0.3.0"},
+      # {:dep_from_git, git: "https://github.com/elixir-lang/my_dep.git", tag: "0.1.0"}
+    ]
+  end
+end