add complex-numbers (#59)

Author: oxe-i

config.json

@@ -377,6 +377,14 @@
         "prerequisites": [],
         "difficulty": 6
       },
+      {
+        "slug": "complex-numbers",
+        "name": "Complex Numbers",
+        "uuid": "5860e836-932b-4eb6-9482-c1ca8bbbce02",
+        "practices": [],
+        "prerequisites": [],
+        "difficulty": 6
+      },
       {
         "slug": "knapsack",
         "name": "Knapsack",

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

@@ -0,0 +1,100 @@
+# Instructions
+
+A **complex number** is expressed in the form `z = a + b * i`, where:
+
+- `a` is the **real part** (a real number),
+
+- `b` is the **imaginary part** (also a real number), and
+
+- `i` is the **imaginary unit** satisfying `i^2 = -1`.
+
+## Operations on Complex Numbers
+
+### Conjugate
+
+The conjugate of the complex number `z = a + b * i` is given by:
+
+```text
+zc = a - b * i
+```
+
+### Absolute Value
+
+The absolute value (or modulus) of `z` is defined as:
+
+```text
+|z| = sqrt(a^2 + b^2)
+```
+
+The square of the absolute value is computed as the product of `z` and its conjugate `zc`:
+
+```text
+|z|^2 = z * zc = a^2 + b^2
+```
+
+### Addition
+
+The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately:
+
+```text
+z1 + z2 = (a + b * i) + (c + d * i)
+        = (a + c) + (b + d) * i
+```
+
+### Subtraction
+
+The difference of two complex numbers is obtained by subtracting their respective parts:
+
+```text
+z1 - z2 = (a + b * i) - (c + d * i)
+        = (a - c) + (b - d) * i
+```
+
+### Multiplication
+
+The product of two complex numbers is defined as:
+
+```text
+z1 * z2 = (a + b * i) * (c + d * i)
+        = (a * c - b * d) + (b * c + a * d) * i
+```
+
+### Reciprocal
+
+The reciprocal of a non-zero complex number is given by:
+
+```text
+1 / z = 1 / (a + b * i)
+      = a / (a^2 + b^2) - b / (a^2 + b^2) * i
+```
+
+### Division
+
+The division of one complex number by another is given by:
+
+```text
+z1 / z2 = z1 * (1 / z2)
+        = (a + b * i) / (c + d * i)
+        = (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
+```
+
+### Exponentiation
+
+Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
+
+```text
+e^(a + b * i) = e^a * e^(b * i)
+              = e^a * (cos(b) + i * sin(b))
+```
+
+## Implementation Requirements
+
+Given that you should not use built-in support for complex numbers, implement the following operations:
+
+- **addition** of two complex numbers
+- **subtraction** of two complex numbers
+- **multiplication** of two complex numbers
+- **division** of two complex numbers
+- **conjugate** of a complex number
+- **absolute value** of a complex number
+- **exponentiation** of _e_ (the base of the natural logarithm) to a complex number

exercises/practice/complex-numbers/.meta/Example.lean

@@ -0,0 +1,49 @@
+namespace ComplexNumbers
+
+structure ComplexNumber where
+  real : Float
+  imag : Float
+  deriving Repr
+
+/- define equality between two complex numbers -/
+instance : BEq ComplexNumber where
+  beq x y := Float.abs (x.real - y.real) <= 0.001 &&
+             Float.abs (x.imag - y.imag) <= 0.001
+
+/- define how a complex number should be constructed out of a literal number -/
+instance {n : Nat} : OfNat ComplexNumber n where
+  ofNat := { real := n.toFloat, imag := 0.0 }
+
+def real (z : ComplexNumber) : Float :=
+  z.real
+
+def imaginary (z : ComplexNumber) : Float :=
+  z.imag
+
+def mul (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  { real := z1.real * z2.real - z1.imag * z2.imag,
+    imag := z1.imag * z2.real + z1.real * z2.imag }
+
+def div (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  let square := z2.real ^ 2 + z2.imag ^ 2
+  { real := (z1.real * z2.real + z1.imag * z2.imag) / square,
+    imag := (z1.imag * z2.real - z1.real * z2.imag) / square }
+
+def sub (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  { real := z1.real - z2.real, imag := z1.imag - z2.imag }
+
+def add (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  { real := z1.real + z2.real, imag := z1.imag + z2.imag }
+
+def abs (z : ComplexNumber) : Float :=
+  Float.sqrt (z.real ^ 2 + z.imag ^ 2)
+
+def conjugate (z : ComplexNumber) : ComplexNumber :=
+  { z with imag := -z.imag }
+
+def exp (z : ComplexNumber) : ComplexNumber :=
+  let expReal := Float.exp z.real
+  { real := expReal * Float.cos z.imag,
+    imag := expReal * Float.sin z.imag }
+
+end ComplexNumbers

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

@@ -0,0 +1,19 @@
+{
+  "authors": [
+    "oxe-i"
+  ],
+  "files": {
+    "solution": [
+      "ComplexNumbers.lean"
+    ],
+    "test": [
+      "ComplexNumbersTest.lean"
+    ],
+    "example": [
+      ".meta/Example.lean"
+    ]
+  },
+  "blurb": "Implement complex numbers.",
+  "source": "Wikipedia",
+  "source_url": "https://en.wikipedia.org/wiki/Complex_number"
+}

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

@@ -0,0 +1,130 @@
+# This is an auto-generated file.
+#
+# Regenerating this file via `configlet sync` will:
+# - Recreate every `description` key/value pair
+# - Recreate every `reimplements` key/value pair, where they exist in problem-specifications
+# - Remove any `include = true` key/value pair (an omitted `include` key implies inclusion)
+# - Preserve any other key/value pair
+#
+# As user-added comments (using the # character) will be removed when this file
+# is regenerated, comments can be added via a `comment` key.
+
+[9f98e133-eb7f-45b0-9676-cce001cd6f7a]
+description = "Real part -> Real part of a purely real number"
+
+[07988e20-f287-4bb7-90cf-b32c4bffe0f3]
+description = "Real part -> Real part of a purely imaginary number"
+
+[4a370e86-939e-43de-a895-a00ca32da60a]
+description = "Real part -> Real part of a number with real and imaginary part"
+
+[9b3fddef-4c12-4a99-b8f8-e3a42c7ccef6]
+description = "Imaginary part -> Imaginary part of a purely real number"
+
+[a8dafedd-535a-4ed3-8a39-fda103a2b01e]
+description = "Imaginary part -> Imaginary part of a purely imaginary number"
+
+[0f998f19-69ee-4c64-80ef-01b086feab80]
+description = "Imaginary part -> Imaginary part of a number with real and imaginary part"
+
+[a39b7fd6-6527-492f-8c34-609d2c913879]
+description = "Imaginary unit"
+
+[9a2c8de9-f068-4f6f-b41c-82232cc6c33e]
+description = "Arithmetic -> Addition -> Add purely real numbers"
+
+[657c55e1-b14b-4ba7-bd5c-19db22b7d659]
+description = "Arithmetic -> Addition -> Add purely imaginary numbers"
+
+[4e1395f5-572b-4ce8-bfa9-9a63056888da]
+description = "Arithmetic -> Addition -> Add numbers with real and imaginary part"
+
+[1155dc45-e4f7-44b8-af34-a91aa431475d]
+description = "Arithmetic -> Subtraction -> Subtract purely real numbers"
+
+[f95e9da8-acd5-4da4-ac7c-c861b02f774b]
+description = "Arithmetic -> Subtraction -> Subtract purely imaginary numbers"
+
+[f876feb1-f9d1-4d34-b067-b599a8746400]
+description = "Arithmetic -> Subtraction -> Subtract numbers with real and imaginary part"
+
+[8a0366c0-9e16-431f-9fd7-40ac46ff4ec4]
+description = "Arithmetic -> Multiplication -> Multiply purely real numbers"
+
+[e560ed2b-0b80-4b4f-90f2-63cefc911aaf]
+description = "Arithmetic -> Multiplication -> Multiply purely imaginary numbers"
+
+[4d1d10f0-f8d4-48a0-b1d0-f284ada567e6]
+description = "Arithmetic -> Multiplication -> Multiply numbers with real and imaginary part"
+
+[b0571ddb-9045-412b-9c15-cd1d816d36c1]
+description = "Arithmetic -> Division -> Divide purely real numbers"
+
+[5bb4c7e4-9934-4237-93cc-5780764fdbdd]
+description = "Arithmetic -> Division -> Divide purely imaginary numbers"
+
+[c4e7fef5-64ac-4537-91c2-c6529707701f]
+description = "Arithmetic -> Division -> Divide numbers with real and imaginary part"
+
+[c56a7332-aad2-4437-83a0-b3580ecee843]
+description = "Absolute value -> Absolute value of a positive purely real number"
+
+[cf88d7d3-ee74-4f4e-8a88-a1b0090ecb0c]
+description = "Absolute value -> Absolute value of a negative purely real number"
+
+[bbe26568-86c1-4bb4-ba7a-da5697e2b994]
+description = "Absolute value -> Absolute value of a purely imaginary number with positive imaginary part"
+
+[3b48233d-468e-4276-9f59-70f4ca1f26f3]
+description = "Absolute value -> Absolute value of a purely imaginary number with negative imaginary part"
+
+[fe400a9f-aa22-4b49-af92-51e0f5a2a6d3]
+description = "Absolute value -> Absolute value of a number with real and imaginary part"
+
+[fb2d0792-e55a-4484-9443-df1eddfc84a2]
+description = "Complex conjugate -> Conjugate a purely real number"
+
+[e37fe7ac-a968-4694-a460-66cb605f8691]
+description = "Complex conjugate -> Conjugate a purely imaginary number"
+
+[f7704498-d0be-4192-aaf5-a1f3a7f43e68]
+description = "Complex conjugate -> Conjugate a number with real and imaginary part"
+
+[6d96d4c6-2edb-445b-94a2-7de6d4caaf60]
+description = "Complex exponential function -> Euler's identity/formula"
+
+[2d2c05a0-4038-4427-a24d-72f6624aa45f]
+description = "Complex exponential function -> Exponential of 0"
+
+[ed87f1bd-b187-45d6-8ece-7e331232c809]
+description = "Complex exponential function -> Exponential of a purely real number"
+
+[08eedacc-5a95-44fc-8789-1547b27a8702]
+description = "Complex exponential function -> Exponential of a number with real and imaginary part"
+
+[d2de4375-7537-479a-aa0e-d474f4f09859]
+description = "Complex exponential function -> Exponential resulting in a number with real and imaginary part"
+
+[06d793bf-73bd-4b02-b015-3030b2c952ec]
+description = "Operations between real numbers and complex numbers -> Add real number to complex number"
+
+[d77dbbdf-b8df-43f6-a58d-3acb96765328]
+description = "Operations between real numbers and complex numbers -> Add complex number to real number"
+
+[20432c8e-8960-4c40-ba83-c9d910ff0a0f]
+description = "Operations between real numbers and complex numbers -> Subtract real number from complex number"
+
+[b4b38c85-e1bf-437d-b04d-49bba6e55000]
+description = "Operations between real numbers and complex numbers -> Subtract complex number from real number"
+
+[dabe1c8c-b8f4-44dd-879d-37d77c4d06bd]
+description = "Operations between real numbers and complex numbers -> Multiply complex number by real number"
+
+[6c81b8c8-9851-46f0-9de5-d96d314c3a28]
+description = "Operations between real numbers and complex numbers -> Multiply real number by complex number"
+
+[8a400f75-710e-4d0c-bcb4-5e5a00c78aa0]
+description = "Operations between real numbers and complex numbers -> Divide complex number by real number"
+
+[9a867d1b-d736-4c41-a41e-90bd148e9d5e]
+description = "Operations between real numbers and complex numbers -> Divide real number by complex number"

exercises/practice/complex-numbers/ComplexNumbers.lean

@@ -0,0 +1,43 @@
+namespace ComplexNumbers
+
+structure ComplexNumber where
+  real : Float
+  imag : Float
+  deriving Repr
+
+/- define equality between two complex numbers -/
+instance : BEq ComplexNumber where
+  beq x y := sorry
+
+/- define how a complex number should be constructed out of a literal number -/
+instance {n : Nat} : OfNat ComplexNumber n where
+  ofNat := sorry
+
+def real (z : ComplexNumber) : Float :=
+  sorry
+
+def imaginary (z : ComplexNumber) : Float :=
+  sorry
+
+def mul (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  sorry
+
+def div (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  sorry
+
+def sub (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  sorry
+
+def add (z1 : ComplexNumber) (z2 : ComplexNumber) : ComplexNumber :=
+  sorry
+
+def abs (z : ComplexNumber) : Float :=
+  sorry
+
+def conjugate (z : ComplexNumber) : ComplexNumber :=
+  sorry
+
+def exp (z : ComplexNumber) : ComplexNumber :=
+  sorry
+
+end ComplexNumbers