Update the description for the complex-numbers exercise

exercises/complex-numbers/description.md

@@ -1,29 +1,101 @@
 # 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 **complex number** is expressed in the form:
 
-`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.
+```math
+z = a + b * i
+```
 
-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`.
+where:
 
-Multiplication result is by definition
-`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`.
+- `a` is the **real part** (a real number),
 
-The reciprocal of a non-zero complex number is
-`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`.
+- `b` is the **imaginary part** (also a real number), and
 
-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`.
+- `i` is the imaginary unit satisfying `i^2 = -1`.
 
-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)`.
+## Key Properties
 
-Implement the following operations:
+### Conjugate
 
-- addition, subtraction, multiplication and division of two complex numbers,
-- conjugate, absolute value, exponent of a given complex number.
+The conjugate of the complex number `z = a + b * i` is given by:
 
-Assume the programming language you are using does not have an implementation of complex numbers.
+```math
+z̅ = a - b * i
+```
+
+### Absolute Value
+
+The absolute value (or modulus) of `z` is defined as:
+
+```math
+|z| = sqrt(a^2 + b^2)
+```
+
+The square of the absolute value, `|z|²`, can be computed as the product of `z` and its conjugate:
+
+```math
+|z|² = z * z̅ = a² + b²
+```
+
+## Operations on Complex Numbers
+
+### 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:
+
+```math
+z₁ + z₂ = (a + c) + (b + d) * i
+```
+
+### Subtraction
+
+The difference of two complex numbers is obtained by subtracting their respective parts:
+
+```math
+z₁ - z₂ = (a - c) + (b - d) * i
+```
+
+### Multiplication
+
+The product of two complex numbers is defined as:
+
+```math
+z₁ * z₂ = (a + b * i) * (c + d * i) = (a * c - b * d) + (b * c + a * d) * i
+```
+
+### Division
+
+The division of one complex number by another is given by:
+
+```math
+z₁ / z₂ = (a + b * i) / (c + d * i) = (a * c + b * d) / (c² + d²) + (b * c - a * d) / (c² + d²) * i
+```
+
+### Reciprocal
+
+The reciprocal of a non-zero complex number is given by:
+
+```math
+1 / z = 1 / (a + b * i) = a / (a² + b²) - b / (a² + b²) * i
+```
+
+### Exponentiation
+
+Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
+
+```math
+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.
+- Calculation of the **conjugate** of a complex number.
+- Calculation of the **absolute value** of a complex number.
+- Calculation of the **exponent** of a given complex number.