# Complex Numbers

## Assignment
Define a new type `Complex` that allows arithmetic calculations in an intuitive fashion. See below for the rules that shall be implemented. Incrementally make sure that all unit tests in `06-complex/complex.hs` are passed.

## Complex Numbers

Complex numbers are numbers of the form: $z = a + bi$

- **`a`**: The real part of the complex number.
- **`b`**: The imaginary part of the complex number.
- **`i`**: The imaginary unit, where $i^2 = -1$.

Complex numbers are used in mathematics, physics, and engineering to extend the real number system and solve equations that have no real solutions, such as $x^2 + 1 = 0$.

Complex numbers are often represented in both rectangular form $a + bi$ and polar/exponential form $r e^{i\theta}$.


## Rules for Computing with Complex Numbers

### 1. **Addition and Subtraction**
- Add or subtract the real and imaginary parts separately.
- Rule:  
  $(a + bi) + (c + di) = (a + c) + (b + d)i$  
  $(a + bi) - (c + di) = (a - c) + (b - d)i$

**Examples**:
1. $(2 + 3i) + (4 + 5i) = 6 + 8i$
2. $(0 + 3i) + (2 + 0i) = 2 + 3i$
3. $(7 + 2i) - (3 + 6i) = 4 - 4i$
4. $(5 + 4i) - (1 + 2i) = 4 + 2i$

### 2. **Multiplication**
- Use the distributive property and the rule $i^2 = -1$.
- Rule:  
  $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

**Examples**:
1. $(1 + 2i)(3 + 4i) = -5 + 10i$
2. $(2 + 3i)(1 - i) = 5 + i$
3. $(4 + i)(2 - 3i) = 11 - 10i$
4. $(3 + 2i)(3 + 2i) = 5 + 12i$

### 3. **Division**
- Multiply numerator and denominator by the conjugate of the denominator.
- Rule:  
  $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$  
  Simplifies to:  
  $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

**Examples**:
1. $\frac{1 + i}{1 - i} = 0 + 1i$
2. $\frac{3 + 2i}{4 + 3i} = \frac{18 + i}{25} = 0.72 - 0.04i$
3. $\frac{2 + i}{1 + i} = \frac{3 - i}{2} = 1.5 - 0.5i$

### 4. **Complex Conjugate**
- The conjugate of $a + bi$ is $a - bi$, used for simplifications and magnitude calculation.

**Examples**:
1. Conjugate of $3 + 4i$ is $3 - 4i$.
2. Conjugate of $5 - i$ is $5 + i$.
3. Conjugate of $-2 + 3i$ is $-2 - 3i$.

### 5. **Magnitude (Modulus)**
- The magnitude is the distance from the origin in the complex plane.
- Rule:  $|a + bi| = \sqrt{a^2 + b^2}$

**Examples**:
1. $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$
2. $|1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$
3. $|0 + 5i| = \sqrt{0^2 + 5^2} = 5$