# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-numbers-Complex%20Numbers.html


## Complex numbers

"The shortest path between two truths in the real domain passes through the complex domain." Jacques Hadamard


#### Complex Numbers


##### What you have to know :

- You must be able to solve quadratic equations (If not, check this lesson)
- How to work with trigonometric properties (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### The complex plane


##### What you have to know :

- You must know the vector properties (If not, check this lesson)
- How to use trigonometry (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### Euler's form / De Moivre Theorem


##### What you have to know :

- How to manipulate exponential expressions (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### Extra


##### Material and references:

- Hodder Book HL(ISBN: 9781510462366) : 4A, 4B, 4C, 4D, 4E

##### Key Points

- 1. For \( z = x + iy \): modulus \( r = \sqrt{x^2 + y^2} \), argument \( \tan \theta = \frac{y}{x} \).
- 2. Polar form: \( z = r(\cos\theta + i\sin\theta) \).
- 3. Modulus–argument rules: \( |zw| = |z||w| \), \( \arg(zw) = \arg z + \arg w \); \( \left|\frac{z}{w}\right| = \frac{|z|}{|w|} \), \( \arg\!\left(\frac{z}{w}\right) = \arg z - \arg w \).
- 4. Euler form: \( z = re^{i\theta} \), \( e^{i\theta} = \cos\theta + i\sin\theta \), \( z^* = re^{-i\theta} \).
- 5. A useful short-cut when working with complex conjugate roots is \[ (x - z)(x - z^*) = x^2 - 2\operatorname{Re}(z)\,x + |z|^2. \]
- 6. De Moivre’s theorem: \( (r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta) \), \( n \in \mathbb{Z} \).
- 7. You should be able to use De Moivre’s theorem to find roots of complex numbers. More generally, the solutions of \( z^n = w \) form a regular \( n \)-gon with vertices on a circle of radius \( |z| \) centred at the origin.
- 8. De Moivre identities: if \( z = \cos\theta + i\sin\theta \), then \( z^n + z^{-n} = 2\cos(n\theta) \), \( z^n - z^{-n} = 2i\sin(n\theta) \).

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