# Further Maths - Exponential Form of Complex Numbers

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/exponential-form-of-complex-numbers/ · Updated: 2021-02-22 · Tags: further-maths, complex-numbers

## See Also
- [Further Maths - Complex Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/complex-numbers/)

## Sergeant
- `further-maths/textbooks/year-2/chapter-1-complex-numbers/ex1a`

## Flashcards

### Euler's relation
##### What is Euler's relation??
$$
e^{i\theta} = \cos \theta + i \sin \theta
$$
^eulers-relation-statement

##### Why can you rewrite $e^{i\theta}$ as $\cos\theta + i\sin\theta$??
Because the Maclaurin series of $\sin x$, $\cos x$ and $e^x$ match up.
^eulers-relation-maclaurin-justification

##### How can you write a complex number with argument $\theta$ and modulus $r$ in exponential form??
$$
re^{i\theta}
$$
^exponential-form-of-complex-number

##### $$e^{\pi i} = -1$$ What is this identity a special case of??
Euler's relation.
^eulers-identity-special-case

### Arithmetic in exponential form
##### $$z_1 = r_1 e^{\theta_1 i} \\ z_2 = r_2 e^{\theta_2 i}$$ What is $z_1 z_2$??
$$
r_1 r_2 e^{(\theta_1 + \theta_2)i}
$$
^exponential-form-multiplication

##### $$z_1 = r_1 e^{\theta_1 i} \\ z_2 = r_2 e^{\theta_2 i}$$ What is $\frac{z_1}{z_2}$??
$$
\frac{r_1}{r_2} e^{(\theta_1 - \theta_2)i}
$$
^exponential-form-division

##### $$z = r e^{\theta i}$$ What is $z^n$??
$$
r^n e^{n\theta i}
$$
^exponential-form-power

### De Moivre's theorem
##### What is De Moivre's Theorem??
If
$$
z = r(\cos\theta + i \sin\theta)
$$

Then

$$
z^n = r^n (\cos n\theta + i \sin n\theta)
$$
^de-moivre-theorem-statement

##### What's the process (but not the actual steps) for proving De Moivre's Theorem using Euler's relation??
Rewrite the modulus-argument form using $e$ and apply the laws of indices.
^de-moivre-proof-via-euler

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