> For the complete documentation index, see [llms.txt](https://jamesbrayy.gitbook.io/atar/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jamesbrayy.gitbook.io/atar/specialist/complex-numbers.md).

# complex numbers

***

## **forms**

#### *cartesian*

* $$z=a+bi$$

#### *polar*

* $$z=r\mathrm{cis}\mathit{\theta}$$
* the principal argument $$\theta$$ occupies the domain $$-\mathit{\pi}<\mathit{\theta}\le \mathit{\pi}$$
* *multiplying and dividing:*
  * ![](/files/WLw3IkYB1h0hRO8VkUdo)
* *de moivre's theorem:*
  * $$\[r \operatorname{cis} \theta]^n = r^n \operatorname{cis} (n \theta), \quad n \in \mathbb{Z}$$
  * *proving triple-angle identities:*
    * ![](/files/c0SfSaLRInMA0b0cUG7c)

## **argand plane**

#### *geometric question*

* ![](/files/b6yvIL5mT8bUH6evtKm1)

#### *loci*

* ![](/files/brl5gudiVemkxIxGb5xl)
* ![](/files/u5Jk2CefwfnF60GJWJty)
* ![](/files/aU7CD4PA9euj9ZGqJhFo)
* ![](/files/UVqHjuxDqDkAUG37EIpv)

## **roots**

#### *roots of unity*

* i.e. determine all 3 solutions to $${\mathrm{z}}^{3}=1$$
  * ![](/files/wCT6WYUeFRjR55gJETc4)
* *sum and product:*
  * ![](/files/39T21BvMkxYeczCeFzkH)
  * i.e. for solutions of $${\mathrm{z}}^{\mathrm{n}}=1$$:
    * $$\sum\_{k=0}^{n-1} z\_k = 0$$
    * $$\prod\_{k=0}^{n-1} z\_k = (-1)^{n-1}$$
* *complex* $$n^{th}$$ *roots:*
  * *i.e.* a solution to $${z}^{5}=-4+4i$$ is $$z=1-i$$. find all solutions.
    * ![](/files/feIqxvAVgarn0YEPKnyu)

## **polynomials**

#### *factor theorem*

* if $$f\left(a\right)=0$$, then $$\left(x-a\right)$$ is a factor $$f\left(x\right)$$

#### *remainder theorem*

* if $$f\left(x\right)$$ is divided by $$(x-a)$$, the remainder will be $$f\left(a\right)$$