> For the complete documentation index, see [llms.txt](https://isubasinghe.gitbook.io/isithas-wiki/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://isubasinghe.gitbook.io/isithas-wiki/math/complex_numbers.md).

# complex\_numbers

## Basic idea

Extension of the reals by adjoining $i$ with $i^2=-1$. Algebraically closed: every non-constant polynomial has a root.

## Key formulas

* $i^2=-1$
* Euler's formula: $e^{i\theta}=\cos\theta+i\sin\theta$
* Modulus: $|a+bi|=\sqrt{a^2+b^2}$
* Conjugate: $\overline{a+bi}=a-bi$