From 4bdac18ae0e098e2a68fe687b5a89b50993f0f79 Mon Sep 17 00:00:00 2001 From: Oleksii Trekhleb Date: Tue, 14 Aug 2018 23:19:30 +0300 Subject: [PATCH] Add polar form of complex number to README. --- src/algorithms/math/complex-number/README.md | 32 ++++++++++++++++++++ 1 file changed, 32 insertions(+) diff --git a/src/algorithms/math/complex-number/README.md b/src/algorithms/math/complex-number/README.md index ec6b4970a0..9a591e51e2 100644 --- a/src/algorithms/math/complex-number/README.md +++ b/src/algorithms/math/complex-number/README.md @@ -39,6 +39,38 @@ a vector on a diagram called an *Argand diagram*, representing the *complex plan imaginary, together form a complex, just like a building complex (buildings joined together). +## Polar Form + +An alternative way of defining a point `P` in the complex plane, other than using +the x- and y-coordinates, is to use the distance of the point from `O`, the point +whose coordinates are `(0, 0)` (the origin), together with the angle subtended +between the positive real axis and the line segment `OP` in a counterclockwise +direction. This idea leads to the polar form of complex numbers. + +![Polar Form](https://upload.wikimedia.org/wikipedia/commons/7/7a/Complex_number_illustration_modarg.svg) + +The *absolute value* (or modulus or magnitude) of a complex number `z = x + yi` is: + +![Radius](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59629c801aa0ddcdf17ee489e028fb9f8d4ea75) + +The argument of `z` (in many applications referred to as the "phase") is the angle +of the radius `OP` with the positive real axis, and is written as `arg(z)`. As +with the modulus, the argument can be found from the rectangular form `x+yi`: + +![Phase](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbbdd9bb1dd5df86dd2b820b20f82995023e566) + +Together, `r` and `φ` give another way of representing complex numbers, the +polar form, as the combination of modulus and argument fully specify the +position of a point on the plane. Recovering the original rectangular +co-ordinates from the polar form is done by the formula called trigonometric +form: + +![Polar Form](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03de1e1b7b049880b5e4870b68a57bc180ff6ce) + +Using Euler's formula this can be written as: + +![Euler's Form](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a087c772212e7375cb321d83fc1fcc715cd0ed2) + ## Basic Operations ### Adding