# Python - Complex Numbers

> Procedural Languages > Python > Python - Data Type

### Table of Contents

## 1 - About

complex numbers in Python

Python uses 'j' for the imaginary unit, not 'i'.

## 2 - Articles Related

## 3 - Snippet

- Type Complex

>>> 1j 1j >>> type(1j) <class 'complex'>

- Addition

>>> (3 + 1j) + (2 + 2j) (5+3j) >>> x = 1+3j >>> x + 1 (2+3j)

## 4 - Complex Plane

The real and imaginary parts of a complex number can be interpreted as x and y coordinates in the complex plane forming a point.

- Coordinates

>>> x = 1+3j >>> x.real # real number coordinates 1.0 >>> x.imag # imaginary number coordinates 3.0

- Plot

import matplotlib.pyplot as plt L=[-2+2j,-1+2j,0+2j,1+2j,2+2j,-1+4j,0+4j,1+4j] X = [x.real for x in L] Y = [x.imag for x in L] plt.scatter(X,Y, color='red') plt.show()

- Absolute value of z = distance from the origin to the point z in the complex plane. (In Mathematics - Mathese, |z|)

>>> abs(1+2j) 2.23606797749979

## 5 - Translation

- Geometric interpretation of f (z) = z + (1+2i)?
- Increase each real coordinate by 1 and increases each imaginary coordinate by 2.
- A translation can “move” the picture anywhere in the complex plane

Translation in general:

f (z) = z + z0

where:

- z0 is a complex number.

## 6 - Arrow

## 7 - Composition (Addition)

## 8 - Multiplication

### 8.1 - Scaling

### 8.2 - Reflection

### 8.3 - By i: rotation by 90 degrees

plot({1j*z for z in L})

### 8.4 - Rotation

Rotating a complex number z means increasing its argument.

Argument of z is the angle in radians between z arrow and the x axis (1 + 0i arrow).

Euler’s formula: For any real number , is the point z on the unit circle with argument .

e = 2.718281828…

When , z = -1

Every complex number can be written in the form where:

- r is the absolute value of z
- is the argument of z

To augment the argument of z, we use exponentiation law <math>e^a * e^b = e^{a+b}</math>

- does a rotation by angle

- Rotation of 45 degrees

from math import e, pi plot({e**(45j)*z for z in L})

- Circle with a rayon of 2

r = 2 circle = 2*pi plot([r*e**(t*circle/20*1j) for t in range(20)])