# Chapter 01: Complex Numbers

Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN.

A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.

For $z_1=(x_1,y_1)$, $z_2=(x_2,y_2)$, we put

- $z_1+z_2= (x_1+x_2, y_1+y_2)$
- $z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$

The set $\mathbb{R}^2$ with operation defined above is denoted by $\mathbb{C}$.

### Contents and summary

- Complex numbers
- Properties of complex numbers
- The Argand's diagram
- De Moivre's theorem
- Roots of the complex numbers
- Basic elementary functions
- Logarithmic functions
- Inverse hyperbolic functions
- Inverse trigonometric functions
- Complex power
- Summation of series