5.1 Cartesian form (Geometric Representation)

5.1 Cartesian form (Geometric Representation)
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation

11.1 Properties of cube roots of unity

Every complex number $z=a+ib$ can be represented by a point on the cartesian plane known as complex plane by ordered pair $(a,b)$ as shown in figure.

The modulus of complex number $z=a+ib$ is equal to the real number $\sqrt {{a^2} + {b^2}} $ and denoted by $\left| z \right|$. In figure, length $OP$ is the modulus of the complex number.

If $(a,b) \ne (0,0)$, then $\theta = {\tan ^{ - 1}}\left( {\frac{b}{a}} \right)$ is called the argument or amplitude of $z$ written as $\arg z$. The argument of the complex number is not unique. $2n\pi + \theta $ ($n \in I$) is also the argument of $z$ for various values of $n$. The unique value of $\theta $ such that $ - \pi < \theta \leqslant \pi $ is called the principal value of the argument unless otherwise stated, amp$z$ implies the principal value of the argument.

By specifying the modulus and argument, a complex number is defined completely.