Introduction to Complex Numbers

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Representation Methods

Representation of a complex number

The methods are:

Cartesian representation
$z=x+iy$
Polar representation
$z=pe^{i\theta}$

Here:

z=pe^{i\theta}

Here:

$x = p\cos{\theta}$: real part
$y=p\sin{\theta}$: imaginary part
$p=\sqrt{x^{2} + y^{2}}$: modulus
$\theta=\tan^{-1}{(\frac{y}{x})}$: arg angle

Euler’s Formula

For $x\in\mathbb{R}$ :

e^{ix} = \cos{x} + i\sin{x}

Euler’s Identity

One of the most beautiful equations in mathematics.

e^{i\pi} + 1 = 0

Written by

Sahithyan K.

November 3, 2026 1 min read

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