Name: Representation of a Complex Number
Uploaded: 2019-07-19
Duration: 6 min 37 s
Description: Learn how to express a complex number and represent it geometrically. Also, understand the concept of modulus and the argument of a complex number.

Question

Convert the complex numbers in polar form and also in exponential form.

$z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i}$

Accepted Answer

Let $z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i}$ be the given complex number

$z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i} \times \frac{5 - \sqrt{3} i}{5 - \sqrt{3} i}$

$= \frac{10 - 2 \sqrt{3} i + 30 \sqrt{3} i - 18 i^{2}}{25 - 3 i^{2}}$

$= \frac{10 + 28 \sqrt{3} i + 18}{25 + 3}$

$= \frac{28 + 28 \sqrt{3} i}{28}$

$∴ z = 1 + \sqrt{3} i = r (\cos θ + i \sin θ)$

This is of the form $a + b i$ , where $a = 1, b = \sqrt{3}$

$∴ r = |z| = \sqrt{1^{2} + (\sqrt{3})^{2}}$

$= 2$

To find the amplitude $θ$ :

$\cos θ = \frac{a}{r} = \frac{1}{2} > 0$ and $\sin θ = \frac{b}{r} = \frac{\sqrt{3}}{2} > 0$

$∴ θ$ belongs to the first quadrant.

$\Rightarrow θ = \frac{π}{3}$ $\dots [∵ \cos \frac{π}{3} = \frac{1}{2} and \sin \frac{π}{3} = \frac{\sqrt{3}}{2}]$

$∴$ Polar form of $z =$ $2 (\cos \frac{π}{3} + i \sin \frac{π}{3})$

and the exponential form of $z = r e^{i θ}$ , where $θ$ is the argument.

$= 2 e^{i (\frac{π}{3})} = 2 e^{\frac{π}{3} i}$

Convert the complex numbers in polar form and also in exponential form.

$z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i}$

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Convert the complex numbers in polar form and also in exponential form. z=2+63i5+3i

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Convert the complex numbers in polar form and also in exponential form.

$z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i}$