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

Express the complex number $\frac{1}{(1 + i)}$ in polar form and exponential form.

Accepted Answer

Given complex number is $\frac{1}{(1 + i)}$

$\Rightarrow \frac{1}{(1 + i)} = \frac{1}{(1 + i)} \times \frac{1 - i}{1 - i} = \frac{1}{2} - \frac{1}{2} i$

Let, $z = \frac{1}{2} - \frac{1}{2} i = a + i b$

Here, $a = \frac{1}{2}, b = - \frac{1}{2}$ and $z = \frac{1}{2} - \frac{1}{2} i$ belongs to the fourth quadrant.

To find the polar form and exponential form of a complex, we need to find the modulus and argument of the complex number.

$∴$ Modulus of $z = |z| = \sqrt{{(\frac{1}{2})}^{2} + {(- \frac{1}{2})}^{2}}$

$\Rightarrow |z| = \sqrt{\frac{1}{2}}$

$\Rightarrow |z| = \frac{1}{\sqrt{2}}$

Argument $θ$ of $z = \frac{1}{2} - \frac{1}{2} i$ is given by

$θ = - \tan^{- 1} (|\frac{- 1}{2} \times \frac{2}{1}|)$ $∵ θ = - \tan^{- 1} (|\frac{Im (z)}{Re (z)}|),$ when $z$ belongs to $4^{th}$ quadrant

$\Rightarrow θ = - \tan^{- 1} (1)$

$\Rightarrow θ = - \frac{π}{4}$

Since, polar form a complex number $z$ is defined as $|z| (\cos θ + i \sin θ)$

$∴$ Polar form of $z = \frac{1}{\sqrt{2}} (\cos (- \frac{π}{4}) + i \sin (- \frac{π}{4})) \Rightarrow \frac{1}{\sqrt{2}} (\cos \frac{π}{4} - i \sin \frac{π}{4})$

Exponential form of complex number is defined $z = |z| e^{i θ}$ , where $|z|$ is the modulus of the complex number and $θ$ is the argument of $z$ .

$∴$ Exponential form of $z = \frac{1}{\sqrt{2}} e^{i (- \frac{π}{4})} \Rightarrow \frac{1}{\sqrt{2}} e^{- i \frac{π}{4}}$

Hence, polar and exponential form of the given complex number is $\frac{1}{\sqrt{2}} (\cos \frac{π}{4} - i \sin \frac{π}{4})$ and $\frac{1}{\sqrt{2}} e^{- i \frac{π}{4}}$ respectively.

Express the complex number $\frac{1}{(1 + i)}$ in polar form and exponential form.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Express the complex number 1(1+i) in polar form and exponential form.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Express the complex number $\frac{1}{(1 + i)}$ in polar form and exponential form.