Question

First simplify each of the following numbers to the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form or to the$\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\frac{\mathbf{1}}{\mathbf{1}\mathbf{+}\mathbf{i}}$.

Short answer

The complex number becomes $\frac{\sqrt{2}}{2}{e}^{-i\frac{\pi }{4}}$.

Step-by-step solution

Given Information

The complex number is $\frac{1}{1+i}$.

Definition of the Complex number

A complex number is a number that is made up of both real and imaginary numbers.

Find the value

Rationalize the complex number.

$\begin{array}{rcl}\frac{1}{1+i}& =& \frac{1}{1+i}·\frac{1-i}{1-i}\\ & =& \frac{1-i}{1^2-i^2}\\ & =& \frac{1-i}{2}\\ & =& \frac{1}{2}-\frac{i}{2}\\ & & \end{array}$

Find x and y

$$\begin{gathered} x=\frac{1}{2} \\ y=\frac{-1}{2} \end{gathered}$$

Find the value of r .

$\begin{array}{rcl}r& =& \sqrt{x^2+y^2}\\ & =& \sqrt{{\left(\frac{1}{2}\right)}^2+{\left(\frac{-1}{2}\right)}^2}\\ & =& \frac{1}{\sqrt{2}}\\ & =& \frac{\sqrt{2}}{2}\\ & & \end{array}$

Find the value of $\theta$$\theta$.

$\begin{array}{rcl}\theta & =& \arctan \left(\frac{\frac{-1}{2}}{\frac{1}{2}}\right)\\ & =& \arctan (-1)\\ & =& \frac{-\pi }{4}\\ & & \end{array}$

$\frac{1}{1+i}=\frac{\sqrt{2}}{2}{e}^{-i\frac{\pi }{4}}$

The complex number becomes $\frac{\sqrt{2}}{2}{e}^{-i\frac{\pi }{4}}$.

The graph is mentioned below.