Q6P - Mathematical Methods in Physical Sciences

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.

${\left(\frac{1+i}{1-i}\right)}^{\mathbf{2}}$ .

Step-by-Step Solution

Answer

The complex number can be written as $e^{i\mathrm{\pi }}$ .

 

The graph is mentioned below:

Step 1: Given Information

The complex number is ${\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)}^2$ .

Step 2: Definition of the Complex number.

A complex number is a number that has real and imaginary numbers and is written in the form as:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}{}_{\mathbf{\iota }}\mathit{b}$ where a and b are real, and z is the complex number.

Step 3: Find the values

$\theta$Factorize the complex number as:

 $$\begin{gathered} z={\left(\frac{1+i}{1-i}\right)}^2 \\ =\left(\frac{{\left(1+i\right)}^2}{{\left(1-i\right)}^2}\right) \\ =\frac{1+2i-1}{1-2i-1} \\ =-1 \end{gathered}$$

 

Find x and y:

 $$\begin{gathered} x=-1 \\ y=0 \end{gathered}$$

 

Find the value of r .

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{1+0} \\ =1 \end{gathered}$$ 

 

Find the value of $\theta$ .

 $$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{x}{r}\right) \\ \theta ={\cos }^{-1}\left(-\frac{1}{1}\right) \\ =\theta \\ {\left(\frac{1+i}{1-i}\right)}^2=e^{i\mathrm{\pi }} \end{gathered}$$

 

 The complex number will be written as $e^{i\mathrm{\pi }}$ .

 

The graph is mentioned below as: