Q29P - Mathematical Methods in Physical Sciences

Question

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

 $\left|e^{\mathrm{i\pi }/2}\right|$

Step-by-Step Solution

Answer

The absolute value of the expression $\left|{\mathrm{e}}^{\mathrm{i\pi }/2}\right|=1.$ 

Step 1: Given information

The given complex number is  $\left|{\mathrm{e}}^{\mathrm{i\pi }/2}\right|.$ 

Step 2: Definition of complex numbers

The numbers that are presented in the form of $\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$ where,   are real numbers and $\mathbf{\text{'}}\mathit{i}\mathbf{\text{'}}$ is an imaginary number called complex numbers.

Step 3: Use the result concluded in problem 27

 

Let the complex number be $z=e^{i\mathrm{\pi }/2}$  .

Represent the given equation as $\mathrm{R}=\left|\mathrm{z}\right|$ .

This can be written as $\mathrm{R}=\left|{\mathrm{e}}^{\mathrm{i\pi }/2}\right|$  .                                                                      

This can be written as $\mathrm{R}=\left|{\mathrm{e}}^{\mathrm{i\pi }/2}\right|$ .                           .…… (1)                                   Also $\left|e^{i\mathrm{\pi }/2}\right|=1$                                                                               ……. (2)                    

The result of  $R=\left|e^{iy}\right|$ is concluded in problem 27.

 

Compare equations (1) and (2) as:

 

R = 1

 

Hence the absolute value of the expression $\left|e^{i\mathrm{\pi }/2}\right|=1$ .