Q18P - Mathematical Methods in Physical Sciences

Question

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt{\mathbf{i}}$.

Step-by-Step Solution

Answer

The value of $\sqrt{i} is \frac{\pm \left(1+i\right)}{\sqrt{2}}$.

The graph used in this question to find the answer is shown below:

Step 1: Given Information

The given expression is $\sqrt{i}$.

Step 2: Definition of Complex Number

Complex numbers have both real numbers and imaginary numbers in them; a complex  can be written in the form of: 

 

z=a+ib

 

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{\mathbf{-}\mathbf{1}}$

Step 3: Find the value of r and θ

The Complex number is in the form 0+i .

 

x=0,y=1

 

The polar coordinates of the point are in the form of $z=r{e}^{i\theta }$  .

 

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{2},or \frac{5\pi }{2},\frac{9\pi }{2},.... \end{gathered}$$

 

The equation $z=r{e}^{i\theta }$can also be written in another form.

 

    $$\begin{gathered} z^{\frac{1}{n}}={\left(r{e}^{i\theta }\right)}^{\frac{1}{n}} \\ {z}^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{\frac{i\theta }{n}} \\ {z}^{\frac{1}{n}}=\sqrt[n]{r}\left(\cos \frac{\theta }{n}+i \sin \frac{\theta }{n}\right) \left(1\right) \end{gathered}$$                                                   

 

When n=2, the equation becomes the 2nd root of the complex number.

 $$\begin{gathered} z^{\frac{1}{2}}=r^{\frac{1}{2}}{e}^{\frac{i\theta }{2}} \\ \theta =\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},......... \end{gathered}$$

Step 4: Plotting the polar coordinate points on the graph

It is clear from the above graph that the points $\left(1,\frac{\pi }{4}\right)$and the point $\left(1,\frac{9\pi }{4}\right)$ are the same.

 

The radius of the circle is 1  and equally spaced $\pi$ apart.

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{4},\frac{5\pi }{4} \end{gathered}$$

 

Hence, the value of $\sqrt{i}=\frac{\pm \left(1+i\right)}{\sqrt{2}}$