Q8P - Mathematical Methods in Physical Sciences

Question

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

1.6-2.7i .

Step-by-Step Solution

Answer

The complex number becomes $3.138e^{-1.035i}$.

 

The graph is mentioned below:

 

Step 1: Given Information

Thecomplex number is 1.6-2.7i .

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:

z = a + ib where a and b are real, and z is the complex number.

Step 3: Find the values

The complex number is 1.6-2.7i .

 

Find x and y:

$$\begin{gathered} x=1.6 \\ y=-2.7 \end{gathered}$$ 

 

Find the value of r :

 $$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(1.6\right)}^2+{\left(2.7\right)}^2} \\ =3.138 \end{gathered}$$

 

Find the value of $\theta$ :

 $$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{x}{r}\right) \\ \theta ={\cos }^{-1}\left(\frac{1.6}{3.138}\right) \\ =-1.035 \\ z=3.138e^{-1.035i} \end{gathered}$$

 

The complex number becomes $3.138e^{-1.035i}$.

 

The graph is mentioned below: