Chapter 8, Problem 34RE

a.

To determine

Graph the complex number in the complex plane.

  $-20$

Answer to Problem 34RE

The graph of complex number in the complex plane $-20$ is,

  

Explanation of Solution

Given:

  $-20$

Concept Used:

Desmos Graph.

According to the given details,

  $z=-20$

  

Conclusion:

Hence, graph is

  

b.

To determine

The modulus and argument.

  $-20$

Answer to Problem 34RE

The modulus and argument of $-20$ is $r=20,\theta =\pi$

Explanation of Solution

Given:

  $-20$

Concept Used:

Modulus and argument: $r=\sqrt{x^2+y^2}$

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

The given complex number is $z=-20$ ,

So, write it in the polar form,

  $z=r\left(\cos \theta +i\sin \theta \right)$

Now evaluate $r$ and $\theta$ , as know that the formula for these terms,

  $r=\sqrt{x^2+y^2}$

So, from the equation,

  $\begin{array}{l}x=-20\\ y=0\end{array}$

Substituting these values in the above formula,

  $\begin{array}{l}r=\sqrt{{\left(-20\right)}^2+{\left(0\right)}^2}\\ r=\sqrt{400+0}\\ r=20\end{array}$

Now evaluate, $\theta$

  $\begin{array}{l}\theta =\arctan \left(\frac{0}{-20}\right)\\ \theta =\arctan \left(0\right)\\ \theta =\pi \end{array}$

Conclusion:

Hence the modulus and argument of $-20$ is $r=20,\theta =\pi$

c.

To determine

The polar form of $-20$ .

Answer to Problem 34RE

The polar form of $-20$ is $20\left(\cos \pi +i\sin \pi \right)$

Explanation of Solution

Given:

  $-20$

Concept Used:

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

  $z=-20$

In polar form,

  $\begin{array}{l}z=r\left(\cos \theta +i\sin \theta \right)\\ r=20\\ \theta =\pi \end{array}$

So now write these calculated values in standard equation of polar form,

  $z=20\left(\cos \pi +i\sin \pi \right)$

Conclusion:

Hence, the polar form of $-20$ is $20\left(\cos \pi +i\sin \pi \right)$