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Chapter 6: Problem 15

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$-4 i$$

Short Answer

Expert verified
The polar form of the complex number \(-4i\) is \(4 (\cos(-\pi/2) + i \sin(-\pi/2))\).

Step by step solution

01

Identify the Complex Number

The complex number is \(-4i\), which can be rewritten as \(0-4i\), meaning that the real part is \(0\) and the imaginary part is \(-4\).
02

Plot the Complex Number

To plot it, make a dot at the point \((0,-4)\) on the complex plane.
03

Calculate the Modulus \(r\)

The modulus \(r\) of a complex number \(z=a+bi\) is given by \(\sqrt{a^2 + b^2}\). In this case, \(r = \sqrt{0^2 + (-4)^2} = 4\).
04

Calculate the Argument \(\Theta\)

The argument \(\Theta\) of a complex number \(z=a+bi\) in radians is calculated as \(\tan^{-1}(b/a)\) or using the signs of \(a\) and \(b\) to determine the correct quadrant.. However, since \(a=0\) in our case, we cannot use \(\tan^{-1}(b/a)\). Instead, we notice that the complex number lies on the negative imaginary axis, which means \(\Theta\) must be \(-\pi/2\) radians or \(-90^\circ\).
05

Write the Complex Number in Polar Form

The polar form of the complex number is \(r(\cos \Theta + i \sin \Theta)\). We substitute \(r = 4\) and \(\Theta = -\pi/2\) into this formula. Thus, the polar form is \(4(\cos(-\pi/2) + i \sin(-\pi/2))\).

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