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Chapter 6: Problem 14

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$2-2 i$$

Short Answer

Expert verified
The complex number plotted on the Argand Plane is a point in the fourth quadrant, at (2, -2). In polar form, it can be expressed as either \(2 \sqrt{2}(\cos(-45^\circ) + i * \sin(-45^\circ))\) or \(2 \sqrt{2}(\cos(-\蟺/4) + i * \sin(-\蟺/4))\).

Step by step solution

01

Identify the real and imaginary parts

The complex number is \(2 - 2i\). In this number, 2 is the real part (a) and -2 is the imaginary part (b). Implies a=2, b=-2.
02

Plot the point on the Argand Plane

Plot the point (2, -2) on the Cartesian plane (Argand Plane). The x-coordinate corresponds to the real part (2), and the y-coordinate corresponds to the imaginary part (-2). It's a point in the fourth quadrants.
03

Calculate the modulus (r)

Now we calculate the modulus or absolute value of the complex number using the formula \( r = |z| = \sqrt{a^2 + b^2} \), which will give us \( r = \sqrt{2^2 + (-2)^2} = 2 \sqrt{2} \)
04

Calculate the argument (胃)

Next, we find the argument or angle using the formula \( 胃 = atan2(b, a) \). Here, it is \( atan2(-2, 2) = -45^\circ \) if deg is preferred, and \( -蟺/4 \) if rad is preferred.
05

Write the polar form

Combining modulus and argument, we write the complex number in polar form as \( r (\cos(胃) + i*sin(胃)) = 2 \sqrt{2} (\cos(-45^\circ) + i * \sin(-45^\circ)) \) or \( 2 \sqrt{2}(\cos(-\蟺/4) + i * \sin(-\蟺/4)) \)

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