Problem 20 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ -4 $$

Step-by-Step Solution

Answer

The polar form of -4 is \( [4 (-cos(\pi) - i.sin(\pi))]\) or in radians, it is \( [4(\cos(\pi) + i.sin(\pi))]\).

Step 1: Plotting the complex number

Plot the complex number on the complex plane (also known as an Argand diagram). This is similar to the Cartesian coordinate system where the horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number \( -4 \) is purely real and has no imaginary part, so it falls on the real axis at the point \(-4,0\).

Step 2: Converting to polar form

The polar form of a complex number is given by \( r(\cos \theta + i\sin \theta)\), where \( r \) is the modulus, and \( \theta \) is the argument or phase. Here, \( r \) is the distance from the origin to the point, which is 4, and \( \theta \) is the angle the line connecting the point and the origin makes with the positive real axis. Since the point is on the negative real axis, \( \theta \) is \( \pi \) (or \( 180^\circ \) if expressed in degrees).

Step 3: Writing the final answer

Write the complex number -4 in polar form using the values of r and \( \theta \) obtained in the previous step. The polar form is \( r(\cos \theta + i\sin \theta)\). Placeholder \( i \) is used to denote that the value is imaginary, however, in this case it is not required because the imaginary part is 0.

Key Concepts

Plotting Complex Numbers

Understanding how to plot complex numbers on a coordinate system helps to visualize and perform operations with these numbers. A complex number like \( -4 \) may seem simple, but plotting it reveals its position relative to other numbers. Imagine a standard X-Y graph. Instead of X and Y, we use the real and imaginary axes.

In this exercise, the number \( -4 \) has no imaginary component, which means it sits on the real (horizontal) axis to the left of the origin. To plot it, you simply move 4 units to the left from the origin. This plotting is the first step in understanding the complex number's properties before converting it to polar form.

Argand Diagram

An Argand diagram is where we plot complex numbers, akin to a Cartesian plane. It's a crucial tool for visualizing complex numbers. The horizontal axis represents the real part of complex numbers, and the vertical axis represents the imaginary part.\
\
\When we plot \( -4 \) on an Argand diagram, we recognize that there's no vertical displacement since the imaginary part is zero; the point lies entirely on the real axis. The diagram can then be used to analyze the complex number further and convert it to polar form.

Polar Coordinates

Polar coordinates offer a different perspective for representing the position of a point, which is essential for polar forms of complex numbers. They use a radius and an angle to indicate location. In the context of complex numbers, the radius is the modulus, and the angle is the argument.\
\
\In polar coordinates, our number \( -4 \) would have a radius (modulus) of 4, because it's four units away from the origin. However, to fully describe its position, you also need the argument (angle), which tells us how the number is oriented with respect to the positive real axis.

Modulus and Argument

These two terms are crucial in transitioning from the rectangular (Cartesian) form to the polar form of a complex number. The modulus of a complex number is its absolute distance from the origin. It's like how far you'd travel in a straight line to reach the point from the center of the graph. For the number \( -4 \) the modulus is 4.

The argument is slightly trickier; it's the angle made with the positive real axis, moving counter-clockwise. Think of it as the direction you're facing if you stood at the origin and looked directly towards our number. Since \( -4 \) is directly to the left of the origin, the argument would be \( \pi \) radians (or 180 degrees). In summary, understanding these two components is the gateway to mastering polar representation of complex numbers.