Problem 19 - 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. $$ -3 $$

Step-by-Step Solution

Answer

The graph of -3 in a complex plane is a point at (-3,0). Its polar form is \(3(\cos\pi + i\sin\pi)\).

Step 1: Plotting the Complex Number

The real part of the complex number is -3 and there is no imaginary part. Thus, it lies on the negative x-axis when plotted on the Argand Plane (Complex plane). The point will be located at (-3,0).

Step 2: Calculating the absolute value/magnitude (r)

Next, the magnitude of a complex number \(z = a + bi\) can be calculated using the formula \(r = \sqrt{a^2 + b^2}\). In our case, a = -3 and b = 0. Therefore, the magnitude will be \(\sqrt{(-3)^2 + 0^2} = 3\).

Step 3: Calculating the Argument (Theta)

The argument \(\Theta\) (the angle made with the positive x axis) of a complex number \(z = a + bi\) can be found using the formula \(\Theta = \arctan(b/a)\). However, because our number is strictly real (-3+0i), our angle would be \(\Theta = 180^\circ\) or \(\Theta = \pi\) radians, because the point lies on the negative x-axis.

Step 4: Writing the Polar Form

The polar form of a complex number is written as \(z = r(\cos(\Theta) + i\sin(\Theta))\). Here, r = 3 and \(\Theta = \pi\) or \(180^\circ\). So, the polar form of -3 will be \(3(\cos\pi + i\sin\pi)\).

Key Concepts

Plotting Complex Numbers

Understanding how to plot complex numbers on the complex plane, also known as the Argand plane, is crucial in grasping the basics of complex numbers. A complex number is composed of a real part and an imaginary part and is often written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).
When plotting a complex number, we consider the real part as the horizontal component and the imaginary part as the vertical component. Therefore, a complex number \( a + bi \) can be represented as a point \( (a, b) \) in the two-dimensional plane. If the imaginary part is zero, the point lies on the real, or horizontal, axis. Conversely, when the real part is zero, the point resides on the imaginary, or vertical, axis.
In graphical terms, plotting a complex number is similar to plotting an ordinary Cartesian coordinate. The exercise given shows the plot of a purely real complex number at \( (-3, 0) \) on the negative x-axis, indicating there is no imaginary counterpart.

Magnitude of Complex Numbers

The magnitude, or absolute value, of a complex number measures how far the number is from the origin on the complex plane. Computationally, this is akin to finding the length of the vector from the origin to the point representing the complex number.
This magnitude is calculated using the formula \( r = \sqrt{a^2 + b^2} \), where \( r \) is the magnitude, and \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively. The magnitude is always a non-negative real number, and it is particularly simple to calculate for purely real or purely imaginary numbers because one component will be zero. In the given exercise, the magnitude of the complex number \( -3 \) is \( 3 \), reflecting the distance from the origin to the point on the negative x-axis.

Argument of Complex Numbers

The argument of a complex number is the angle the line makes with the positive x-axis, measured in a counterclockwise direction. To find this angle, we usually use the arctangent function in the formula \( \Theta = \arctan(b/a) \). However, for points lying directly on the axes, we apply specific rules: \( 0^\circ \) or \( 0 \) radians for the positive x-axis, \( \pm 180^\circ \) or \( \pm \pi \) radians for the negative x-axis, \( 90^\circ \) or \( \pi/2 \) radians for the positive y-axis, and \( -90^\circ \) or \( -\pi/2 \) radians for the negative y-axis.
For example, a complex number with a negative real part and no imaginary part, such as \( -3 \) from our exercise, has an argument of \( 180^\circ \) or \( \pi \) radians because it lies on the negative x-axis. Understanding the argument helps us see the direction in which the number is pointing in the complex plane.

Polar Form Conversion

Complex numbers can also be expressed in polar form, which uses the magnitude (r) and argument (\( \Theta \)) to represent the number. The standard equation for the polar form is \( z = r(\cos(\Theta) + i\sin(\Theta)) \).
This form is particularly useful for multiplication and division of complex numbers as well as raising them to powers or finding square roots.
The exercise solution converts the complex number -3 into polar form using the previously calculated magnitude of 3 and argument of \( \pi \) radians. As a result, we get \( 3(\cos\pi + i\sin\pi) \), which is the polar representation of the number. In essence, the polar form highlights the geometric interpretation of a complex number by focusing on its magnitude (or 'how far') and argument (or 'in which direction') from the origin.