Problem 16 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1-\sqrt{3} i $$

Step-by-Step Solution

Answer

Polar: 2(\cos(-\frac{\pi}{3}) + i\sin(-\frac{\pi}{3})), Exponential: 2 e^{-i \frac{\pi}{3}}.

Step 1 - Determine the real and imaginary parts

Let’s identify the real and imaginary parts of the complex number. Given the complex number is \(1 - \sqrt{3}i\). Here, the real part (a) is 1 and the imaginary part (b) is -\(\sqrt{3}\).

Step 2 - Plot the number in the Complex Plane

Plot the point (1, -√3) on the complex plane. This corresponds to moving 1 unit to the right on the real axis and \(-\sqrt{3}\) units downward on the imaginary axis.

Step 3 - Calculate the modulus (r)

The modulus of the complex number is calculated as: \[ r = \sqrt{a^2 + b^2} \] Substituting the values, we get: \[ r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

Step 4 - Calculate the argument (θ)

The argument is calculated using the arctangent function: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] Substituting the values, we get: \[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3}) \]. Since the number is in the fourth quadrant, θ = \(-\frac{\pi}{3}\).

Step 5 - Write the polar form

The polar form of a complex number is given by: \[ z = r (\cos \theta + i \sin \theta) \] Substituting the values, we get: \[ z = 2 (\cos(-\frac{\pi}{3}) + i \sin(-\frac{\pi}{3})) \]

Step 6 - Write the exponential form

The exponential form of a complex number is given by: \[ z = r e^{i\theta} \] Substituting the values, we get: \[ z = 2 e^{-i\frac{\pi}{3}} \]

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane. Here, a complex number is represented graphically. The horizontal axis represents the real part of the number while the vertical axis represents the imaginary part. For example, the number \(1 - \sqrt{3}i\) can be plotted by moving 1 unit to the right (along the real axis) and \(-\sqrt{3}\) units downwards (along the imaginary axis).

• Real Axis: Horizontal axis
• Imaginary Axis: Vertical axis
• Coordinate: (Real, Imaginary)

Polar Form

Polar form expresses a complex number in terms of modulus and argument. This form is very useful in understanding the magnitude and direction of a complex number.

• Modulus (r): Distance from origin
• Argument (θ): Angle with the positive real axis

For the number \(1 - \sqrt{3}i\), we first calculate the modulus as \(r = 2\) and then the argument as \(\theta = -\frac{\pi}{3}\). The polar form is then: \[ z = 2(\cos(-\frac{\pi}{3}) + i \sin(-\frac{\pi}{3})) \]

Exponential Form

The exponential form of a complex number is another way to represent it, using Euler's formula. It ties together the polar form with an exponent involving the imaginary unit, i.

• Formula: \(z = re^{i\theta}\)
• Relationship: Links the magnitude and phase angle

For our example, with modulus 2 and argument \(-\frac{\pi}{3}\), the exponential form becomes: \[ z = 2e^{-i\frac{\pi}{3}} \]

Modulus

The modulus of a complex number is its distance from the origin on the complex plane. It's calculated using the Pythagorean theorem: \[ r = \sqrt{a^2 + b^2} \].

• For \(a = 1\) and \(b = -\sqrt{3}\): \( r = 2 \)
• Interpretation: Magnitude of the number

Argument

The argument of a complex number is the angle it makes with the positive real axis. It's calculated using the arctangent function: \[ \theta = \tan^{-1}(\frac{b}{a}) \].

• Placement: Fourth quadrant
• Value: \(\theta = -\frac{\pi}{3}\)

It helps in determining the direction of the complex number in the plane.