Question

Find the polar and exponential forms of the following complex numbers: \((a) \frac{3}{2}+\frac{3 \sqrt{3}}{2} i\) (b) \(4(\sqrt{3}+i)\)

Short answer

(a) Polar: \(3\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)\), Exponential: \(3e^{i \frac{\pi}{3}}\). (b) Polar: \(8\left(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6}\right)\), Exponential: \(8e^{i \frac{\pi}{6}}\).

Step-by-step solution

Identify the Real and Imaginary Parts

For the complex number \( z = \frac{3}{2} + \frac{3 \sqrt{3}}{2} i \), the real part is \( a = \frac{3}{2} \) and the imaginary part is \( b = \frac{3 \sqrt{3}}{2} \).

Calculate the Magnitude

Calculate the magnitude of \( z \) using the formula \( |z| = \sqrt{a^2 + b^2} \). Substitute \( a \) and \( b \): \[ |z| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{3 \sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{27}{4}} = \sqrt{\frac{36}{4}} = 3. \]

Calculate the Argument of the Complex Number

The argument \( \theta \) is found using \( \tan \theta = \frac{b}{a} \). Here, \( \tan \theta = \frac{\frac{3\sqrt{3}}{2}}{\frac{3}{2}} = \sqrt{3} \), so \( \theta = \frac{\pi}{3} \).

Express in Polar Form

The polar form of a complex number is \( r(\cos \theta + i \sin \theta) \). Using \( r = 3 \) and \( \theta = \frac{\pi}{3} \), the polar form is: \[ 3 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right). \]

Express in Exponential Form

The exponential form is \( re^{i\theta} \). Substituting \( r = 3 \) and \( \theta = \frac{\pi}{3} \), the exponential form is: \[ 3 e^{i \frac{\pi}{3}}. \]

Repeat Steps 1-5 for Second Complex Number

For the complex number \( z = 4(\sqrt{3} + i) \), identify the parts: Real part: \( a = 4 \sqrt{3} \), Imaginary part: \( b = 4 \). Calculate the magnitude: \[ |z| = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8. \] Calculate the argument: \( \tan \theta = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}} \), so \( \theta = \frac{\pi}{6} \). The polar form is: \[ 8 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right). \] The exponential form is: \[ 8 e^{i \frac{\pi}{6}}. \]

Key concepts

Polar Form

The Polar Form of a complex number beautifully combines its magnitude and direction. The number is described in terms of its distance from the origin, called the magnitude, and the angle it forms with the positive real axis, known as the argument. Imagine plotting a complex number on a plane: the Polar Form is like tracing out a path that leads directly to the point, considering both its length and direction.

For any complex number represented as \( z = a + bi \), the Polar Form is expressed as:

• \( z = r(\cos \theta + i \sin \theta) \)

Here, \( r \) denotes the magnitude \( |z| \), calculated using the formula \( \sqrt{a^2 + b^2} \). \( \theta \), the argument, can be found through \( \tan \theta = \frac{b}{a} \). This expression gives a complete description of a complex number's position in the plane. Such a transformation helps in simplifying multiplication and division of complex numbers, making it incredibly useful in both mathematical and real-world applications.

To take the example from the original exercise, for the complex number \( \frac{3}{2} + \frac{3 \sqrt{3}}{2} i \), we calculated a magnitude \( r = 3 \) and an argument \( \theta = \frac{\pi}{3} \). Thus, the Polar Form becomes \( 3\left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\right) \). This representation not only looks cleaner but also informs you about how far and in which direction to go from the origin on the complex plane.

Exponential Form

The Exponential Form of complex numbers offers a compact and elegant way to express these numbers, especially useful for dealing with powers and roots. This form leverages Euler's formula, which states that for any real number \( \theta \),

• \( e^{i\theta} = \cos \theta + i \sin \theta \)

Based on this relationship, any complex number in Polar Form \( r(\cos \theta + i \sin \theta) \), can be conveniently represented as

• \( re^{i\theta} \)

In this expression, \( r \) is the magnitude and \( \theta \) is the argument, as in the Polar Form.

This explanatory prowess of the Exponential Form simplifies many operations, such as multiplication and division of complex numbers, which boils down to multiplying/dividing their magnitudes and adding/subtracting their angles. Additionally, Exponential Form elegantly covers the exploration of powers and roots through De Moivre's Theorem. This makes it an essential tool in various scientific computations, especially in fields such as engineering and physics.

Referring back to the exercise, the given complex number \( \frac{3}{2} + \frac{3\sqrt{3}}{2} i \), translated into Exponential Form becomes \( 3e^{i \frac{\pi}{3}} \). This concise expression is powerful, reducing time and effort in complex operations and offering deeper insights into the nature of the number.

Magnitude and Argument

At the heart of both the Polar and Exponential Forms is the understanding of two crucial concepts: Magnitude and Argument. They form the bridge that connects the rectangular coordinate system (real and imaginary parts) to these alternative forms, revealing a deeper structure of complex numbers.

**Magnitude**: Simply put, the magnitude (or modulus) of a complex number is its "distance" from the origin when plotted on the complex plane. It is akin to the length of the vector that the complex number represents. The formula used to calculate this is:

• \( |z| = \sqrt{a^2 + b^2} \)

where \( a \) and \( b \) are the real and imaginary components, respectively.

**Argument**: The argument provides the "direction" of the complex number. It is the angle formed between the positive real axis and the line representing the complex number in the plane, calculated using the arctangent function:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

The argument must always be considered in the correct quadrant as determined by the signs of \( a \) and \( b \).

In our exercise, these have specific numerical values for the examples given. For example, the complex number \( \frac{3}{2} + \frac{3\sqrt{3}}{2} i \) has a magnitude of 3, and an argument of \( \frac{\pi}{3} \). These components play a fundamental role in transforming the complex number into both Polar and Exponential Forms, enabling a clearer understanding and easier operations with complex numbers.