Problem 22 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 2+\sqrt{3} i $$

Step-by-Step Solution

Answer

Polar form: \sqrt{7}(\cos\tan^{-1}\left(\frac{\sqrt{3}}{2}\right) + i\sin\tan^{-1}\left(\frac{\sqrt{3}}{2}\right)).Exponential form: \sqrt{7} e^{i\tan^{-1}\left(\frac{\sqrt{3}}{2}\right)}.

Step 1 - Identify the components

Given the complex number: oindent$$ 2 + \sqrt{3} i $$ oindent Real part \( (a) = 2 \) and Imaginary part \( (b) = \sqrt{3} \)

Step 2 - Plot on the complex plane

In the complex plane, plot the point \((2, \sqrt{3})\): oindent The horizontal axis represents the real part (2) and the vertical axis represents the imaginary part (\sqrt{3}).

Step 3 - Calculate the modulus

The modulus \( r \) is the distance from the origin to the point \((2, \sqrt{3})\) and is given by: oindent \[ r = \sqrt{a^2 + b^2} = \sqrt{2^2 + (\sqrt{3})^2} = \sqrt{4 + 3} = \sqrt{7} \]

Step 4 - Calculate the argument

The argument \( \theta \) is the angle with the positive real axis and can be computed using: oindent \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

Step 5 - Convert to polar form

The polar form of a complex number is given by: oindent $$ r(\cos\theta + i\sin\theta) $$ oindent Using the modulus \( r = \sqrt{7} \) and argument \( \theta = \tan^{-1}\left(\frac{\sqrt{3}}{2}\right) \), the polar form is: oindent \[ \sqrt{7}(\cos\tan^{-1}\left(\frac{\sqrt{3}}{2}\right) + i\sin\tan^{-1}\left(\frac{\sqrt{3}}{2}\right)) \]

Step 6 - Convert to exponential form

The exponential form of a complex number is given by: oindent $$ re^{i\theta} $$ oindent Using the modulus \( r = \sqrt{7} \) and argument \( \theta = \tan^{-1}\left(\frac{\sqrt{3}}{2}\right) \), the exponential form is: oindent \[ \sqrt{7} e^{i\tan^{-1}\left(\frac{\sqrt{3}}{2}\right)} \]

Key Concepts

Complex Plane

Complex numbers are represented in the complex plane. The complex plane is similar to the Cartesian plane, but here we plot complex numbers. The horizontal axis (x-axis) represents the real part of the number, while the vertical axis (y-axis) represents the imaginary part. For example, if we have the complex number \(2 + \sqrt{3}i\), it is represented as the point (2, \(\sqrt{3}\)) in the complex plane. This helps in visualizing complex numbers geometrically.

Polar Form

Complex numbers can also be expressed in polar form, which is very useful, especially in calculations involving multiplication and division. The polar form uses the modulus (distance from origin) and the argument (angle with the positive real axis).
The polar form of a complex number \( a + bi \) is written as:
\[ r(\cos\theta + i\sin\theta) \] where:

• \( r \) is the modulus.
• \( \theta \) is the argument.

For the example \(2 + \sqrt{3}i\):
- Modulus = \(\sqrt{7}\)
- Argument = \(\tan^{-1}(\frac{\sqrt{3}}{2})\)
The polar form becomes:
\[ \sqrt{7}(\cos \tan^{-1}(\frac{\sqrt{3}}{2}) + i\sin \tan^{-1}(\frac{\sqrt{3}}{2})) \]

Exponential Form

In addition to the polar form, complex numbers can be expressed in exponential form, which is compact and useful for computations. The exponential form of a complex number uses Euler's formula:
\[ e^{i\theta} = \cos\theta + i\sin\theta \] This allows us to write the complex number in the form:
\[ re^{i\theta} \] For our example \(2 + \sqrt{3}i\), using the modulus \(\sqrt{7}\) and argument \(\tan^{-1}(\frac{\sqrt{3}}{2})\), the exponential form is:
\[ \sqrt{7}e^{i \tan^{-1}(\frac{\sqrt{3}}{2})} \]

Modulus

The modulus of a complex number is like the distance from the origin to the point representing the number in the complex plane. It's a measure of the size of the complex number. The modulus (\( r \)) is calculated using the formula:
\[ r = \sqrt{a^2 + b^2} \] For the example \(2 + \sqrt{3}i\):

• Real part (\( a \)) = 2
• Imaginary part (\( b \)) = \(\sqrt{3}\)

The modulus is:
\[ r = \sqrt{2^2 + (\sqrt{3})^2} = \sqrt{4 + 3} = \sqrt{7} \]

Argument

The argument of a complex number is the angle formed with the positive real axis. It helps in determining the direction of the complex number from the origin. It is usually denoted by \(\theta\). The argument is calculated using the arc tangent function (\(\tan^{-1}\)):
\[ \theta = \tan^{-1}(\frac{b}{a}) \] For the example \(2 + \sqrt{3}i\):

• Real part (\( a \)) = 2
• Imaginary part (\( b \)) = \(\sqrt{3}\)

The argument is:
\[ \theta = \tan^{-1}(\frac{\sqrt{3}}{2}) \] Understanding this angle is crucial in both the polar and exponential forms.