Problem 14 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ -1+i $$

Step-by-Step Solution

Answer

The polar form is \( \sqrt{2}( \cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}) ) \), and the exponential form is \( \sqrt{2} e^{i\frac{3\pi}{4}} \).

Step 1: Identify the Real and Imaginary Parts

Given the complex number ewline -1+iewline , identify the real and imaginary parts. The real part is ewline -1ewline , and the imaginary part is ewline 1ewline .

Step 2: Plot the Complex Number

To plot ewline -1+iewline in the complex plane, locate the point ewline (-1, 1)ewline , where ewline -1ewline is on the real axis and ewline 1ewline is on the imaginary axis. This point lies in the second quadrant.

Step 3: Find the Magnitude

The magnitude ewline rewline of the complex number is found using the formula:ewline \( r = \sqrt{(-1)^2 + 1^2} \)ewline This simplifies to:ewline \( r = \sqrt{1 + 1} = \sqrt{2} \)ewline

Step 4: Find the Argument

The argument ewline \( \theta \)ewline of the complex number is found using the formula:ewline \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)ewline , where ewline bewline is the imaginary part and ewline aewline is the real part.ewline In this case:ewline \( \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = \frac{3\pi}{4} \)ewline (since the point is in the second quadrant)

Step 5: Write the Polar Form

The polar form of the complex number is given by:ewline \( r ( \cos(\theta) + i\sin(\theta) ) \)ewline So, the polar form is:ewline \( \sqrt{2} ( \cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}) ) \)ewline

Step 6: Write the Exponential Form

The exponential form of the complex number is given by:ewline \( re^{i\theta} \)ewline So, the exponential form is:ewline \( \sqrt{2} e^{i\frac{3\pi}{4}} \)ewline

Key Concepts

Polar Form

Polar form is a way to represent complex numbers using their magnitude and angle relative to the positive real axis. This representation is especially useful for multiplication and division of complex numbers, as the magnitudes are multiplied or divided and the angles are added or subtracted.
To convert a complex number into polar form, you need to find its magnitude (also known as modulus) and argument (also known as the angle).

• Magnitude (r): This is the distance from the origin to the point representing the complex number in the complex plane. It's calculated using the formula \[ r = \sqrt{a^2 + b^2} \, \] where \ a \ is the real part, and \ b \ is the imaginary part.


• Argument (θ): This is the angle formed with the positive real axis. It's calculated using the inverse tangent function \[ θ = \tan^{-1}\left(\frac{b}{a}\right). \]

Using our example, for the complex number -1+i:

• Magnitude: \[ r = \sqrt{(-1)^2 + 1^2} = \sqrt{2} \]
• Argument: \[ θ = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = \frac{3\pi}{4} \] (since the point lies in the second quadrant).

Hence, the polar form is \[ \sqrt{2} \( \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right) \). \]

Exponential Form

Exponential form is another convenient way to express complex numbers, particularly helpful when dealing with complex exponentials and simplification in advanced mathematics. In this representation, Euler's formula comes in handy:
Euler's formula states \[ e^{iθ} = \cos(θ) + i\sin(θ). \]
Using this, any complex number in polar form can be expressed in exponential form.

• Exponential form: Given the polar form \[ r(\cos(θ) + i\sin(θ)), \] it translates to exponential form as \[ re^{iθ}. \]

So, for our example, the complex number -1+i in polar form was \[ \sqrt{2} \( \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right) \). \]
Translating this to exponential form, we get
\[ \sqrt{2} e^{i\frac{3\pi}{4}}. \]
This form simplifies many operations, especially those involving multiplication, division, and finding powers or roots of complex numbers.

Complex Plane

The complex plane, also called the Argand plane, is a two-dimensional plane used to visually represent complex numbers. In this plane:

• Real Axis: The horizontal axis represents the real part of complex numbers.
• Imaginary Axis: The vertical axis represents the imaginary part.

Each complex number corresponds to a unique point on this plane, determined by its real and imaginary parts. For example, the complex number -1+i is represented by the point (-1, 1), where:

• -1 is the value on the real axis (left of origin).
• 1 is the value on the imaginary axis (above origin).

Plotting complex numbers helps us see operations like addition, subtraction, and multiplication graphically. It's particularly useful when converting to polar form since the magnitude is the distance from the origin, and the argument is the angle made with the positive real axis.
Overall, understanding the complex plane offers a deep, visual insight into the nature and behavior of complex numbers.