Problem 19 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 4-4 i $$

Step-by-Step Solution

Answer

Polar form: \[ 4\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \] Exponential form: \[ 4\sqrt{2} e^{-i\frac{\pi}{4}} \]

Step 1: Identify the Complex Number

The given complex number is 4 - 4i with a real part of 4 and an imaginary part of -4.

Step 2: Plot the Complex Number

To plot 4 - 4i in the complex plane, locate the point (4, -4) where 4 is on the real axis (horizontal axis) and -4 is on the imaginary axis (vertical axis).

Step 3: Calculate the Magnitude

The magnitude (or modulus) of the complex number is found using the formula: \[ |z| = \sqrt{a^2 + b^2} \] For 4 - 4i, \[ |4 - 4i| = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

Step 4: Calculate the Argument

The argument (or angle) of the complex number is found using the formula: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] For 4 - 4i, \[ \theta = \tan^{-1}\left(\frac{-4}{4}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \]

Step 5: Write in Polar Form

The polar form of a complex number is \[ r(\cos\theta + i\sin\theta) \] Substitute the magnitude and the argument values, \[ 4\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \]

Step 6: Write in Exponential Form

The exponential form of a complex number is \[ re^{i\theta} \] Substitute the magnitude and the argument values, \[ 4\sqrt{2} e^{-i\frac{\pi}{4}} \]

Key Concepts

complex plane

To understand complex numbers better, start by visualizing them on the complex plane. This is a two-dimensional plane where the horizontal axis represents the real part of the number, and the vertical axis represents the imaginary part.
For example, in the complex number **4 - 4i**, the real part is 4, and the imaginary part is -4. You plot this as the point (4, -4).

• Real Axis: Horizontal line representing real numbers
• Imaginary Axis: Vertical line representing imaginary numbers
• Point: Combination of both real and imaginary parts, plotted accordingly

This graphical representation helps to better understand and solve complex number problems.

polar form

The polar form is another way to represent complex numbers, using a combination of a distance from the origin (magnitude) and an angle (argument).
For the complex number **4 - 4i**:

• Magnitude, denoted as **r**, represents the distance from the origin (0,0) to the point (4, -4).
• Argument, denoted as **θ**, is the angle between the positive real axis and the line connecting the origin to the point.

In polar form, a complex number is expressed as **r(cos θ + i sin θ)**.
For **4 - 4i**, the polar form is **4√2 (cos(-π/4) + i sin(-π/4))**.

exponential form

The exponential form is a compact way to write complex numbers using Euler's formula: \( re^{iθ} \).
This representation combines the magnitude and the argument in a sleek exponential format.
For the complex number **4 - 4i**:

• Magnitude (**r**): 4√2
• Argument (**θ**): -π/4

In exponential form, **4 - 4i** becomes **4√2 e^{-iπ/4}**.
This form is very efficient for multiplication and division of complex numbers.

magnitude

The magnitude (or modulus) of a complex number is a measure of its size, calculated as the distance from the origin to the point in the complex plane.
For a complex number **a + bi**, the formula for magnitude is:
\[ |z| = \sqrt{a^2 + b^2} \]
For **4 - 4i**:

• a = 4
• b = -4
• Magnitude: \(\sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)

Understanding the magnitude helps in converting to both polar and exponential forms.

argument

The argument of a complex number is the angle formed with the positive real axis, measured in radians.
To find the argument of a complex number **a + bi**, use the formula:
\[ θ = \tan^{-1}(\frac{b}{a}) \]
For **4 - 4i**:

• a = 4
• b = -4
• Argument: \(\tan^{-1}(\frac{-4}{4}) = \tan^{-1}(-1) = -\frac{π}{4} \)

The argument is crucial for representing the complex number in polar and exponential forms.