Problem 1 - Algebra and Trigonometry

Question

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ 4 i $$

Step-by-Step Solution

Answer

The complex number is graphed at (0, 4), and its modulus is 4.

Step 1: Understanding Complex Numbers

A complex number is in the form \( a + bi \) where \( a \) is the real part and \( bi \) is the imaginary part. In the given number, \( 4i \), the real part \( a = 0 \) and the imaginary part \( b = 4 \).

Step 2: Graphing the Complex Number

To graph the complex number \( 4i \), plot the point on the complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. Since \( a = 0 \) and \( b = 4 \), plot the point at \( (0, 4) \).

Step 3: Finding the Modulus

The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Here, \( a = 0 \) and \( b = 4 \), so the modulus is \( \sqrt{0^2 + 4^2} = \sqrt{16} = 4 \).

Key Concepts

Imaginary Number

A complex number can be broken into two parts: the real part and the imaginary part. The imaginary part includes the unit 'i', which is the square root of -1. Imaginary numbers are essential as they allow equations that have no real solutions to be solved. For instance, the number 4i consists purely of an imaginary part, where the real number part is zero.

• The real part: This is the 'a' in the complex number form of \( a + bi \). In 4i, it is 0.
• The imaginary part: This is the 'bi' in the complex number, where 'i' is the imaginary unit. For 4i, 'b' equals 4.

Imaginary numbers often seem abstract, but they serve vital functions in electrical engineering, physics, and control theory. Understanding them as shifts along the vertical axis of a complex plane can make them more relatable.

Modulus of a Complex Number

When you want to understand how far a complex number is from the origin on the complex plane, you calculate its modulus. Think of this like finding the length of the hypotenuse in a right triangle where one side is the real part and the other is the imaginary part. For a complex number \( a + bi \), the formula for the modulus is given by:
\[ \text{Modulus} = \sqrt{a^2 + b^2} \]
Let's break it down for the complex number 4i:

• Substitute the values: Here, \( a = 0 \) and \( b = 4 \), so we use \( \sqrt{0^2 + 4^2} \).
• Simplify: This gives us \( \sqrt{16} \), which equals 4.

The modulus of a complex number gives its absolute value or magnitude, providing insight into its size on the complex plane.

Graphing Complex Numbers

Graphing complex numbers helps visualize them using the complex plane, which resembles a coordinate system but with real numbers on the x-axis and imaginary numbers on the y-axis.
To graph the complex number 4i, you'll follow these steps:

• Identify the real and imaginary components. For 4i, the real part is 0, and the imaginary part is 4.
• Locate the point \((0, 4)\) on the complex plane: Exactly where the real part is 0 and imaginary part is 4.

Plotting this point gives a visual representation of 4i, situated four units up from the origin on the imaginary (vertical) axis. This visual approach can make working with complex numbers much more intuitive.