Problem 13 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ \frac{3+4 i}{5} $$

Step-by-Step Solution

Answer

The modulus is 1, and it is plotted at (0.6, 0.8) on the complex plane.

Step 1: Write the Complex Number

The given complex number is \( \frac{3 + 4i}{5} \). To simplify it, divide both the real and imaginary parts by 5. This results in \( \frac{3}{5} + \frac{4}{5}i \).

Step 2: Plot the Complex Number on the Complex Plane

On a complex plane, the real part is plotted on the x-axis and the imaginary part on the y-axis. Here, the real part is \( \frac{3}{5} \) and the imaginary part is \( \frac{4}{5} \). Plot the point \( \left( \frac{3}{5}, \frac{4}{5} \right) \).

Step 3: Calculate the Modulus

The modulus of a complex number \( a + bi \) is calculated as \( \sqrt{a^2 + b^2} \). Substitute \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \): \[ \text{Modulus} = \sqrt{\left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = \sqrt{1} = 1 \].

Key Concepts

Understanding the Complex Plane

When working with complex numbers, it's essential to visualize them on the complex plane. The complex plane, similar to a Cartesian plane, consists of two axes. The horizontal axis, or x-axis, represents the real part of a complex number, while the vertical axis, or y-axis, represents the imaginary part. Together, these form a coordinate system where each point corresponds to a complex number.

• The real number component is placed along the x-axis.
• The imaginary number component is situated along the y-axis.

In our example, the simplified complex number is \( \frac{3}{5} + \frac{4}{5}i \). This means you will plot the point where the x-coordinate is \( \frac{3}{5} \) and the y-coordinate is \( \frac{4}{5} \). Thus, the complex plane acts as a powerful tool to graphically represent and analyze complex numbers.

Unraveling the Modulus of Complex Numbers

The modulus of a complex number is a measure of its magnitude or distance from the origin on the complex plane. Much like finding the distance between two points in standard Euclidean geometry, the formula to determine the modulus of a complex number \( a + bi \) is derived from the Pythagorean theorem:\[ \text{Modulus} = \sqrt{a^2 + b^2} \]This calculation provides the 'length' of the vector represented by the complex number. In simpler terms, it's the line connecting the point \((a, b)\) to the origin \((0, 0)\). For our complex number \( \frac{3}{5} + \frac{4}{5}i \), the modulus computation is:

• First, square each part: \( \left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2 \).
• Next, add these squares: \( \frac{9}{25} + \frac{16}{25} = \frac{25}{25} \).
• Finally, take the square root of the result: \( \sqrt{1} = 1 \).

Thus, the modulus is 1, indicating the point lies exactly one unit away from the origin.

Plotting Complex Numbers with Ease

Plotting complex numbers might sound intimidating, but it's quite straightforward once you grasp the coordinate representation. Here's a simple way to plot these on the complex plane:

• Identify the real part of the complex number, \( a \), and locate this value on the x-axis.
• Identify the imaginary part, \( b \), and mark that on the y-axis.
• Draw a dot where these two meet, forming the point \((a, b)\).

For instance, the complex number \( \frac{3}{5} + \frac{4}{5}i \) translates to the exact point \( \left( \frac{3}{5}, \frac{4}{5} \right) \). Think of it like plotting coordinates in a graph. This visualization offers insight into the behavior and relationships between complex numbers, aiding in deeper mathematical exploration and understanding.