Problem 9 - Precalculus Mathematics for Calculus

Question

Graph the complex number and find its modulus. $$5+2 i$$

Step-by-Step Solution

Answer

The complex number is represented by the point (5, 2) and its modulus is \(\sqrt{29}\).

Step 1: Identify the Real and Imaginary Parts

Recognize that the complex number is given as \(5 + 2i\). Here, \(5\) is the real part, and \(2\) is the imaginary part.

Step 2: Plot the Complex Number on a Graph

On the complex plane, plot the real part \(5\) on the horizontal axis (x-axis) and the imaginary part \(2\) on the vertical axis (y-axis). The point \((5, 2)\) represents the complex number \(5 + 2i\).

Step 3: Use the Pythagorean Theorem to Find the Modulus

The modulus of a complex number \(a + bi\) is given by the formula \(|a + bi| = \sqrt{a^2 + b^2}\). Substitute \(a = 5\) and \(b = 2\) into the formula: \[|5 + 2i| = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}\].

Key Concepts

Graphing Complex Numbers

Graphing complex numbers helps visualize their structure on a special type of plot known as the complex plane. Complex numbers are made up of a real part and an imaginary part, similar to how coordinates work in traditional geometry.

To graph a complex number, you need to identify its real part, which will be plotted on the horizontal axis, and its imaginary part, which will go on the vertical axis. For example, in the complex number 5 + 2i:

• 5 is the real part.
• 2 is the imaginary part.

Therefore, to represent 5 + 2i on the complex plane, you find 5 on the horizontal axis and 2 on the vertical axis. The point (5, 2) is where you plot this complex number.

This graphing approach allows us to see not just the location of a complex number, but also its behavior in relation to other points on the plane.

Modulus of a Complex Number

The modulus of a complex number gives us a way to measure its 'size' or 'magnitude'. This concept is akin to finding the length of a vector in standard geometry.

To calculate the modulus of a complex number represented as \(a + bi\), use the formula:\[|a + bi| = \sqrt{a^2 + b^2}\]In this example, for the complex number 5 + 2i:

• \( a = 5 \)
• \( b = 2 \)

Substitute these values into the modulus formula:\[|5 + 2i| = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}\]The modulus \( \sqrt{29} \) indicates the distance of the point (5, 2) from the origin in the complex plane, thus providing a sense of how 'far' the complex number is from the center.

Complex Plane

The complex plane is a two-dimensional plot used to represent complex numbers. Unlike the traditional XY-plane, the complex plane uses the horizontal axis to represent real components and the vertical axis for imaginary components.

Every complex number corresponds to a specific point on this plane, much like coordinates in a mapping system. This plane enables you to visualize complex addition, subtraction, and even other more advanced operations such as multiplication and division.

Think of the complex plane as an extended geometric space for both real and imaginary numbers. The origin of this plane is where the real and imaginary axes meet, representing the complex number 0 + 0i (or just 0).

• The horizontal axis is labeled as the "real axis."
• The vertical axis is labeled as the "imaginary axis."

In our previous example, the complex number 5 + 2i is plotted as the point (5, 2) in this plane. This visual approach can simplify a lot of problems involving complex numbers."