Problem 13 - College Algebra

Question

For the following exercises, plot the complex numbers on the complex plane. \(i\)

Step-by-Step Solution

Answer

The complex number \( i \) is plotted at (0, 1) on the complex plane.

Step 1: Understand the Complex Number

Identify the real part and the imaginary part of the complex number. For the given complex number, \( i \), we observe that the real part is 0 and the imaginary part is 1.

Step 2: Identify the Axes of the Complex Plane

The complex plane consists of a horizontal axis, known as the real axis, and a vertical axis, known as the imaginary axis. Each complex number can be represented as a point on this plane.

Step 3: Plot the Real Part

Since the real part of \( i \) is 0, we place the point right on the imaginary axis.

Step 4: Plot the Imaginary Part

The imaginary part of \( i \) is 1. Move up from the origin by 1 unit along the imaginary axis to place the point.

Step 5: Mark the Point

Place the point on the complex plane at the intersection of the position identified in Step 3 and Step 4. That is, at \( (0, 1) \) on the complex plane.

Step 6: Annotate the Complex Plane

Label the plotted point as \( i \) to clearly identify it as the complex number being plotted.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane that allows us to visually represent complex numbers. Reimagine it as an enhanced version of the coordinate plane you might have encountered in mathematics classes. Instead of just "x" and "y" axes, the complex plane consists of a "real" axis and an "imaginary" axis. Each point on this plane corresponds to a complex number. For example, a complex number like \(a + bi\) can be visualized as the point \((a, b)\) on the complex plane. This simplification enables you to understand complex numbers not just arithmetically but also geometrically.

• The horizontal axis is called the "real axis".
• The vertical axis is called the "imaginary axis".

The complex plane helps in observing relationships between complex numbers by visual plot rather than through equations alone. It simplifies comparisons and operations like addition or multiplication.

Imaginary Axis

The vertical line in the complex plane is referred to as the imaginary axis. This axis is crucial for plotting and understanding the imaginary component of a complex number.

• It represents all numbers of the form \(bi\).
• Numbers on this axis have a real part equal to zero.

For instance, the pure imaginary number \(i\) is located directly on this axis at the point \((0, 1)\). The imaginary axis enables you to understand how complex numbers extend beyond traditional real numbers, offering a broader spectrum for representing mathematical concepts.

Imaginary Part

The imaginary part of a complex number is the component that emphasizes its deviation from standard real numbers. For a generic complex number of the form \(a + bi\), the imaginary part is represented by \(b\), which is always multiplied by the imaginary unit \(i\).

Here are some essential features and details:

• The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).
• The imaginary part contributes to the vertical movement along the imaginary axis.

In simple terms, any number that involves \(i\) contributes to the imaginary part of a complex number. For example, in the complex number \(2 + 3i\), \(3\) is the imaginary part. This part determines the vertical position of the number on the complex plane.

Real Part

In the world of complex numbers, the real part is akin to the ordinary numbers we are most familiar with. It forms the fundamental component of a complex number. Given a complex number in the form \(a + bi\), the real part is \(a\).

Let's break it down:

• The real part is visualized on the horizontal real axis of the complex plane.
• It dictates the position along this horizontal axis.

Imagine the complex number \(4 + 5i\), here, \(4\) is the real part. It decides where the complex number sits across the horizontal direction on the complex plane. Thus, the real part maps directly to traditional x-values you've plotted in real-world scenarios.