Problem 13 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Plot the point at (0, 1) on the complex plane and label it as \( i \).

Step 1: Understand the Complex Number

The given complex number is \( i \), which is equivalent to \( 0 + 1i \). This means the real part is zero and the imaginary part is one.

Step 2: Identify Coordinates on the Complex Plane

On the complex plane, the horizontal axis (x-axis) represents the real part of a complex number and the vertical axis (y-axis) represents the imaginary part. For \( i \), the coordinates are (0, 1).

Step 3: Plot the Point

Find the point where the x-coordinate is 0 and the y-coordinate is 1 on the graph. Place a dot or an 'X' at this position to represent the complex number \( i \).

Step 4: Label the Point

Label the point with the complex number \( i \) to indicate this is the number being plotted on the complex plane.

Key Concepts

Understanding the Complex Plane

The complex plane, also known as the Argand plane, is a visual representation of complex numbers. It is a two-dimensional plane where each point represents a complex number. This concept is similar to how we use the Cartesian coordinate system to plot real numbers, but with a twist.

On the complex plane:

• The horizontal axis, or x-axis, is used for the real part of a complex number.
• The vertical axis, or y-axis, is used for the imaginary part.

To plot a complex number, such as the given number in the exercise, which is\(i\), you need to locate its real and imaginary parts on this plane and mark the corresponding point. For the number\(i\), which can be rewritten as \(0 + 1i\), you would find the point at position (0,1).

This graphical interpretation helps in understanding the behavior of complex numbers, especially when performing operations like addition, subtraction, and multiplication.

Exploring the Imaginary Part

The imaginary part of a complex number is the coefficient of the imaginary unit\(i\). In mathematics,\(i\) is defined as the square root of -1. This might sound tricky at first, but think of\(i\) as a tool that allows us to expand number systems to solve problems that real numbers alone cannot handle.

For the complex number\(i\), the imaginary part is 1 because it is equivalent to\(0 + 1i\). Other examples:

• In the complex number\(3 + 2i\), the imaginary part is 2.
• For\(-4 - 5i\), the imaginary part is -5.

When plotting on the complex plane, the imaginary part represents the vertical movement from the origin. A positive imaginary part like\(+1\) moves up on the y-axis, while a negative part moves down. In our case, at point (0,1), the imaginary part is simply the distance above the x-axis.

Defining the Real Part

The real part of a complex number is the component without the imaginary unit\(i\). It behaves just like a regular real number and determines the horizontal position of the complex number on the complex plane.

In the expression\(0 + 1i\), the real part is 0. If you think of the complex number\(3 + 2i\), the real part is 3, and in\(-4 - 5i\), it is -4. This tells us how far left or right the corresponding point is from the imaginary axis.

• A real part of zero means the point lies directly on the imaginary axis.
• A positive real part shifts the point to the right of the origin.
• A negative real part moves it to the left.

Understanding both the real and imaginary parts is essential for accurately plotting complex numbers like\(i\) on the complex plane, ensuring they are correctly positioned relative to both axes.