Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$2$$

Step-by-Step Solution

Answer

The complex number 2 is graphed as a vector from (0,0) to (2,0).

Step 1: Identify the Complex Number

The given complex number to graph is \(2\). In standard form, a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here, \(a = 2\) and \(b = 0\).

Step 2: Locate on the Complex Plane

On the complex plane, the x-axis represents the real part and the y-axis represents the imaginary part. Since the complex number \(2\) has a real part of \(2\) and an imaginary part of \(0\), it corresponds to the point (2, 0) on the complex plane.

Step 3: Draw the Vector

To represent the complex number as a vector, start at the origin (0, 0) and draw a line to the point (2, 0). This line is the vector representation of the complex number \(2\) on the complex plane.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. It is similar to the Cartesian coordinate system you may be familiar with, but it has a unique twist. Instead of just displaying two real values, it showcases one real and one imaginary value.

• The horizontal axis, known as the real axis, displays the real part of the complex number.
• The vertical axis, called the imaginary axis, displays the imaginary part of the complex number.

In this setup, each complex number can be plotted as a point in this plane. For example, the complex number 2 is plotted on the real axis because it has no imaginary component. This position on the complex plane helps in visualizing how complex numbers can behave similarly to vectors.

Real and Imaginary Parts

To understand complex numbers, it's essential to grasp the concept of their real and imaginary parts. A complex number is typically expressed in the form \(a + bi\), where:

• \(a\) is the real part, a real number found along the real axis of the complex plane.
• \(b\) is the imaginary part, also a real number, but it is multiplied by \(i\)—the imaginary unit \((i^2 = -1)\).

For the complex number \(2\), the real part is 2, and the imaginary part is 0. This means it lies entirely on the real axis at the point (2, 0) in the complex plane. When b equals 0, the complex number simplifies to a real number. This simplification helps clarify that regardless of the presence of an imaginary part, the method of graphing on the complex plane remains consistent.

Vector Representation

In mathematics, vectors are quantities that have both magnitude and direction. Complex numbers can be elegantly represented as vectors, allowing us to utilize vector operations and insights. When graphing a complex number, think of it as drawing a vector in the complex plane.To represent the complex number \(2\) as a vector:

• Start at the origin, which is the point (0, 0) on the complex plane.
• Draw a straight line from the origin to the point (2, 0), which corresponds to the complex number \(2\).

This line is the vector representation of the complex number. The direction of the vector is purely along the real axis, and its length corresponds to the magnitude of the real part, which is 2 in this case. This geometric interpretation aids in understanding operations involving complex numbers, such as addition and multiplication, which can also be visualized through vector transformations.