Problem 74 - Precalculus

Question

Determine whether each statement is true or false. In the complex plane, any point that lies along the vertical axis is an imaginary number.

Step-by-Step Solution

Answer

True, any point on the vertical axis of the complex plane is an imaginary number.

Step 1: Understand the complex plane

The complex plane consists of a horizontal axis (real axis) and a vertical axis (imaginary axis). Each point represents a complex number in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, \(i\) is the imaginary unit, and \(a\) represents the real part of the complex number while \(b\) is the imaginary part.

Step 2: Define numbers on the vertical axis

Any point that lies along the vertical axis of the complex plane has no real part, implying \(a = 0\). Therefore, such a point can be represented as \(bi\), where \(b\) is a real number and \(b eq 0\). This is the definition of an imaginary number.

Step 3: Evaluate the statement

Since numbers on the vertical axis have the form \(bi\) and \(bi\) is an imaginary number by definition, any point that lies along the vertical axis must be an imaginary number.

Key Concepts

Imaginary Numbers

Imaginary numbers are a crucial part of understanding complex numbers. They were invented to address the square roots of negative numbers, which cannot be expressed on the real number line.
In mathematics, the symbol used for the imaginary unit is \(i\), defined by the property that \(i^2 = -1\). This means that any imaginary number can be written as a real number multiplied by \(i\). For instance, \(3i\) and \(-2.5i\) are both examples of imaginary numbers.

• Imaginary numbers are essential to form complex numbers, which are numbers in the format \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part.
• When describing properties, note that the coefficient of \(i\) in a complex number could be zero, in which case the number is purely real.

This unique definition gives imaginary numbers their own place on the complex plane as numbers that vary along the vertical axis only.

Complex Plane

The complex plane is a visual representation of complex numbers. It helps us to understand and operate with these numbers by providing a geometric perspective.
The complex plane features two axes:

• The horizontal axis, known as the real axis, represents the real part \(a\) of a complex number \(a + bi\).
• The vertical axis, called the imaginary axis, represents the imaginary part \(bi\).

Each point on the complex plane corresponds to a complex number. The beauty of the complex plane is in its ability to represent both the real and imaginary components together, allowing for operations like addition, subtraction, and multiplication to be visualized as geometric transformations.
If you were to plot the complex number \(4 + 3i\), this would be located at a point 4 units along the real axis and 3 units up the imaginary axis from the origin.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is foundational. Every complex number has two parts:

• The **real part**: This is the component without the imaginary unit \(i\), denoted by \(a\) in the number \(a + bi\). It’s plotted on the complex plane’s horizontal axis.
• The **imaginary part**: This involves the coefficient of \(i\), represented as \(b\) in \(a + bi\). It lies along the vertical axis.

When expressed graphically, a complex number appears as a vector or point in the two-dimensional space of the complex plane.
An interesting scenario arises when \(a = 0\), meaning our complex number is purely imaginary (like \(bi\)). In this case, it lies directly on the imaginary axis itself.
Conversely, if \(b = 0\), the number is purely real and falls directly on the real axis. Understanding these relationships ensures a comprehensive grasp of how complex numbers work and interact within the complex plane.