Problem 2 - Precalculus Student Solutions Manual 5th

Question

Determine whether each statement is true or false. If it is false, tell why. No real number is a pure imaginary number.

Step-by-Step Solution

Answer

True, no real number can be a pure imaginary number as they are defined distinctly.

Step 1 - Understand Definitions

First, recall the definitions of 'real number' and 'pure imaginary number.' A real number is any number that can be found on the number line, including both positive and negative numbers, as well as zero. A pure imaginary number is a number of the form bi, where b is a real number and i is the imaginary unit with the property that i² = -1.

Step 2 - Clarify Each Type of Number

Given the definitions, analyze whether any number can be both a real number and a pure imaginary number. A pure imaginary number always includes the imaginary unit i, differentiating it from being purely real.

Step 3 - Consider Examples

Examine examples: 5 is a real number; it has no imaginary part. 3i is a pure imaginary number; it cannot be placed on the standard real number line because it involves the imaginary unit i.

Step 4 - Conclusion

Since no number can simultaneously be purely real (no imaginary unit) and purely imaginary (involves the imaginary unit), the initial statement is true.

Key Concepts

Real Numbers

Real numbers are the everyday numbers you see and use frequently. They can be found on the number line and include:

• Positive numbers (e.g., 1, 2, 3)
• Negative numbers (e.g., -1, -2, -3)
• Whole numbers and integers (e.g., 0, 12, -5)

Real numbers also include fractional and decimal quantities (e.g., 1/2, 0.75, -2.3). Each real number has a specific location on the number line, making them practical and intuitive to understand.
Real numbers can be contrasted with other types of numbers, such as complex and imaginary numbers, which we will discuss next.

Pure Imaginary Numbers

Pure imaginary numbers introduce the concept of an imaginary unit 'i'. These numbers are not located on the real number line. Instead, they exist in a different dimension.
A pure imaginary number will have the form bi where 'b' is a real number and 'i' stands for the imaginary part. The imaginary unit 'i' has a unique property: \[ i^2 = -1 \]. This property differentiates them from real numbers.
Examples of pure imaginary numbers include:

• 3i
• -7i
• 0.5i

These numbers can't be placed on a standard number line because they require an imaginary axis. However, they are crucial in complex number theory and have applications in engineering and physics.

Imaginary Unit

The imaginary unit 'i' is central to understanding pure imaginary numbers. It is defined by its most unique property: \[ i^2 = -1 \]. This definition means that when 'i' is squared, it gives a negative result, something no real number can do.
In mathematics, 'i' allows for solutions to equations that would otherwise have no solution within the realm of real numbers. For example, the equation \[ x^2 + 1 = 0\] has no real solutions because no real number squared gives -1. However, using the imaginary unit, we can say \[ x = i \] or \[ x = -i \].
Understanding the imaginary unit is essential because it serves as the foundation for more advanced concepts in mathematics, particularly in the study of complex numbers where both real and imaginary parts are involved.