Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each statement is true or false. If it is false, tell why. Every pure imaginary number is a complex number.

Step-by-Step Solution

Answer

The statement is true because pure imaginary numbers are a subset of complex numbers.

Step 1: Review Definitions

A pure imaginary number is of the form \( bi \), where \( b \) is a non-zero real number and \( i \) is the imaginary unit with the property \( i^2 = -1 \). A complex number is of the form \( a + bi \), where both \( a \) and \( b \) are real numbers. When \( a = 0 \), \( a + bi \) simplifies to \( bi \), which is a pure imaginary number.

Step 2: Compare Pure Imaginary and Complex Numbers

Given the definitions, notice that every pure imaginary number \( bi \) can be considered a complex number \( a + bi \) with \( a = 0 \). Therefore, each pure imaginary number satisfies the general form of complex numbers.

Step 3: Determine the Truth of the Statement

Since pure imaginary numbers are a special case of complex numbers (with \( a = 0 \)), the statement that every pure imaginary number is a complex number is true.

Key Concepts

Pure Imaginary Numbers

Pure imaginary numbers are an interesting subset of numbers in mathematics. They are numbers of the form \( bi \), where \( b \) is a non-zero real number, and \( i \) is known as the imaginary unit. Here, the real part of the number is zero, and the entire value is composed of the imaginary component. This sets pure imaginary numbers apart from real numbers, which do not include the imaginary unit.
It might seem strange at first that we have these numbers "imaginary," but they are fundamental in solving equations that real numbers alone cannot. Pure imaginary numbers arise naturally in many scientific and engineering problems when dealing with phenomena like alternating current or waveforms.
Just remember:

• A pure imaginary number has no real part.
• It takes the form \( bi \).
• The value of \( b \) is a real number, but not zero.

Imaginary Unit

The imaginary unit, denoted by \( i \), is a fundamental concept in the realm of complex numbers. Its primary property is that \( i^2 = -1 \). This property is what distinguishes \( i \) from real numbers and gives rise to the concept of pure imaginary and complex numbers.
Imagine trying to find the square root of a negative number in real number terms - it simply does not exist. This is where \( i \) fills the gap. With \( i \), it becomes possible to solve equations that involve negative square roots. For example, the square root of \(-1\) is represented as \( i \).

• \( i \) is not a real number but an essential part of complex number calculations.
• Any complex number can be expressed with \( i \), such as \( a + bi \), where both \( a \) and \( b \) are real numbers.
• The presence of \( i \) in an equation indicates the involvement of an imaginary component.

Real Numbers

Real numbers are probably the type of numbers you're most familiar with. These include all the numbers you encounter on a daily basis—like -3, 0, 7.2, and even \( \pi \). They encompass all rational and irrational numbers and can be positive, negative, or zero.
Real numbers play a crucial role in multiple areas of math such as algebra, geometry, and calculus. However, when it comes to discussing complex numbers, the real number component is just one part of the story. Complex numbers are made by combining real numbers with imaginary units in the form \( a + bi \). When \( b = 0 \), the complex number becomes a real number.

• Real numbers can be thought of as a line—a continuous line that extends in both directions without end.
• They include whole numbers, fractions, and numbers like square roots that don't result in a neat fraction.
• In a complex number \( a + bi \), the real number is represented by \( a \).

Understanding real numbers and their interaction with imaginary numbers helps pave the way for mastering complex numbers.