Problem 20 - A Graphical Approach to College Algebra

Question

Determine whether each statement is true or false. If is false, tell why. A complex number might not be a pure imaginary number.

Step-by-Step Solution

Answer

True, complex numbers can have nonzero real parts, making them not purely imaginary.

Step 1: Define Complex Numbers

A complex number is any number of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

Step 2: Define Pure Imaginary Numbers

A pure imaginary number is a complex number of the form \( 0 + bi \) where \( b eq 0 \). It means that the real component is zero, and only the imaginary part is non-zero.

Step 3: Analyze the Statement

The statement suggests that a complex number could potentially not be a pure imaginary number. We already know that any complex number takes the form \( a + bi \). If \( a eq 0 \), the number has a real part and it is not purely imaginary.

Step 4: Conclusion: True or False

The statement "A complex number might not be a pure imaginary number" is true because a complex number can have a nonzero real part, making it not purely imaginary.

Key Concepts

Imaginary Unit

Complex numbers may seem intimidating at first glance, but they can be easily understood once we break them down.
The imaginary unit, denoted as \( i \), is a crucial building block of complex numbers.
It is defined by the property \( i^2 = -1 \). This definition is what makes it stand out from real numbers, as it allows us to solve equations like \( x^2 + 1 = 0 \), which do not have solutions within the realm of real numbers.
By introducing \( i \), we can extend our number system and tackle equations that were previously unsolvable. This expands the possibilities of what we can calculate and analyze in mathematics.
Understanding \( i \) is key to grasping the broader concept of complex numbers, as it forms their imaginary component.

Pure Imaginary Numbers

Pure imaginary numbers are a special subset of complex numbers. They take the specific form \( 0 + bi \), where \( b \) is a real number and is not equal to zero.
Essentially, these numbers have no real part—in other words, their real component is zero. Instead, they consist solely of an imaginary part, characterized by the term \( bi \).
For example, \( 3i \) and \( -5i \) are pure imaginary numbers because they follow the structure \( 0 + bi \).
Understanding pure imaginary numbers is crucial because they highlight the flexibility and versatility of complex numbers. While they appear distinct from real numbers, they integrate smoothly into mathematical operations, allowing for comprehensive mathematical solutions.

Real Component

Every complex number can be dissected into two key parts: the real part and the imaginary part.
In a complex number expressed as \( a + bi \), \( a \) represents the real component, which is just a normal real number.
The real component is crucial because it defines the position of the complex number on the real axis when graphed on the complex plane.

• If \( a \) is zero, the complex number is purely imaginary.
• If both \( a \) and \( b \) are non-zero, the number is a true mix of real and imaginary parts.

This demonstrates the full spectrum of numbers that complex numbers can describe, extending from real numbers all the way to pure imaginary numbers.
The inclusion of a real component gives complex numbers the flexibility to represent a wider range of values and scenarios in mathematics.