Problem 1 - A Graphical Approach to College Algebra

Question

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$-9 i$$

Step-by-Step Solution

Answer

Real part: 0, Imaginary part: -9, Pure imaginary number.

Step 1: Identify the Real Part

A complex number is generally written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part coefficient. For the complex number \(-9i\), we can rewrite it as \(0 + (-9)i\). Hence, the real part \(a = 0\).

Step 2: Identify the Imaginary Part

Continuing from the expression \(0 + (-9)i\), the imaginary part coefficient \(b\) is \(-9\). The imaginary part is the expression without \(-9i\). Thus, the imaginary part is \(-9\).

Step 3: Classify the Complex Number

Since the real part \(a\) is 0 and the imaginary part coefficient \(b\) is \(-9\), the number is a pure imaginary number. A pure imaginary number has no real part. Therefore, \(-9i\) is classified as a pure imaginary number.

Key Concepts

Real Part

The real part of a complex number is a fundamental concept in understanding how these numbers are structured. Complex numbers are generally expressed in the form \(a + bi\), where \(a\) represents the real part and \(b\) is associated with the imaginary part, usually accompanied by \(i\), the imaginary unit.
In the example \(-9i\), we can rewrite it as \(0 + (-9)i\) to clearly see the structure. Here, the real component \(a = 0\).

• When the real part is zero, it means that the number lies entirely along the imaginary axis on the complex plane.
• Understanding real parts helps in distinguishing parts of a complex number that align with real numbers on the number line.

Recognizing and extracting the real part from complex numbers is critical, especially when performing operations like addition, subtraction, and when plotting on complex planes.

Imaginary Part

The imaginary part of a complex number is what extends real numbers into the two-dimensional complex plane. It is the component that distinguishes complex numbers from ordinary real numbers. In a complex number like \(a + bi\), the imaginary part is given by \(b\). This coefficient is paired with the imaginary unit \(i\), which satisfies \(i^2 = -1\).
For the number \(-9i\), the imaginary part is \(-9\), with the expression as \(0 + (-9)i\).

• The imaginary part influences where the number sits vertically on the complex plane, apart from the real axis.
• It allows representation of complex numbers that can express solutions to equations that do not yield real number results.

Understanding the imaginary part opens up broader perspectives on handling mathematical problems involving roots of negative numbers, which are pivotal in fields like engineering and physics.

Pure Imaginary Numbers

Pure imaginary numbers hold a unique place in the family of complex numbers. They are characterized by having no real component at all. A complex number qualifies as pure imaginary if it can be written as \(0 + bi\), without a real part \(a\).
In our example, \(-9i\) is considered a pure imaginary number because its format is \(0 + (-9)i\). The real part is zero and the imaginary part is non-zero.

• These numbers lie entirely on the imaginary axis of the complex plane.
• Pure imaginary numbers are significant when analyzing oscillations, waves, and other phenomena in physics.
• They help in understanding the "point" or "distance" from the origin in an entirely vertical direction.

Mastering the concept of pure imaginary numbers enhances comprehension of how different combinations of real and imaginary parts influence complex systems.