Problem 1 - A Graphical Approach to Precalculus with Limits

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

The real part is 0, the imaginary part is -9, and it's a pure imaginary number.

Step 1: State the Real Part

For a complex number in the form \( a + bi \), the real part is \( a \). For the complex number \(-9i\), there is no visible real part, so the real part is \( 0 \).

Step 2: State the Imaginary Part

For the complex number in the form \( a + bi \), the imaginary part is \( b \). The given complex number is \(-9i\), so the imaginary part is \(-9\).

Step 3: Identify the Number Type

A complex number is categorized as follows:- Real if the imaginary part is \( 0 \).- Pure imaginary if the real part is \( 0 \) and the imaginary part is non-zero.- Nonreal complex if neither part is zero.Since \(-9i\) has a real part of \( 0 \) and an imaginary part of \(-9\), it is a pure imaginary number.

Key Concepts

Real Part

In the world of complex numbers, every number is expressed as a combination of a real part and an imaginary part. A complex number takes the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

Let's break it down a bit. The real part \( a \) is a number you would typically see on a number line. It's what you'd consider a typical number without any fancy extensions.

If you encounter a complex number like \(-9i\), you might notice that it lacks a conventional number standing alone; this means the real part is actually \( 0 \).

• The real part helps anchor your complex number in the "real world".
• It tells us what portion of our complex number doesn't require the square root of negative numbers.

Imaginary Part

When we talk about the imaginary part of a complex number, we're focusing on the segment that involves the imaginary unit \( i \), where \( i \) is defined as \( \sqrt{-1} \).

Yet again, consider a general complex number in the form \( a + bi \). Here, the imaginary part is represented by \( b \).

For our specific number, \(-9i\), we look to the term \(-9\), concluding that it’s the figure associated with \( i \). Thus, the imaginary part is \(-9\).

• The imaginary part can sometimes seem a little "out there" because it deals with numbers we don't see in everyday arithmetic.
• However, it's essential in extending our number system to solve equations that have no solutions in the real numbers alone.

Number Classification

Classifying a complex number involves identifying which categories it fits into based on its real and imaginary parts. There are three potential classifications to consider:

• Real Number: If a complex number has zero as its imaginary part (like \( 5 + 0i \)), it's purely real.
• Pure Imaginary Number: If a complex number has zero as its real part and a non-zero imaginary part (such as \(-9i\)), it's purely imaginary.
• Nonreal Complex Number: If both the real and imaginary parts are non-zero, it's classified as a nonreal complex number (e.g., \( 3 + 4i \)).

So, for \(-9i\), it’s easy to see it fits neatly into the pure imaginary category. Understanding these classifications helps demystify the often confusing realm of complex numbers.