Problem 8 - 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. $$-i \sqrt{3}$$

Step-by-Step Solution

Answer

Real part: 0, Imaginary part: \(-\sqrt{3}\), Classification: Pure Imaginary.

Step 1: Identify the Complex Number

The given complex number is \[-i \sqrt{3}\]. It can be written in the standard form of a complex number as \[0 + (- \sqrt{3})i\].

Step 2: State the Real Part

In the complex number \[0 + (- \sqrt{3})i\], the real part is \[0\]. This is because there is no term that doesn't include \(i\).

Step 3: State the Imaginary Part

The imaginary part of the complex number is \[- \sqrt{3}\]. This is the coefficient of \(i\) in the expression.

Step 4: Classify the Complex Number

Since the real part is \[0\] and there is a non-zero imaginary part, this number is classified as a pure imaginary number. It is not a real number or a nonreal complex number as the latter requires both real and imaginary parts to be non-zero.

Key Concepts

Real Part of Complex Numbers

Complex numbers are expressions that have both a real part and an imaginary part, and they are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The **real part** of a complex number is the coefficient \(a\).

It is simply the number that stands alone without the imaginary unit \(i\). Occasionally, like in our example \(-i \sqrt{3}\), the real part is zero, meaning the complex number doesn’t have a separate non-imaginary component. Understanding the real part is a crucial step in working with complex numbers as it helps identify the type of complex number you are dealing with.

Imaginary Part of Complex Numbers

The **imaginary part** of a complex number involves the imaginary unit \(i\), which represents \(\sqrt{-1}\). In a complex number expressed as \(a + bi\), the imaginary part is \(b\).

This is the coefficient of \(i\), and it determines how the imaginary part affects the complex number. For the complex number \(-i \sqrt{3}\), the imaginary part is \(-\sqrt{3}\). This shows that while there isn’t a standard real number component here, we have a substantial imaginary component.

• The imaginary part can be positive, negative, or zero.
• It's instrumental in calculations involving complex numbers.

Recognizing the imaginary part is pivotal to classifying and working with complex numbers.

Classification of Complex Numbers

When classifying complex numbers, it’s important to understand the roles of both the real and imaginary parts. Complex numbers can fall into one of several categories:

• **Real numbers**: A number entirely real with an imaginary part of zero, like \(5 + 0i\).
• **Pure imaginary numbers**: A number where the real part is zero and there is a non-zero imaginary part, like our example \(-i \sqrt{3}\).
• **Nonreal complex numbers**: A number having both a non-zero real part and a non-zero imaginary part, such as \(3 + 4i\).

For the number \(-i \sqrt{3}\), the classification is **pure imaginary** because there is no real part and the imaginary part \(-\sqrt{3}\) is non-zero. Identifying these categories helps in understanding and solving problems involving complex numbers.