Problem 12 - Precalculus Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$i \sqrt{3}$$

Step-by-Step Solution

Answer

The real part is 0 and the imaginary part is \(\sqrt{3}\).

Step 1: Identify the Complex Number Format

A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The given number is \( i \sqrt{3} \).

Step 2: Rewrite in Complex Number Format

To rewrite \( i \sqrt{3} \) in the complex number format, observe that there is no explicit real part shown. Therefore, \( i \sqrt{3} \) can be rewritten as \( 0 + i \sqrt{3} \). This indicates that \( a = 0 \) and \( b = \sqrt{3} \).

Step 3: Extract the Real Part

Since the complex number \( i \sqrt{3} \) is rewritten as \( 0 + i \sqrt{3} \), the real part is \( a = 0 \).

Step 4: Extract the Imaginary Part

In the rewritten form, the coefficient of \( i \) is \( \sqrt{3} \), so the imaginary part is \( b = \sqrt{3} \).

Key Concepts

Real Part

When you encounter a complex number, it's essential to dissect it into its components. One of these components is the real part. The real part of any complex number refers to the numerical part that does not involve the imaginary unit, usually represented by \( i \). In the standard complex number format \( a + bi \), \( a \) is the real part.

To identify the real part, you examine the number in front of \( i \). If there's no number written, it often means the real part is zero. In our example, \( i \sqrt{3} \), rewritten as \( 0 + i \sqrt{3} \), signifies that there is no real number involved besides zero. Thus, the real part, \( a \), is simply \( 0 \).

Understanding the real part's role is crucial as it represents the horizontal axis value in the complex plane, indicating the placement from an arithmetic point of view. Recognizing this helps you solve more complex equations involving real and imaginary numbers.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit, \( i \). This part is crucial for solving various mathematical problems involving complex numbers. It gives insight into how far the number lies along the imaginary axis in the complex plane.

In a standard complex number, \( a + bi \), the imaginary part is represented by \( b \), which is the coefficient of \( i \). In our example \( i \sqrt{3} \), rewritten as \( 0 + i \sqrt{3} \), it's clear that the imaginary part is \( \sqrt{3} \). The number \( \sqrt{3} \) is the factor modifying \( i \), indicating the distance from the origin along the vertical or imaginary axis.

The imaginary part does not exist in isolation mathematically but is an integral portion of the larger complex number. It has applications in fields such as engineering, physics, and complex analysis. Remembering that the imaginary part directly alters the behavior of equations involving \( i \) is important for problem-solving.

Complex Number Format

The complex number format \( a + bi \) is a standardized way of representing complex numbers. Understanding this format is vital for interpreting and manipulating complex numbers correctly.

A complex number consists of two elements:

• The real part, \( a \), which is a real number.
• The imaginary part, \( bi \), where \( b \) is a real number, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

Complex numbers are represented in the complex plane, where the real part lies on the horizontal axis and the imaginary part on the vertical axis. The format \( a + bi \) helps distinguish these parts, allowing problems to be solved systematically.

In the exercise example, \( i \sqrt{3} \) was given. Here, the lack of a visible real component means \( a = 0 \), and the imaginary component is clearly \( \sqrt{3} \). Recognizing and rewriting complex numbers in the \( a + bi \) format not only helps in mathematical discussions but is also crucial in fields that use complex numbers regularly, ensuring clarity and precision in solutions.