Problem 9 - Precalculus: Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$\frac{-2-5 i}{3}$$

Step-by-Step Solution

Answer

Real part = \(\frac{-2}{3}\); Imaginary part = \(-\frac{5}{3}\).

Step 1: Understand the Complex Number

A complex number is typically expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this problem, we are given \(\frac{-2-5i}{3}\). This means both the real part and the imaginary part of the complex number are divided by 3.

Step 2: Separate the Real and Imaginary Parts

To separate the real and imaginary components given in the fraction, express \(\frac{-2-5i}{3}\) as \(\frac{-2}{3} + \frac{-5}{3} i\). This representation allows us to easily identify the distinct real and imaginary parts.

Step 3: Identify the Real Part

The real part of our complex number is \(\frac{-2}{3}\). This is obtained directly from the component that does not involve \(i\).

Step 4: Identify the Imaginary Part

The imaginary part of our complex number is \(-\frac{5}{3}\), which is the coefficient of \(i\).

Key Concepts

Real Part

When dealing with complex numbers, the real part is the component without the imaginary unit \( i \). This component represents the actual 'real' number within a complex number, much like a regular number that we're accustomed to. For instance, in a complex number expressed in the form \(a + bi\), \(a\) is the real part.

To understand better, consider the complex number \(-2 - 5i\). If you divide this entire complex number by 3, it becomes \(-\frac{2}{3} - \frac{5}{3}i\). The real part here is clearly \(-\frac{2}{3}\), as it stands alone without being multiplied by \(i\).

Understanding the real part is essential because it helps in distinguishing the non-imaginary ingredient of complex numbers. This is particularly useful in real-world applications like electrical engineering, where complex numbers are used to describe circuits.

Imaginary Part

The imaginary part of a complex number is the portion that is multiplied by the imaginary unit \( i \). In any complex number \(a + bi\), \(b\) is referred to as the imaginary part. Though it sounds fictitious, the imaginary part has real applications, especially in engineering disciplines.

Consider our given fraction of a complex number, \(-2 - 5i\), divided by 3 to get \(-\frac{2}{3} - \frac{5}{3}i\). The imaginary part here is \(-\frac{5}{3}\). This component might look slightly involved but is simply the coefficient of \(i\).

Understanding the imaginary part helps us in performing operations such as addition, subtraction, and division of complex numbers. Plus, the imaginary part allows for the depiction of multi-dimensional constructs like waves and oscillations in physics and engineering.

Complex Number Division

Complex number division may sound tricky but it is quite straightforward once you grasp the concept of separating real and imaginary parts. When you divide a complex number by a real number, as in \(-2 - 5i\) divided by 3, you're essentially dividing each part of the complex number independently by that real number.

The main steps to perform such a division are:

• Separate real and imaginary parts: Write the complex number as \(a + bi\), and divide both \(a\) and \(b\) by the divisor.
• Simplify the expression: This results in two separate fractions, one for the real part and one for the imaginary part.
• Combine: Write the simplified real and imaginary parts back into a complex number form.

In our example, we separate \(-2 - 5i\) to form \(\frac{-2}{3} + \frac{-5}{3}i\).

Dividing complex numbers allows engineers and scientists to solve equations that involve multiple dimensions, enabling more comprehensive analyses in fields like electromagnetism and signal processing.