Problem 29 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{6-2 i}{3} $$

Step-by-Step Solution

Answer

The simplified complex number is \( 2 - \frac{2}{3}i \).

Step 1: Separate Real and Imaginary Parts

The given complex expression is \( \frac{6 - 2i}{3} \). Separate the real and imaginary parts in the numerator: the real part is 6, and the imaginary part is \(-2i\).

Step 2: Perform Division on Real Part

Divide the real part 6 by 3: \( \frac{6}{3} = 2 \).

Step 3: Perform Division on Imaginary Part

Divide the imaginary part \(-2i\) by 3: \( \frac{-2i}{3} = -\frac{2}{3}i \).

Step 4: Combine Results

Combine the results from the division operations of both the real and imaginary parts: \( 2 - \frac{2}{3}i \). This is the simplified form of the complex number.

Key Concepts

Imaginary Part

In the world of complex numbers, the imaginary part is a crucial component that defines how these numbers work. Complex numbers are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). The term \( bi \) represents the imaginary part of the complex number.
Understanding what the imaginary part means will help demystify complex numbers:

• The imaginary part is not a real number — it's a multiplication of a real number and the imaginary unit \( i \).
• In the expression \( 6 - 2i \), the imaginary part is \(-2i\).
• The presence of the imaginary unit \( i \) signifies that this part of the complex number extends into the 'imaginary' dimension of the number system.

Getting a grip on the imaginary part helps in visualizing and working with complex numbers, especially when it comes to operations like addition, subtraction, and division as seen in this exercise.

Real Part

The real part of a complex number, as you might have guessed, is the part without the imaginary unit \( i \). For a complex number of the form \( a + bi \), \( a \) is considered the real part. It can be visualized on the horizontal axis of the complex plane.

• In our given expression \( 6 - 2i \), the real part is \( 6 \).
• The real part is just like any number you deal with in daily life, and it doesn't involve the imaginary unit \( i \).
• It can be added, subtracted, multiplied, or divided like any other real number.

Understanding the distinction between the real and imaginary parts is critical when performing operations on complex numbers. It is a foundation concept when expressing complex numbers in a simplified form.

Division of Complex Numbers

Dividing complex numbers can initially seem daunting, but it's just a methodical process.In the expression \( \frac{6 - 2i}{3} \), each component of the complex number \( 6 - 2i \) (the real and imaginary parts) is divided separately by the number 3.Steps to perform division:

• Divide the real part: \( \frac{6}{3} = 2 \).
• Divide the imaginary part: \( \frac{-2i}{3} = -\frac{2}{3}i \).
• Combine these results to find the simplified complex number: \( 2 - \frac{2}{3}i \).

The division in this case involves simplifying each component by the denominator, ultimately leading to a standard form of the complex number. Recognizing how to manipulate each part separately helps in achieving a simplified and accurate solution to problems like these.