Problem 15 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (3+2 i)+(5-3 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(8 - i\).

Step 1: Identify Real and Imaginary Parts

Recognize the real and imaginary components of each complex number: - The first number is \(3 + 2i\), with a real part of 3 and an imaginary part of 2i.- The second number is \(5 - 3i\), with a real part of 5 and an imaginary part of -3i.

Step 2: Add Real Parts

Add the real parts from both complex numbers together:\[3 + 5 = 8\] This means the real part of the sum is 8.

Step 3: Add Imaginary Parts

Add the imaginary parts from both complex numbers:\[2i + (-3i) = -i\]This means the imaginary part of the sum is \(-i\).

Step 4: Combine Results

Combine the results from Steps 2 and 3 to express the sum as a complex number:\[ 8 - i \]This is the final simplified complex number.

Key Concepts

Addition of Complex Numbers

When performing the addition of complex numbers, you combine similar components from each number. Complex numbers are of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. For example, given \((3 + 2i) + (5 - 3i)\), you need to treat the real and imaginary parts separately.

• Real parts: These are the numbers without the 'i'. In this expression, they are 3 and 5.


• Imaginary parts: These include the 'i', so here they are \(2i\) and \(-3i\).

To add these complex numbers, simply add the real parts together and then add the imaginary parts. It's similar to adding like terms in algebra. In this exercise:

• Real addition: \(3 + 5 = 8\)


• Imaginary addition: \(2i + (-3i) = -i\)

Combine these results to get the final sum: \(8 - i\).

Real and Imaginary Parts

Understanding the breakdown of complex numbers into their real and imaginary parts can simplify the addition process. The real part refers to the portion of the complex number without the imaginary unit \(i\). Imaginary parts, on the other hand, include the term \(i\). Recognizing these components helps in systematically organizing and solving complex number operations.
To illustrate, in the expression \((3 + 2i) + (5 - 3i)\), each complex number is split into two parts:

• Real Parts: \(3\) from the first complex number and \(5\) from the second.


• Imaginary Parts: \(2i\) from the first and \(-3i\) from the second.

When handling these parts, always maintain the sign in front of each component. Correctly identifying and manipulating the real and imaginary parts ensures accurate mathematical operations.

Simplifying Complex Expressions

Simplification is the final step when working with complex numbers. After adding the real and imaginary parts separately, the next task is to combine these results into a single expression. In most cases, you'll want to write the final answer in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers.
For our exercise, after adding the real and imaginary parts, we arrived at a result of \(8 - i\). This is already in the simplified form. Here's how you ensure an expression is simplified:

• Make sure each part (real and imaginary) is a single term.


• Combine like terms where possible.


• Ensure the expression is presented as \(a + bi\), even if the imaginary part \(b\) might be negative, as seen here with \(-i\).

Expressing complex numbers in their simplest form helps in making them easy to understand and compare with others.