Problem 19 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-4+4 i)-(-6+9 i) $$

Step-by-Step Solution

Answer

The simplified result is \(2 - 5i\).

Step 1: Identify the Complex Numbers

We begin with the complex numbers given in the expression: \(-4 + 4i\) and \(-6 + 9i\). Complex numbers are generally expressed in the form \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Distribute the Negative Sign

Next, distribute the negative sign in front of the second complex number. The operation becomes \(-4 + 4i + 6 - 9i\). Distributing the negative sign changes the signs of the real and imaginary parts of the second complex number.

Step 3: Combine Like Terms

We then group and combine the real and imaginary parts separately. For the real parts: \(-4 + 6\) gives \(2\). For the imaginary parts: \(4i - 9i\) gives \(-5i\).

Step 4: Write the Simplified Complex Number

The combination of the real and imaginary parts gives us the simplified complex number. Thus, the result is \(2 - 5i\).

Key Concepts

Real and Imaginary Parts

Complex numbers might feel tricky at first, but they become clear once you understand their components. Imagine any complex number as being made up of two parts: the real part and the imaginary part. We often write them in the form of \(a + bi\), where:

• \(a\) is called the real part. It's just like a regular number you use every day.
• \(b\) is the imaginary part. To stand out, it’s paired with \(i\), the imaginary unit. Remember, \(i\) stands for \(\sqrt{-1}\).

By separating each component, you can work with them just like regular numbers.
For instance, in the expression \(-4 + 4i\), \(-4\) is the real part, while \(4\) is the imaginary part. Similarly, for \(-6 + 9i\), the real part is \(-6\), and the imaginary part is \(9\). When dealing with complex numbers, you handle the real and imaginary parts separately before combining them again.

Complex Number Subtraction

Subtracting complex numbers is very similar to subtracting binomials in algebra. Let's see how it works using our example expression \((-4 + 4i) - (-6 + 9i)\). The first step is to distribute any negative signs through subtraction, converting the operation into adding the opposite. This step is crucial as it ensures the correct signs for both the real and imaginary parts of the number you're subtracting.
Here's how you do it:

• For the real parts: Change the sign of the second real component. So, \(-4 - (-6)\) becomes \(-4 + 6\).
• For the imaginary parts: Do the same. Thus, \(4i - 9i\) remains untouched as its subtraction is direct.

Ensure each time you perform operations on real and imaginary parts separately, keeping them in mind to get an accurate simplification.

Simplifying Complex Expressions

Simplifying complex expressions is the last step in getting a neat result. After separating and dealing with real and imaginary parts from subtraction, you need to combine your results properly. Let's walk through this for the problem at hand.

• First, work with the real components: \(-4 + 6 = 2\).
• Next, handle the imaginary components: \(4i - 9i = -5i\).

Finally, bring together these simplified parts to form a single expression. So, our simplified complex number becomes \(2 - 5i\).
This combination introduces the complete simplified result. Remember, handling real and imaginary parts separately makes complex expressions less intimidating and wraps up operations neatly.