Problem 16 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-2-4 i)+(1+6 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-1 + 2i\).

Step 1: Identify Real and Imaginary Parts

We start by identifying the real and imaginary parts of each complex number in the expression. The first complex number is \(-2 - 4i\), where \(-2\) is the real part and \(-4i\) is the imaginary part. The second complex number is \(1 + 6i\), where \(1\) is the real part and \(6i\) is the imaginary part.

Step 2: Add the Real Parts

Next, we perform addition on the real parts of the complex numbers. The real part from \(-2 - 4i\) is \(-2\), and from \(1 + 6i\) is \(1\). Therefore, the sum of the real parts is \(-2 + 1 = -1\).

Step 3: Add the Imaginary Parts

Now, we need to add the imaginary parts of the complex numbers. The imaginary part from \(-2 - 4i\) is \(-4i\), and from \(1 + 6i\) is \(6i\). So, the sum of the imaginary parts is:\((-4i) + (6i) = (2i)\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to form the final complex number. The real part is \(-1\) and the imaginary part is \(2i\), so the simplified complex number is:\(-1 + 2i\).

Key Concepts

Real and Imaginary Parts

Every complex number can be split into its real and imaginary components. Think of it like a coordinate on a 2-dimensional plane. The real part is like the "x-coordinate" and the imaginary part is like the "y-coordinate".
The number in front of the imaginary unit, denoted as "i", is the imaginary part. So in the complex number \(-2 - 4i\), \(-2\) is the real part, and the imaginary part is \(-4i\).
In another example, for the complex number \(1 + 6i\), the real part is simply \(1\), and the imaginary part is \(6i\). It's essential to identify these parts for any operations you perform with complex numbers.

Complex Numbers

Complex numbers are numbers that include both a real part and an imaginary part. Imaginary numbers are expressed with the imaginary unit "i", where \(i\) satisfies the equation \(i^2 = -1\).
A complex number is usually written in the form \(a + bi\), where "a" represents the real part and "bi" is the imaginary part.

• They are denoted by their components, e.g., \(-2 - 4i\) or \(1 + 6i\).
• Complex numbers extend our traditional idea of numbers beyond just the real number line.

Understanding how to work with complex numbers opens up a broader world of mathematical possibilities, especially in fields like engineering and physics.

Simplifying Complex Numbers

Simplifying complex numbers involves combining their real parts and imaginary parts separately and then putting them together. It's like cooking; first, gather ingredients, mix them separately, and then combine everything for the final dish.

• First, combine the real parts. For example, if you have \(-2\) and \(1\), you add them: \(-2 + 1 = -1\).
• Then, combine the imaginary parts. Here, we take \(-4i\) and \(6i\), which add up as \((-4i) + (6i) = 2i\).

Once these parts are summed, the whole of the complex number is like a combined result. Therefore, the complex number from the exercise becomes \(-1 + 2i\) after simplifying. Understanding this process allows for easier manipulation of complex numbers in future equations and solutions.