Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Multiply or divide as indicated. Simplify each answer. $$(7+9 i)+(1-2 i)+(-8-7 i)$$

Step-by-Step Solution

Answer

The simplified expression is \(0\).

Step 1: Group Real Parts

Identify and group the real parts of the complex numbers: \(7\), \(1\), and \(-8\). Calculate their sum: \(7 + 1 - 8 = 0\).

Step 2: Group Imaginary Parts

Identify and group the imaginary parts of the complex numbers: \(9i\), \(-2i\), and \(-7i\). Calculate their sum: \(9 - 2 - 7 = 0\).

Step 3: Combine Real and Imaginary Parts

Combine the sums of the real parts and the imaginary parts from Steps 1 and 2. The resulting complex number is \(0 + 0i\).

Key Concepts

Addition of Complex Numbers

Complex numbers can appear intimidating at first, but they follow a structured process for addition, similar to elementary arithmetic. When adding complex numbers, we can break it down to the sum of their real and imaginary parts separately.

Consider the complex numbers given by the exercise: \((7+9i)\), \((1-2i)\), and \((-8-7i)\). The objective is to perform addition on them. Here’s how you proceed:

• First, group all the real parts: 7, 1, and -8.
• Then, group all the imaginary parts: \(9i\), \(-2i\), and \(-7i\).

After organizing them, add the real parts together and then the imaginary parts separately. This method ensures clarity and prevents errors, especially when more complex or larger numbers are involved.

Imaginary Parts

Imaginary parts of a complex number involve the unit \(i\), the imaginary unit, where \(i^2 = -1\). This imaginary component allows us to perform mathematical operations beyond real numbers.

In our exercise, the imaginary parts are: \(9i\), \(-2i\), and \(-7i\). Adding imaginary components is straightforward, just as you would with whole numbers:

• Add \(9i\) and \(-2i\) to get \(7i\).
• Next, add \(7i\) and \(-7i\) to finally reach a net imaginary sum of \(0i\).

Understanding this process highlights how imaginary numbers are both intriguing and manageable, allowing us to see beyond the limitations of real numbers.

Real Parts

The real part of a complex number is akin to the numbers we deal with daily. They have no imaginary component, making them intuitive to work with when part of a complex number addition.

To find the sum of the real parts of the complex numbers from the exercise, you have:

• Real numbers 7, 1, and -8.
• Proceed by adding these real numbers one by one: \(7 + 1 = 8\) and \(8 - 8 = 0\).

The result shows that the real components can nullify each other, resulting in a total sum of zero. Real parts are the foundation, offering a sense of normalcy when dealing with the complex world of numbers.