Problem 38 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (8+3 i)+(-7-2 i) $$

Step-by-Step Solution

Answer

The result is \(1 + 1i\).

Step 1: Identify the Complex Numbers

The complex numbers in the expression are \((8 + 3i)\) and \((-7 - 2i)\). Our goal is to add these two complex numbers.

Step 2: Add the Real Parts

Separate the real parts from the imaginary parts. The real parts are 8 and -7. Add these numbers: \[8 + (-7) = 1\]

Step 3: Add the Imaginary Parts

Next, deal with the imaginary parts, which are \(3i\) and \(-2i\). Combine them as follows: \[3i + (-2i) = 1i\]

Step 4: Combine the Results

Now, we combine both the real and imaginary results from the previous steps. The real part is 1, and the imaginary part is \(1i\). Therefore, our final expression is: \[1 + 1i\]

Key Concepts

Addition of Complex Numbers

Complex numbers are expressed in the form \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part. The addition of complex numbers essentially involves combining like terms.

When you add two complex numbers, you follow a straightforward process:

• Identify the real parts and the imaginary parts separately.
• Add the real parts together.
• Add the imaginary parts together.
• Combine these two results to form a new complex number.

For example, consider adding the complex numbers \((8 + 3i)\) and \((-7 - 2i)\). Follow these steps:

• Add the real parts: \(8 + (-7) = 1\).
• Add the imaginary parts: \(3i + (-2i) = 1i\).
• Combine them: the result is \(1 + 1i\).

This method ensures the process is consistent and easy to apply to any pair of complex numbers.

Real and Imaginary Parts

Every complex number consists of two components: the real part and the imaginary part. Understanding how to identify and work with these parts is crucial when performing operations like addition.

The real part in the complex number \(a + bi\) is simply the number \(a\), and the imaginary part is the term \(bi\).

Let's break it down further with the example: in \((8 + 3i) + (-7 - 2i)\):

• The real parts are 8 and -7.
• The imaginary parts are \(3i\) and \(-2i\).

By separately handling the real and imaginary parts, we make the arithmetic straightforward:

• Real part: \(8 - 7 = 1\)
• Imaginary part: \(3i - 2i = 1i\)

So, the complex number forming from these operations combines to become \(1 + 1i\).

This approach is a common and essential technique in dealing with complex numbers.

Mathematical Operations

Performing mathematical operations on complex numbers, like addition or subtraction, involves the manipulation of both the real and imaginary components.

The key is to separate these parts and deal with like terms individually. By following this method, you can simplify complex expressions without confusion:

• When adding, combine the real numbers with other real numbers.
• Combine imaginary numbers accordingly.

For the operation \((8 + 3i) + (-7 - 2i)\), this involves:

• Adding 8 and -7 to get the new real part, 1.
• Adding \(3i\) and \(-2i\) to form the new imaginary part, \(1i\).

The result of these combined operations gives the final complex number \(1 + 1i\). Remember that every time you perform a mathematical operation on complex numbers, maintaining the order of operations and carefully separating real from imaginary will lead to correct and simplified results.