Problem 37 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (3+4 i)+(5-6 i) $$

Step-by-Step Solution

Answer

The result is \(8 - 2i.\)

Step 1: Identify Real and Imaginary Parts

First, recognize the real and imaginary components of each complex number in the equation. The first complex number is \(3 + 4i\) and the second is \(5 - 6i\). The real parts are 3 and 5, and the imaginary parts are \(4i\) and \(-6i\), respectively.

Step 2: Add Real Parts

Add the real components of both complex numbers together. This gives us: \[3 + 5 = 8.\]

Step 3: Add Imaginary Parts

Add the imaginary components of both complex numbers together. Consider the imaginary unit \(i\) in your calculation: \[4i + (-6i) = -2i.\]

Step 4: Combine Results

Combine the results from steps 2 and 3 to form the new complex number. Thus, we have: \[8 - 2i.\]

Key Concepts

Understanding Real and Imaginary Parts

Complex numbers are fascinating entities that consist of two main components: the real part and the imaginary part. A typical complex number is written in the form \(a + bi\), where \(a\) represents the real part and \(bi\) is the imaginary part.

The real part is simply the number preceding the term \(bi\), and it behaves just like the real numbers you know well. For example, in the complex number \(3 + 4i\), 3 is the real part. In another example, the number \(5 - 6i\) has its real part as 5.

• The real part does not involve the imaginary unit \(i\).
• The imaginary part includes the coefficient of \(i\), which in \(3 + 4i\) is 4, and in \(5 - 6i\) is -6.

Recognizing these parts is crucial in performing operations such as addition and subtraction, where you will deal with each component separately.

What is the Imaginary Unit?

The imaginary unit, denoted by \(i\), is fundamental to complex numbers. It is defined by the property that \(i^2 = -1\).

This means that \(i\) is not a real number. Instead, it provides a way to extend the real numbers to cover more complex situations. The imaginary unit allows mathematicians and engineers to solve problems that cannot be tackled with real numbers alone.

• Think of \(i\) as an entirely new dimension, separate from the real number line.
• When operating with complex numbers, \(i\) stays intact until we perform specific multiplications or other operations involving \(i^2\).

Understanding \(i\) is vital to grasping the essence of complex number operations, especially when working with expressions like \(4i + (-6i)\). Here, you add or subtract the coefficients of \(i\) together, acknowledging that \(i\) itself remains unchanged unless squared.

Performing Complex Number Operations

Adding complex numbers is a straightforward process involving separate handling of the real and imaginary parts. In the exercise, you have two complex numbers: \(3 + 4i\) and \(5 - 6i\). Let's break down the operation:

• First, add the real parts: \(3 + 5 = 8\).
• Second, add the imaginary parts: \(4i + (-6i)\). Focus on the coefficients: \(4 - 6 = -2\), so you get \(-2i\).

Finally, combine these results to form the resultant complex number, \(8 - 2i\).

This method applies not only to addition but also to subtraction of complex numbers. Always treat the real and imaginary parts separately, and remember to keep the imaginary unit \(i\) consistent throughout your calculations. With this approach, you can confidently handle complex number operations in your future math challenges.