Problem 49 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(3-8 i)+(2 i+4)$$

Step-by-Step Solution

Answer

The sum is \(7 - 6i\).

Step 1: Identify the Parts

The given expression is \((3 - 8i) + (2i + 4)\). Identify the real and imaginary components within each parentheses: \(3\) and \(4\) are real parts, while \(-8i\) and \(2i\) are imaginary parts.

Step 2: Combine the Real Parts

Add the real numbers from each group. Combine \(3\) and \(4\):\[3 + 4 = 7\].

Step 3: Combine the Imaginary Parts

Add the imaginary numbers from each group. Combine \(-8i\) and \(2i\):\[-8i + 2i = -6i\].

Step 4: Write the Result in Standard Form

Combine the results from Steps 2 and 3 to write the final answer in standard form, which is \(a + bi\). The sum is:\[7 - 6i\].

Key Concepts

Standard Form

In the realm of complex numbers, the standard form is essential for proper mathematical communication and understanding. A complex number is typically expressed as \(a + bi\), where \(a\) represents the real part and \(b\) is the imaginary coefficient. The term \(i\) denotes the imaginary unit, defined as \(\sqrt{-1}\). This form allows mathematicians to distinguish clearly between the two components of a complex number and simplify operations such as addition and subtraction.

Expressing complex numbers in standard form is crucial because it provides consistency, making it easy to perform and visualize mathematical operations. For example, writing the expression \(7 - 6i\) in standard form denotes that 7 is the real part and \(-6\) is the imaginary coefficient. This clarity becomes even more important when performing arithmetic operations involving multiple complex numbers.

Real and Imaginary Parts

Understanding the real and imaginary parts of complex numbers is the foundation for complex number operations. A complex number \(a + bi\) has two distinct components:

• The "real part" \(a\) is simply a real number, just like numbers you are familiar with on the number line.


• The "imaginary part" \(bi\) has the imaginary unit \(i\), which satisfies the equation \(i^2 = -1\).

These parts can interact with each other under addition, subtraction, or any other arithmetic operation to achieve meaningful results.

During calculations, you treat the real and imaginary parts separately, as seen in the original exercise. First, the real numbers are combined, and then the imaginary parts are added together. This method maintains the integrity of both components of the expression. For example, in the given problem, you have real parts 3 and 4 and imaginary parts \(-8i\) and \(+2i\). Handling each separately ensures clarity and accuracy in your calculations.

Addition and Subtraction of Complex Numbers

Adding or subtracting complex numbers involves handling their real and imaginary parts separately. This step-by-step process ensures accurate results and highlights the independence of the real and imaginary components of a complex number. Here's how it's executed:

• Identify the Parts: Start by identifying which numbers form the real parts and which numbers make up the imaginary parts.


• Add or Subtract Real Parts: Sum or subtract the real numbers as you would with ordinary arithmetic. In our example, you add the numbers 3 and 4.


• Add or Subtract Imaginary Parts: Similarly, treat the imaginary numbers, remembering to include their respective \(i\) terms. In this exercise, \(-8i + 2i\) gives \(-6i\).


• Combine the Results: After performing these independent operations, combine the outcomes into the standard form \(a + bi\).

This addition and subtraction process is foundational and highlights the unique properties of complex numbers, facilitating their broad application in real-world scenarios where both real and imaginary components are significant.