Problem 39 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (6-i)+(9+3 i) $$

Step-by-Step Solution

Answer

The result of the operation is \(15 + 2i\).

Step 1: Identify Real and Imaginary Parts

In the expression \((6-i) + (9+3i)\), identify the real and imaginary parts. \(6-i\) has a real part of \(6\) and an imaginary part of \(-1i\). \(9+3i\) has a real part of \(9\) and an imaginary part of \(3i\).

Step 2: Add the Real Parts

Add the real parts of the two expressions. This means adding \(6\) and \(9\) to get the resulting real part: \[6 + 9 = 15\].

Step 3: Add the Imaginary Parts

Add the imaginary parts of the two expressions, which are \(-1i\) and \(3i\). Combine these to get the resulting imaginary part: \[-1i + 3i = 2i\].

Step 4: Combine Real and Imaginary Parts

Express the answer in the form of \(a + bi\). From Steps 2 and 3, combine the real and imaginary parts: \[15 + 2i\].

Key Concepts

Understanding the Real Part of Complex Numbers

Complex numbers have two components: the real part and the imaginary part. The real part of a complex number is the real number that is not multiplied by the imaginary unit, which is denoted as 'i'. For example, in the complex number - "6 - i", the real part is 6 - "9 + 3i", the real part is 9. Identifying the real part is essential when performing operations such as addition or subtraction on complex numbers. In mathematical operations, we handle the real and imaginary parts separately, ensuring clarity and correctness in the results.

Deciphering the Imaginary Part of Complex Numbers

The imaginary part of a complex number is associated with the imaginary unit 'i', which satisfies the equation \(i^2 = -1\). Unlike real numbers, the imaginary part introduces a dimension that can be visualized in a plane rather than a line. This component appears as a multiple of 'i'. Looking at our example: - "6 - i", the imaginary part is \(-1i\).- "9 + 3i", the imaginary part is \(3i\).Calculating with imaginary numbers involves treating 'i' as a variable, while understanding it has unique properties, particularly \(i^2 = -1\). This distinction helps simplify complex number operations and is critical in both basic arithmetic with complex numbers and more advanced mathematics.

Mastering the Addition of Complex Numbers

Adding complex numbers involves combining their real and imaginary parts separately. Let's look at the calculation of \((6 - i) + (9 + 3i)\). To perform the addition:

• Add the real parts: \(6 + 9 = 15\)


• Add the imaginary parts: \(-1i + 3i = 2i\)

After calculating both components, the sum of the complex numbers is constructed in the form \(a + bi\). Therefore, combining the results gives us the final answer of \(15 + 2i\). This process of addition illustrates how each part operates independently, yet the results combine to form a new, complete complex number, showcasing the elegance and utility of complex arithmetic in mathematical applications.