Problem 38 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The result of the operation is 1+1i.

Step 1: Identify Real and Imaginary Parts

For the complex numbers (8+3i) and (-7-2i) , identify the real and imaginary parts. The first term has a real part of 8 and an imaginary part of 3i, and the second term has a real part of -7 and an imaginary part of -2i.

Step 2: Combine Real Parts

Add the real parts of the complex numbers: 8 + (-7) . The operation results in: 8 - 7 = 1.

Step 3: Combine Imaginary Parts

Add the imaginary parts of the complex numbers: 3i + (-2i) . The operation results in: 3i - 2i = 1i.

Step 4: Reconstruct the Complex Number

Combine the results from the previous steps to form the final complex number. The real part is 1 and the imaginary part is 1i. Thus, the answer is: 1 + 1i.

Key Concepts

Real and Imaginary Parts

Complex numbers are fascinating as they combine both real and imaginary parts, forming a unique way to express mathematical values. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is known as the real part, while \(bi\) represents the imaginary part. In this context, \(i\) is the imaginary unit, which is defined as \(i^2 = -1\).
Identifying these parts is the foundational step when working with complex numbers. For example, in the complex number \(8 + 3i\), the real part is 8, and the imaginary part is 3, corresponding to \(3i\). Understanding and distinguishing between these parts allows us to efficiently perform arithmetic operations on complex numbers.

Addition of Complex Numbers

The addition of complex numbers involves separately adding their real parts and imaginary parts. This principle makes the calculation straightforward while respecting the properties of both real numbers and imaginary units.

To add the complex numbers \((8 + 3i)\) and \((-7 - 2i)\), first, focus on their respective parts. Sum the real parts: \(8 + (-7) = 1\). Then, sum the imaginary parts: \(3i + (-2i) = 1i\).

This step-by-step addition maintains coherence and respects the structure of complex numbers. Ultimately, the addition results in a new complex number \(1 + 1i\), effectively combining the results of these arithmetic operations.

Complex Number Representation

Complex numbers have a distinctive form represented by \(a + bi\), which allows them to express solutions that may not be captured by standard real numbers. This form is crucial, as it reveals both the magnitude and direction of numbers on the complex plane.

The complex number \(1 + 1i\) lies in the first quadrant of this plane. Here, the real part \(1\) determines the horizontal placement, while the imaginary part \(1i\) determines the vertical placement. The importance of this representation extends to various fields such as engineering and physics, where complex numbers model phenomena like electrical circuits or wave functions.

• The real part aligns with the horizontal axis (the x-axis).
• The imaginary part aligns with the vertical axis (the y-axis).

The clarity of this representation empowers students to visualize complex numbers as more than abstract concepts, providing a comprehensive view of their application and utility.