Problem 22 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (8-3 i)+(-8+3 i) $$

Step-by-Step Solution

Answer

The answer is \(0 + 0i\).

Step 1: Identify the real parts

Look at the given complex numbers, \((8 - 3i) + (-8 + 3i)\). Identify the real components from each complex number. Here, the real parts are 8 (from the first complex number) and -8 (from the second complex number).

Step 2: Add the real parts

Add the real parts identified in Step 1 together: \(8 + (-8) = 0\).

Step 3: Identify the imaginary parts

In the given complex numbers, the imaginary components are -3i (from the first complex number) and +3i (from the second complex number).

Step 4: Add the imaginary parts

Add their imaginary components: \(-3i + 3i = 0i\).

Step 5: Combine the results

Combine the results obtained from adding the real and imaginary parts. The final result is \(0 + 0i\).

Key Concepts

Addition of Complex Numbers

When you encounter complex numbers, it's important to remember they have both a real part and an imaginary part. Adding these kinds of numbers is quite straightforward once you get the hang of it. Complex numbers are often expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

For this addition process, simply add the real parts together and the imaginary parts together. Consider our example \((8 - 3i) + (-8 + 3i)\):

• Identify the real parts: 8 and -8.
• Add them: \(8 + (-8) = 0\).
• Identify the imaginary parts: -3i and 3i.
• Add them: \(-3i + 3i = 0i\).

The result of this addition is \(0 + 0i\). Even though it feels like an algebraic jigsaw, sticking to this simple process makes handling complex numbers quite manageable.

Imaginary Unit

The concept of the imaginary unit is key to understanding complex numbers. The imaginary unit is denoted by "i," where \(i\) is defined as the square root of -1. It isn't something you can visualize in the same way as real numbers, but its properties allow us to handle calculations involving negative roots.

When you're working with the imaginary unit, remember these basic properties of \(i\):

• \(i^2 = -1\).
• \(i^3 = -i\).
• \(i^4 = 1\).

Using these properties, you can solve complex equations and simplify expressions that involve complex numbers.

Real and Imaginary Parts

Understanding the structure of complex numbers means knowing about real and imaginary parts. A complex number is written in standard form as \(a + bi\). In this form, \(a\) is called the real part, and \(bi\) is the imaginary part.

To clarify, let's break down our example \((8 - 3i) + (-8 + 3i)\):

• The first number \(8 - 3i\) has a real part \(a = 8\) and an imaginary part \((-3i)\).
• The second number \(-8 + 3i\) has a real part \(a = -8\) and an imaginary part \((+3i)\).

To combine complex numbers, as we've done in this exercise, treat them like separate entities: add real to real, and imaginary to imaginary. This systematic approach helps avoid errors and ensures accuracy in your calculations.