Problem 52 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (-2-4 i)-(6-8 i) $$

Step-by-Step Solution

Answer

-8 + 4i

Step 1: Distribute the negative sign

First, distribute the negative sign to the second complex number. The expression becomes: \((-2 - 4i) - (6 - 8i) = (-2 - 4i) - 6 + 8i\).

Step 2: Combine Like Terms

Now, combine the real components and the imaginary components separately. \[(-2 - 6) + (-4i + 8i) = -8 + 4i\].

Step 3: Write the Final Answer

The result is written in the form \(a + bi\), which in this case is \(-8 + 4i\).

Key Concepts

Complex Number Subtraction

When subtracting complex numbers like \((-2-4 i)-(6-8 i)\), it's helpful to visualize each part of the complex numbers separately. Complex numbers have two components: the real and the imaginary. When we perform subtraction, it's important to address both components independently.

Let's see how we subtract complex numbers step-by-step:

• First, treat the subtraction between two complex numbers as distributing a negative sign across the second number. This converts the subtraction into addition of negative terms: \((-2-4i)-(6-8i) = (-2-4i) + (-6+8i)\).
• Next, combine "like terms." This means adding together the real components and the imaginary components separately. Real parts: \(-2 - 6 = -8\); imaginary parts: \(-4i + 8i = 4i\).
• The final step is to write the answers from the above operations in a standard complex number form, resulting in \(-8 + 4i\).

This method ensures that each component of the complex number is addressed properly.

Imaginary Numbers

Imaginary numbers often seem a bit mysterious at first, but they're quite straightforward once you get accustomed to them. Imaginary numbers are based on the imaginary unit, \(i\), which is defined as \(i = \sqrt{-1}\).

This means that whenever you see \(i\) in a math equation, it indicates that this part of the number doesn't exist on the traditional number line. Imaginary numbers let us work with square roots of negative numbers, which is impossible with only real numbers.

• \(i^2 = -1\), which is foundational in working out operations with \(i\).
• Imaginary numbers are added and subtracted like real numbers; they follow their own rules often in parallel to real parts.

Understanding the essence of \(i\) helps us approach equations with confidence, knowing that it's another tool to solve equations.

Real and Imaginary Components

Complex numbers are unique because they consist of two parts: the real component and the imaginary component. When represented in the form \(a + bi\), \(a\) is the real part, while \(b\) together with \(i\) is the imaginary part.

• The real component is the "traditional" numeric part that you'd normally see on the number line. It's always a real number, and operations on this component are done like usual arithmetic.
• The imaginary component involves the imaginary unit \(i\). It can only be properly worked with when recognizing the property of \(i\).

For example, in the complex number \(-8 + 4i\), \(-8\) is the real component and \(4i\) is the imaginary component. In calculations, real and imaginary parts are handled separately to maintain clarity and correctness of the result. Breaking them down simplifies complex number operations, aiding comprehension of what each element represents.