Problem 71 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (5-6 i)-4 i $$

Step-by-Step Solution

Answer

5 - 10i

Step 1: Identify the Complex Numbers

The problem involves the complex number \(5 - 6i\) and the imaginary number \(- 4i\). Both numbers include imaginary components represented by \(i\), where \(i\) represents the square root of \(-1\).

Step 2: Distribute Negative Sign

Recognize that subtracting \(4i\) is equivalent to adding \(-4i\). The expression becomes \((5 - 6i) + (-4i)\).

Step 3: Combine Like Terms

Focus on combining the imaginary parts: \(-6i + (-4i) = -10i\). This involves simply adding the coefficients of \(i\) (\(-6\) and \(-4\)).

Step 4: Write the Result

The expression now simplifies to \(5 - 10i\). This is already in the standard form \(a + bi\), where \(a = 5\) and \(b = -10\).

Key Concepts

Complex Number Operations

Complex number operations are essential for understanding how to work with numbers that consist of a real part and an imaginary part. A complex number is typically written in the form \(a + bi\), where \(a\) represents the real part, and \(b\) represents the imaginary part. Imaginary numbers are defined by the presence of \(i\), which, by definition, is the square root of \(-1\).

When performing operations on complex numbers, similar rules apply as with ordinary numbers, but we also account for the imaginary component. Here’s what to keep in mind:

• Addition: Combine the real parts together and the imaginary parts together.
• Subtraction: Similarly, subtract the real parts and the imaginary parts separately.
• Multiplication: Use the distributive property and simplify using \(i^2 = -1\).
• Division: Multiply the numerator and the denominator by the conjugate of the denominator to eliminate \(i\) from the denominator.

In our exercise, we are primarily dealing with subtraction, transformed into addition by the sign change, and thus simplifying our process.

Imaginary Numbers

Imaginary numbers might sound a bit fanciful, but they are grounded in mathematics and have real-world applications. The imaginary unit \(i\) is crucial here. It provides a way to deal with numbers we can't otherwise represent on the real number line. Specifically, \(i\) is defined so that \(i^2 = -1\).

Whenever you see an \(i\), it indicates that you’re dealing with an imaginary part of a complex number. Complex numbers, therefore, are a combination of both real and imaginary parts, presenting a more comprehensive set of numbers than real numbers alone.

In our exercise, the

• Complex number \(5 - 6i\) reveals \(5\) as the real part and \(-6i\) as the imaginary part.
• Imaginary number \(-4i\) means we have only the imaginary component to consider.

Understanding imaginary numbers is essential for exploring more advanced topics like electrical engineering and quantum physics.

Combining Like Terms

Combining like terms is an essential skill in algebra, and it remains important when dealing with complex numbers. The principle is straightforward: terms that contain the same variable can be combined to simplify expressions.

For complex numbers, this process involves separately combining the real components and the imaginary components. For example:

• If you have \(-6i\) and \(-4i\), you treat \(i\) as the common variable, and add the coefficients \(-6\) and \(-4\).
• This results in \(-10i\), effectively merging the like terms.

In our example, the real part \(5\) doesn't have another term to combine with, so it remains unchanged. By combining like terms, we ensure our expression is in its simplest form, \(a + bi\), which in this case is \(5 - 10i\). This practice of combining terms helps organize expressions logically and makes computations and further operations more straightforward.