Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (-4+4 i)-(-6+9 i) $$

Step-by-Step Solution

Answer

The simplified result is \(2 - 5i\).

Step 1: Understand the operation

We need to perform the subtraction of the complex numbers: \((-4 + 4i)\) and \((-6 + 9i)\). Subtraction should be performed separately on real and imaginary parts.

Step 2: Subtract the real parts

Subtract the real part of the second complex number from the first: \[ (-4) - (-6) = -4 + 6 = 2 \] The real part of the result is \(2\).

Step 3: Subtract the imaginary parts

Subtract the imaginary part of the second complex number from the first: \[ 4i - 9i = -5i \] The imaginary part of the result is \(-5i\).

Step 4: Form the simplified complex number

Combine the results from Steps 2 and 3 to get the simplified complex number: \[ 2 - 5i \]

Key Concepts

Complex Number Operations

Complex numbers are a crucial part of mathematics, especially when dealing with equations that don't easily solve within the realm of real numbers. A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). When we talk about operations with complex numbers, we refer to addition, subtraction, multiplication, and division.

To perform these operations:

• Addition: Add the real parts and imaginary parts separately.
• Subtraction: Subtract the real parts and imaginary parts separately.
• Multiplication: Use the distributive property and substitute \( i^2 \) with \(-1\).
• Division: Multiply by the conjugate to eliminate the imaginary part in the denominator.

The mentioned operations simplify how complex combinations work together, making it easier to solve equations that appear complicated initially.

Subtraction of Complex Numbers

Subtraction of complex numbers follows a straightforward approach. You perform subtraction by dealing with the real and imaginary parts separately. Consider the problem \((-4 + 4i) - (-6 + 9i)\).

Here's how you perform the subtraction:

• Real Part Calculation: You subtract the real part of the second number from the real part of the first number: \((-4) - (-6) = 2\).
• Imaginary Part Calculation: Similarly, subtract the imaginary part of the second number from the first: \(4i - 9i = -5i\).

By following these simple steps, you can easily subtract complex numbers, yielding the result in a simplified complex form. This makes the numbers more manageable and ready for further operations if needed.

Simplifying Complex Numbers

Once you perform operations on complex numbers, like subtraction, the result must often be simplified.

Simplifying a complex number entails expressing it neatly in the standard form \(a + bi\). Using our example \(2 - 5i\), this form clearly shows the real component \(a = 2\) and the imaginary component \(b = -5\).

The simplification process might include:

• Combining like terms so that real and imaginary parts are distinct and separate.
• Expressing the result in simplest form without radicals or fractions, if possible.
• Ensuring the imaginary unit \(i\) appears only once, typically after the 'b' coefficient.

This neat arrangement is essential for solving problems further, like finding the magnitude or angle when considering complex numbers in polar form. Simplification helps improve clarity and understanding, which are highly valuable in mathematical discussions and applications.