Problem 16 - Intermediate Algebra

Question

Add or subtract as indicated. $$ (-8+4 i)-(9-4 i) $$

Step-by-Step Solution

Answer

The result is \(-17 + 8i\).

Step 1: Identify the Operation

The given problem requires us to subtract two complex numbers: \(-8+4i\) and \(9-4i\). This means we need to perform subtraction between the real parts and the imaginary parts separately.

Step 2: Subtract the Real Parts

To subtract the real parts, take the real number from the first complex number, -8, and subtract it from the real number of the second complex number, 9. Calculate:\(-8 - 9 = -17\).

Step 3: Subtract the Imaginary Parts

Next, subtract the imaginary part of the second complex number from the imaginary part of the first complex number.Calculate:\(4i - (-4i) = 4i + 4i = 8i\).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to write the difference as a new complex number.So we have:\(-17 + 8i\).

Key Concepts

Subtracting Complex Numbers

Complex numbers consist of a real part and an imaginary part. Subtracting them involves working with these components separately. In this exercise, we are presented with two complex numbers:

• First number: \(-8 + 4i\)
• Second number: \(9 - 4i\)

To subtract complex numbers, you handle the real parts and imaginary parts independently. For the real parts, subtract the real component of the second number from the first. Similarly, subtract the imaginary component of the second from the first. This separation allows us to deal with potentially complicated expressions more easily. Remember, subtraction in complex numbers follows similar rules to algebra but carefully handles the imaginary unit, \(i\), which represents \(\sqrt{-1}\). Breaking the components allows for systematic and error-free calculations.

Real and Imaginary Parts

Each complex number can be expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In our exercise, for the complex numbers \(-8 + 4i\) and \(9 - 4i\):

• The real parts are \(-8\) and \(9\).
• The imaginary parts are \(4i\) and \(-4i\).

Identifying parts correctly is crucial. An error at this stage can lead to wrong calculations. The real part is just the number itself and doesn't interact with \(i\). The imaginary part multiplies with \(i\), which makes handling negative signs and addition of like terms important. A common pitfall is confusing subtraction of negative imaginary parts, where double negatives turn to positive. For instance, \(4i - (-4i)\) simplifies to \(4i + 4i\). Recognizing these can save time and prevent errors.

Steps in Solving Algebraic Problems

Solving problems involving algebra and complex numbers becomes straightforward with systematic steps.Step 1: Identify the Operation
Determine whether you need to add or subtract. Clearly understanding this sets the foundation.Step 2: Separate and Operate on Real and Imaginary Parts
Subtract the real parts of the numbers separately. Then, subtract the imaginary parts, being mindful of negative signs, especially with the imaginary parts where negatives can appear tricky.Step 3: Combine the Results
Meld the components together to express the final result as a single complex number. This reassembly combines the results into a neat form, such as \(-17 + 8i\) in our case.
While following these structured steps, reassess your intermediate calculations for accuracy. Understanding these steps ensures precision and helps build confidence in handling similar problems efficiently in future exercises.