Question

is the work shown below correct? explain your answer.

\\((11 + 2i) - (3 - 10i)\\)
\\(= 11 + 2i - 3 - 10i\\)
\\(= (11 - 3) + (2i - 10i)\\)
\\(= 8 - 8i\\)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Adding and subtracting real parts with real parts, and imaginary parts with imaginary parts.

Step 1: Analyze the original expression and the first step

The original problem is:

$$ (11 + 2i) - (3 - 10i) $$

The first step shown in the image is:

$$ = 11 + 2i - 3 - 10i $$

When subtracting a complex number in parentheses, we must distribute the negative sign to both terms inside the parentheses:

$$ -(3 - 10i) = -3 + 10i $$

The work shown failed to distribute the negative sign to the imaginary term \( -10i \), incorrectly leaving it as \( -10i \) instead of changing it to \( +10i \).

Step 2: Identify the subsequent errors

Because of the sign error in the first step, the grouping in the second step is also incorrect:

$$ = (11 - 3) + (2i - 10i) $$

Using the correct distribution, the terms should be grouped as:

$$ (11 - 3) + (2i + 10i) $$

Step 3: Calculate the correct simplified form

Let's find the correct result by distributing and combining like terms properly:

$$ (11 + 2i) - (3 - 10i) = 11 + 2i - 3 + 10i $$

$$ = (11 - 3) + (2i + 10i) $$

$$ = 8 + 12i $$

The work in the image incorrectly concluded with \( 8 - 8i \).

Answer:

No, the work shown is incorrect.

When subtracting the complex number \( (3 - 10i) \), the negative sign must be distributed to both terms inside the parentheses. This should result in \( -3 + 10i \), but the work incorrectly shows \( -3 - 10i \).

The correct steps are:

$$ (11 + 2i) - (3 - 10i) = 11 + 2i - 3 + 10i $$

$$ = (11 - 3) + (2i + 10i) $$

$$ = 8 + 12i $$