Question

which subtraction expression has the difference \\(1 + 4i\\)?

\\((-2 + 6i) - (1 - 2i)\\)
\\((-2 + 6i) - (-1 - 2i)\\)
\\((3 + 5i) - (2 - i)\\)
\\((3 + 5i) - (2 + i)\\)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Combining real parts and imaginary parts separately.

Step 1: Understand the target difference

We want to find which subtraction expression results in the complex number:

$$ 1 + 4i $$

To subtract complex numbers, we subtract their real parts and their imaginary parts separately:

$$ (a + bi) - (c + di) = (a - c) + (b - d)i $$

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Step 2: Evaluate the first option

Let's simplify \( (-2 + 6i) - (1 - 2i) \):

$$ (-2 - 1) + (6 - (-2))i $$

$$ -3 + (6 + 2)i = -3 + 8i $$

This does not equal \( 1 + 4i \).

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Step 3: Evaluate the second option

Let's simplify \( (-2 + 6i) - (-1 - 2i) \):

$$ (-2 - (-1)) + (6 - (-2))i $$

$$ (-2 + 1) + (6 + 2)i = -1 + 8i $$

This does not equal \( 1 + 4i \).

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Step 4: Evaluate the third option

Let's simplify \( (3 + 5i) - (2 - i) \):

$$ (3 - 2) + (5 - (-1))i $$

$$ 1 + (5 + 1)i = 1 + 6i $$

This does not equal \( 1 + 4i \).

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Step 5: Evaluate the fourth option

Let's simplify \( (3 + 5i) - (2 + i) \):

$$ (3 - 2) + (5 - 1)i $$

$$ 1 + 4i $$

This matches our target difference of \( 1 + 4i \).

Answer:

The correct option is the fourth one:

$$ (3 + 5i) - (2 + i) $$