Operations of complex numbers worksheet - Free Printable

Since the user has not specified which option to use (ExplainAnswer, TeachTopic, or GeneratePractice), and all are false, I will proceed to solve a representative selection of problems from the worksheet step by step, verifying each calculation carefully before presenting the final answers.

I’ll choose one problem from each major section to demonstrate:

1. Addition/Subtraction: Problem 1 → (4 + i) + (7 - i)
2. Multiplication/Simplification: Problem 7 → 5(-8i)(1 + i)
3. Challenge Problem: Problem 10 → -6 - (-2 - 2i) - (5 - 4i)
4. Rationalizing Denominator: Problem 14 → (3i)/(-3 + i)
5. Simplify Complex Fraction: Problem 17 → (-7 - 3i)/(2 + 6i)

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Problem 1: (4 + i) + (7 - i)

Step 1: Remove parentheses:
4 + i + 7 - i

Step 2: Combine real parts: 4 + 7 = 11
Combine imaginary parts: i - i = 0

Final result: 11 + 0i → just 11

✔ Verified.

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Problem 7: 5(-8i)(1 + i)

Step 1: Multiply constants first: 5 × (-8i) = -40i

Step 2: Now multiply -40i by (1 + i):
-40i × 1 = -40i
-40i × i = -40i²

Recall: i² = -1 → so -40i² = -40(-1) = +40

Step 3: Add results: 40 - 40i

Final answer: 40 - 40i

✔ Verified.

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Problem 10: -6 - (-2 - 2i) - (5 - 4i)

Step 1: Distribute the negatives:
-6 + 2 + 2i - 5 + 4i

(Why? Because minus a negative is plus; minus a positive is minus.)

Step 2: Combine real parts: -6 + 2 - 5 = -9
Combine imaginary parts: 2i + 4i = 6i

Final answer: -9 + 6i

✔ Verified.

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Problem 14: (3i)/(-3 + i)

We need to rationalize the denominator — multiply numerator and denominator by the conjugate of the denominator.

Conjugate of (-3 + i) is (-3 - i)

So:

Numerator: 3i × (-3 - i) = 3i×(-3) + 3i×(-i) = -9i -3i²
But i² = -1 → -3i² = -3(-1) = +3
So numerator = 3 - 9i

Denominator: (-3 + i)(-3 - i) = (-3)² - (i)² = 9 - (-1) = 9 + 1 = 10

So overall: (3 - 9i)/10 = 3/10 - (9/10)i

Final answer: \frac{3}{10} - \frac{9}{10}i

✔ Verified.

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Problem 17: (-7 - 3i)/(2 + 6i)

Rationalize by multiplying numerator and denominator by conjugate of denominator: (2 - 6i)

Numerator: (-7 - 3i)(2 - 6i)
= -7×2 + (-7)×(-6i) + (-3i)×2 + (-3i)×(-6i)
= -14 + 42i -6i + 18i²
i² = -1 → 18i² = -18
So: -14 -18 + (42i -6i) = -32 + 36i

Denominator: (2 + 6i)(2 - 6i) = 2² - (6i)² = 4 - 36i² = 4 - 36(-1) = 4 + 36 = 40

So: (-32 + 36i)/40 = -32/40 + 36i/40 = simplify fractions:

Divide numerator and denominator by 4:
-8/10 + 9i/10 → further reduce: -4/5 + (9/10)i

Wait — let’s check simplification again:

-32 ÷ 8 = -4, 40 ÷ 8 = 5 → -4/5
36 ÷ 4 = 9, 40 ÷ 4 = 10 → 9/10

Actually, better to divide numerator and denominator by 4 directly:

-32/40 = -8/10 = -4/5
36/40 = 9/10

So final: -4/5 + (9/10)i

Alternatively, write as single fraction: (-32 + 36i)/40 → but standard form is separate real and imaginary.

✔ Verified.

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Final Answer:
1) 11
7) 40 - 40i
10) -9 + 6i
14) $\frac{3}{10} - \frac{9}{10}i$
17) $-\frac{4}{5} + \frac{9}{10}i$