Sure! Let’s solve each problem step by step. The key idea is that
complex numbers are added or subtracted by combining their
real parts and
imaginary parts separately.
A complex number has the form:
>
a + bi, where *a* is the real part, and *b* is the coefficient of the imaginary unit *i* (where *i² = -1*).
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✔ Problem 1: (i) + (3i)
- Both are purely imaginary.
- Add coefficients: 1i + 3i =
4i
Answer: 4i
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✔ Problem 2: (15i) – (7i)
- Subtract imaginary parts: 15i – 7i =
8i
Answer: 8i
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✔ Problem 3: (–5 + 15i) – (–3 + 3i)
- Distribute the minus sign: –5 + 15i + 3 – 3i
- Combine real parts: –5 + 3 =
–2
- Combine imaginary parts: 15i – 3i =
12i
- Final answer:
–2 + 12i
Answer: –2 + 12i
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✔ Problem 4: –8 + (2i) + (–8 + 6i)
- Group real and imaginary:
- Real: –8 + (–8) =
–16
- Imaginary: 2i + 6i =
8i
- Final answer:
–16 + 8i
Answer: –16 + 8i
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✔ Problem 5: (–6i) – (6i)
- Subtract: –6i – 6i =
–12i
Answer: –12i
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✔ Problem 6: (–5 + 3i) – (4 – 5i)
- Distribute minus: –5 + 3i – 4 + 5i
- Real: –5 – 4 =
–9
- Imaginary: 3i + 5i =
8i
- Final:
–9 + 8i
Answer: –9 + 8i
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✔ Problem 7: (5 + 3i) – (–2 – 5i)
- Distribute minus: 5 + 3i + 2 + 5i
- Real: 5 + 2 =
7
- Imaginary: 3i + 5i =
8i
- Final:
7 + 8i
Answer: 7 + 8i
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✔ Problem 8: (–1 + i) – (–7 + 4i) – 5
- First, handle subtraction: –1 + i + 7 – 4i – 5
- Real: –1 + 7 – 5 =
1
- Imaginary: i – 4i =
–3i
- Final:
1 – 3i
Answer: 1 – 3i
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✔ Problem 9: (–7i) – (3i)
- Subtract: –7i – 3i =
–10i
Answer: –10i
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✔ Problem 10: (i) + (8 – i)
- Combine: 8 + i – i =
8 + 0i = 8
Answer: 8
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✔ Problem 11: (–5 – 3i) – (–6 + 8i)
- Distribute minus: –5 – 3i + 6 – 8i
- Real: –5 + 6 =
1
- Imaginary: –3i – 8i =
–11i
- Final:
1 – 11i
Answer: 1 – 11i
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✔ Problem 12: (–1 + i) – (–3 – 11i)
- Distribute minus: –1 + i + 3 + 11i
- Real: –1 + 3 =
2
- Imaginary: i + 11i =
12i
- Final:
2 + 12i
Answer: 2 + 12i
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✔ Problem 13: (–2 – i) – (4 + i)
- Distribute minus: –2 – i – 4 – i
- Real: –2 – 4 =
–6
- Imaginary: –i – i =
–2i
- Final:
–6 – 2i
Answer: –6 – 2i
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✔ Problem 14: (–6 + 6i) – (–5 + i)
- Distribute minus: –6 + 6i + 5 – i
- Real: –6 + 5 =
–1
- Imaginary: 6i – i =
5i
- Final:
–1 + 5i
Answer: –1 + 5i
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## 📝 Final Answers Summary:
1)
4i
2)
8i
3)
–2 + 12i
4)
–16 + 8i
5)
–12i
6)
–9 + 8i
7)
7 + 8i
8)
1 – 3i
9)
–10i
10)
8
11)
1 – 11i
12)
2 + 12i
13)
–6 – 2i
14)
–1 + 5i
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✔ You’re all set! Just remember:
combine like terms — real with real, imaginary with imaginary. That’s the golden rule for adding/subtracting complex numbers.
Parent Tip: Review the logic above to help your child master the concept of complex number worksheet.
