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Short answer
With z₁ = 3 + 4i and z₂ = 1 − 2i: the sum is 4 + 2i, the difference is 2 + 6i, the product is 11 − 2i and the quotient is −1 + 2i. Division uses the conjugate of the denominator.
1. The four rules
- Add or subtract: combine the real parts and the imaginary parts separately.
- Multiply: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, using i² = −1.
- Divide: multiply top and bottom by the conjugate of the denominator.
2. Worked results for 3 + 4i and 1 − 2i
| Operation | Working | Result |
|---|---|---|
| Add | (3 + 1) + (4 − 2)i | 4 + 2i |
| Subtract | (3 − 1) + (4 + 2)i | 2 + 6i |
| Multiply | 3 − 6i + 4i − 8i² | 11 − 2i |
| Divide | (3 + 4i)(1 + 2i) ÷ 5 | −1 + 2i |
3. Division in full
The conjugate of 1 − 2i is 1 + 2i. Top: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i. Bottom: (1 − 2i)(1 + 2i) = 1² + 2² = 5. Result: (−5 + 10i) ÷ 5 = −1 + 2i.
4. Check your answer
Multiply the quotient by the divisor. (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = −1 + 4i + 4 = 3 + 4i, which is the original numerator.
5. Common mistakes
- Forgetting that i² = −1, which flips the sign of the 8i² term.
- Dividing real and imaginary parts separately instead of using the conjugate.
- Changing the sign of the whole denominator instead of only its imaginary part.
6. Frequently asked questions
How do you divide complex numbers?expand_more
Multiply top and bottom by the conjugate of the denominator, then simplify.
What is i squared?expand_more
i² = −1, which is why imaginary terms multiplied together become real.
What is the conjugate of 1 − 2i?expand_more
1 + 2i. Only the sign of the imaginary part changes.
How can I check a division?expand_more
Multiply the answer by the divisor and compare with the numerator.
7. Sources
- menu_bookWolfram MathWorld: Complex Number
- menu_bookWolfram MathWorld: De Moivre's Identity