Complex Number Calculator
Multiply complex numbers like binomials and use i² = −1: (3 + 4i)(1 − 2i) = 11 − 2i.
11 − 2i
- Modulus |result|
- 11.18033989
- Argument (degrees)
- -10.30484647 °
- Conjugate
- 11 + 2i
- |z₁|
- 5
Step-by-step solution
- (a + bi)(c + di) = (ac − bd) + (ad + bc)i, using i² = −1
- = 11 − 2i
Complex Number Formula
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
i² = −1; divide by multiplying by the conjugate
Worked Example
(3 + 4i) × (1 − 2i):
- (a + bi)(c + di) = (ac − bd) + (ad + bc)i, using i² = −1
- = 11 − 2i
Argument (degrees): -10.30484647 ° · Conjugate: 11 + 2i · |z₁|: 5
Test-day tip
Complex numbers appear on both tests, usually as one multiplication or a division by a conjugate; write the answer in a + bi form to match the choices.
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Frequently asked questions
What are the powers of i?
They repeat every four: i, −1, −i, 1. Divide the exponent by 4 and use the remainder: i²⁷ = i³ = −i.
How do I divide by a complex number?
Multiply the top and bottom by the conjugate of the denominator, which makes the denominator the real number c² + d².
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Complex Number Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/complex-number-calculator/
IEEE
Scientific Converters, "Complex Number Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/complex-number-calculator/