Complex number calculator
Enter two complex numbers, get the results.
How do I multiply complex numbers?
(a + bi)(c + di) = (ac − bd) + (ad + bc)i, since i² = −1. (3 + 4i)(1 − 2i) = 11 − 2i.
How do I divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom: (3 + 4i) ÷ (1 − 2i) = (3 + 4i)(1 + 2i) ÷ 5 = −1 + 2i.
What are the modulus and argument?
The modulus |z| = √(a² + b²) is the distance from 0; the argument is the angle from the positive real axis. For 3 + 4i: |z| = 5 and the angle is 53.13°.
How do I raise a complex number to a power?
By De Moivre’s formula: raise the modulus to the power and multiply the angle by it. (3 + 4i)² = −7 + 24i.
What is the square root of a complex number?
The calculator gives the principal root: √(3 + 4i) = 2 + i. The other root is its negative, −2 − i.
What is the polar form of a complex number?
z = r(cos θ + i sin θ), where r is the modulus and θ the argument. It is also written r·e^(iθ), by Euler's formula. Multiplying two complex numbers in polar form multiplies the moduli and adds the angles, which is why powers and roots are easy there.
Why does the argument depend on the quadrant?
The angle must point to the right quadrant. For −1 + i the plain arctan(1 ÷ −1) gives −45°, but the number is in the second quadrant, so the argument is 135°. The calculator uses the two-argument arctangent, which gets this right.
What is the conjugate for?
The conjugate of a + bi is a − bi. A number times its conjugate is a² + b², a real number, which is how a division by a complex number is turned into a division by a real one.
How the calculation works
Complex numbers a + bi are added and subtracted part by part. Multiplication uses i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division multiplies top and bottom by the conjugate of the divisor, so the denominator becomes the real number c² + d². The modulus is √(a² + b²) and the argument is the angle of the point (a, b) from the positive real axis, found with atan2. Powers use De Moivre's formula: the modulus is raised to the power and the angle multiplied by it. The square root returned is the principal root, with the other root being its negative.
Worked example
Let z = 2 − 3i and w = 4 + 1i. The sum is 6 − 2i and the difference is −2 − 4i. The product is (2 · 4 − (−3) · 1) + (2 · 1 + (−3) · 4)i = 11 − 10i. The quotient z ÷ w is 0.294 − 0.824i, using the denominator 4² + 1² = 17. For z, the modulus is √(2² + 3²) = 3.61, the argument is −56.31° (fourth quadrant) and the conjugate is 2 + 3i. The cube is −46 − 9i, and the principal square root is 1.674 − 0.896i.
Common mistakes
- Forgetting i² = −1. When multiplying, the product of the two imaginary parts is real and has the opposite sign: bi · di = −bd.
- Dividing without the conjugate. Splitting the quotient into a ÷ c and b ÷ d is wrong. Multiply by the conjugate of the divisor.
- Reading the angle from arctan alone. The result of arctan(b ÷ a) can be 180° off. Check the quadrant of the point.
Sources and related calculators
Definitions and rules: OpenStax Algebra and Trigonometry 2e, the sections on complex numbers and on the polar form of complex numbers. Related: quadratic formula calculator, polar coordinates calculator and circle equation calculator.