Complex Number Calculator
Add, subtract, multiply and divide complex numbers, or find a conjugate, polar form, integer power or all nth roots, with modulus and argument in degrees or radians.
Interactive calculator · Method COMPLEX-NUMBER-1.0 · Last reviewed: 2026-11-04
Calculate your scenario
(a + bi) ± (c + di) = (a ± c) + (b ± d)i; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²).
Your inputs
Default worked example. Edit the inputs to calculate your scenario.
Default worked example
(3 + 4i) × (1 − 2i) = 11 − 2i; |w| = √(11² + -2²) = 11.18034, θ = atan2(-2, 11) = -10.304846°.
How to use the calculator
Enter the real and imaginary parts of z₁ (and z₂ for the four arithmetic operations), choose an operation, and set n for powers or roots.
What the result means
(3 + 4i)(1 − 2i) = 11 − 2i, a point at distance √125 ≈ 11.18 from the origin, just below the real axis at about −10.3°.
Formula and units
(a + bi) ± (c + di) = (a ± c) + (b ± d)i; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²). Conjugate a − bi. Modulus r = √(a² + b²), argument θ = atan2(b, a) in (−180°, 180°]. Powers use repeated multiplication (negative n inverts); the n-th roots are r^(1/n) ∠ (θ + 360° k) ÷ n for k = 0 … n − 1.
Worked examples
(3 + 4i) × (1 − 2i): z₁ real part: 3 dimensionless; z₁ imaginary part: 4 dimensionless; z₂ real part: 1 dimensionless; z₂ imaginary part: -2 dimensionless; n (power or root; roots need n ≥ 1): 3 dimensionless; Operation: mul; Angle unit: deg.
Example result: Rectangular form: 11 − 2i; Polar form: 11.18034 ∠ -10.304846°; Modulus |w|: 11.18034; Argument θ: -10.304846°; Operation: (3 + 4i) × (1 − 2i).
(3 + 4i) ÷ (1 − 2i): z₁ real part: 3 dimensionless; z₁ imaginary part: 4 dimensionless; z₂ real part: 1 dimensionless; z₂ imaginary part: -2 dimensionless; n (power or root; roots need n ≥ 1): 3 dimensionless; Operation: div; Angle unit: deg.
Example result: Rectangular form: -1 + 2i; Polar form: 2.236068 ∠ 116.565051°; Modulus |w|: 2.236068; Argument θ: 116.565051°; Operation: (3 + 4i) ÷ (1 − 2i) = (3 + 4i) × (1 + 2i) ÷ 5.
Limitations and common mistakes
Results display to 6 decimal places, and values within 10⁻¹² of the modulus are shown as 0 to hide floating-point noise. The argument is the principal value; 0 has no defined argument. Powers are limited to |n| ≤ 20, and the roots table lists at most 12 roots.
Forgetting that i² = −1 when multiplying by hand, or reading the argument from atan(b/a), which puts numbers in the second and third quadrants on the wrong side.
Sources and assumptions
Method COMPLEX-NUMBER-1.0; last formula review 2026-11-04. Sources and their supported claims. Calculation methodology.