About the Complex Number Calculator
This complex number calculator performs arithmetic on numbers of the form a + bi, where i is the imaginary unit (i² = −1). Enter the real and imaginary parts of z₁ and z₂, choose an operation — add, subtract, multiply, divide, raise to a power, take nth roots, conjugate or reciprocal — and get the result in rectangular form together with its modulus, argument and polar and exponential forms.
It is designed for algebra, precalculus and engineering students, and for anyone working with AC circuits (impedance and phasors), signal processing or control systems where complex arithmetic and polar conversion come up constantly. A table shows both inputs and the result side by side in every form.
Angles are given in degrees and radians using the principal argument between −180° and 180°. Powers and roots use De Moivre’s theorem, so all n roots of a number are listed, not just the principal one.
How to use the complex number calculator
- 1Enter the real and imaginary parts of z₁.
- 2Pick an operation.
- 3Enter z₂ for two-number operations, or n for powers and roots.
- 4Read the result in rectangular form (a + bi).
- 5Use the modulus, angle and polar form for phasor or geometric work.
Formula and method
Addition and subtraction work part by part: real with real, imaginary with imaginary. Multiplication expands like two binomials and uses i² = −1. Division multiplies the numerator and denominator by the conjugate of the divisor (c − di), which makes the denominator the real number c² + d².
Any complex number can also be written in polar form r(cos θ + i sin θ) = r·e^(iθ), where r = √(a² + b²) is the modulus and θ = atan2(b, a) is the argument. De Moivre’s theorem then gives powers as rⁿ(cos nθ + i sin nθ), and the n distinct nth roots as r^(1/n) at angles (θ + 2πk)/n for k = 0 … n − 1.
- i
- Imaginary unit, i² = −1
- a, b
- Real and imaginary parts of z₁
- c, d
- Real and imaginary parts of z₂
- r, θ
- Modulus and argument (angle) in polar form
Worked examples
Multiply (3 + 2i)(1 + 7i)
(3)(1) − (2)(7) = −11 for the real part and (3)(7) + (2)(1) = 23 for the imaginary part, so the product is −11 + 23i. Its modulus is √(121 + 529) ≈ 25.495.
Divide (3 + 2i) ÷ (1 + 7i)
Multiply by the conjugate 1 − 7i: the numerator becomes 17 − 19i and the denominator 1² + 7² = 50, giving 0.34 − 0.38i.
Raise (1 + i) to the 8th power
1 + i has modulus √2 and angle 45°. By De Moivre, (√2)⁸ = 16 and 8 × 45° = 360°, which points along the positive real axis, so the answer is exactly 16.
Cube roots of −8
−8 has modulus 8 and angle 180°. Its three cube roots have modulus 2 at 60°, 180° and 300°: 1 + 1.732i, −2 and 1 − 1.732i. The principal root (smallest angle) is shown first.
Frequently asked questions
How do you divide complex numbers?+
Multiply the numerator and denominator by the conjugate of the denominator. For (a + bi) ÷ (c + di), multiply by (c − di)/(c − di); the denominator becomes c² + d², a real number, and you can then split the result into real and imaginary parts.
How do I convert a complex number to polar form?+
The modulus is r = √(a² + b²) and the angle is θ = atan2(b, a), which places the angle in the correct quadrant. Then a + bi = r(cos θ + i sin θ) = r∠θ. For example 1 + i = √2∠45°.
What is the modulus of a complex number?+
The modulus |a + bi| = √(a² + b²) is the distance of the point (a, b) from the origin in the complex plane. It behaves like absolute value: |z₁z₂| = |z₁||z₂|.
What is the conjugate of a complex number?+
The conjugate of a + bi is a − bi — the reflection across the real axis. A number times its conjugate is always real: (a + bi)(a − bi) = a² + b², which is why conjugates are used for division.
How many nth roots does a complex number have?+
Every non-zero complex number has exactly n distinct nth roots, spaced evenly around a circle of radius r^(1/n) at angles 360°/n apart. For example, 1 has three cube roots: 1 and −0.5 ± 0.866i.