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Complex Number Calculator

Work with complex numbers in a + bi form. Add, subtract, multiply or divide two numbers, convert to polar and exponential form, or raise to a power and find all n-th roots with De Moivre's theorem, each with steps and a plot on the complex plane.

Operation

Angle unit

First number z₁ = a + bi

a (real part of z₁)

b (imaginary part of z₁)

Second number z₂ = c + di

c (real part of z₂)

d (imaginary part of z₂)

n (power or root)

Roots need a whole number from 1 to 100

Enter your values to see the result.

How to use the Complex Number Calculator

Choose the operation: arithmetic, polar form, a power or roots.
Enter z₁ = a + bi (and z₂ = c + di for arithmetic, or n for powers and roots).
Pick degrees or radians for angles.
Press Calculate to see the result in rectangular and polar form.
Read the steps and the complex plane plot, or copy the results.

Complex number formulas

Operation	Formula
Addition	(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction	(a + bi) − (c + di) = (a − c) + (b − d)i
Multiplication	(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Division	(a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
Modulus	\|z\| = √(a² + b²)
Argument	θ = atan2(b, a)
Polar / exponential form	z = r(cos θ + i sin θ) = r·e^(iθ)
Power (De Moivre)	zⁿ = rⁿ(cos nθ + i sin nθ)
n-th roots	w_k = r^(1/n)·(cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)), k = 0 … n − 1
Conjugate	z̄ = a − bi, and z·z̄ = a² + b²

Powers of i

The powers of i repeat in a cycle of four, so iⁿ depends only on n mod 4.

n mod 4	0	1	2	3
iⁿ	1	i	−1	−i
Example	i⁴ = 1	i⁵ = i	i¹⁰ = −1	i⁷ = −i

Common angles in polar form

z	Modulus r	Argument θ	Polar form
1 + i	√2 ≈ 1.4142	45° (π/4)	√2 ∠ 45°
i	1	90° (π/2)	1 ∠ 90°
−1	1	180° (π)	1 ∠ 180°
3 + 4i	5	53.1301°	5 ∠ 53.13°
1 + √3 i	2	60° (π/3)	2 ∠ 60°
−2 − 2i	2√2 ≈ 2.8284	−135° (−3π/4)	2.8284 ∠ −135°

Frequently asked questions

How do you multiply complex numbers?

Expand like binomials and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 4i)(1 − 2i) that gives (3 + 8) + (−6 + 4)i = 11 − 2i.

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²). The denominator becomes a real number.

How do I convert a complex number to polar form?

The modulus is r = √(a² + b²) and the argument is θ = atan2(b, a), which places the angle in the correct quadrant. Then z = r(cos θ + i sin θ) = r·e^(iθ). For 1 + i, r = √2 ≈ 1.4142 and θ = 45°.

What is De Moivre's theorem?

For z = r(cos θ + i sin θ) and a whole number n, zⁿ = rⁿ(cos nθ + i sin nθ). It makes powers easy: (1 + i)⁸ = (√2)⁸(cos 360° + i sin 360°) = 16.

How many n-th roots does a complex number have?

Every non-zero complex number has exactly n distinct n-th roots. They have modulus r^(1/n) and arguments (θ + 2πk)/n for k = 0 … n − 1, so they sit evenly spaced on a circle. The cube roots of 8 are 2 and −1 ± 1.732051i.

What range does the argument use?

The principal argument returned here is between −180° and 180° (−π to π radians), as given by atan2. Adding any multiple of 360° gives the same complex number.

What is the complex conjugate?

The conjugate of a + bi is a − bi. Multiplying a number by its conjugate gives the real number a² + b² = |z|², which is why conjugates are used for division.

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