Complex Number Calculator
Add, subtract, multiply or divide two complex numbers; find the modulus, argument, conjugate and polar form of one; or compute integer powers and nth roots with De Moivre's theorem — with a plot on the complex plane and every step shown.
Written by Suhaib Hassan. Every identity is re-derived by hand. Our methodology.
Arithmetic uses the standard algebraic rules (i² = −1); powers and roots use De Moivre's theorem on the polar form, r(cos θ + i sin θ).
It plots every number on the Argand plane, so the geometry behind the algebra is never hidden.
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Argand plane
Each number plotted as a point (Re, Im).
Step-by-step working
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Worked examples
Multiplying
(3+4i) × (1−2i)
= 11 − 2i
Modulus & argument
Find |3+4i| and Arg(3+4i).
θ = atan2(4,3) ≈ 53.13°
De Moivre power
(1+i)⁸
(1+i)⁸ = 16
Key formulas
- Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
- Division: multiply by the conjugate: z₁/z₂ = z₁z̄₂ / |z₂|².
- Modulus: |z| = √(a²+b²). Argument: θ = atan2(b, a), in (−π, π].
- De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
- nth roots: ⁿ√z has n values, at angles (θ+2πk)/n for k = 0, …, n−1, each with modulus |z|^(1/n).
By Suhaib Hassan · Reviewed 29 September 2026 · How CalculatePilot verifies formulas →
Sources
- Churchill, R. V., & Brown, J. W. (2013). Complex Variables and Applications (9th ed.). McGraw-Hill, chapters 1–2.
- ISO 80000-2:2019, clause 8 (complex numbers).
How to divide complex numbers: worked example
Multiply the top and bottom by the conjugate of the denominator. For (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i.
Top: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i. Bottom: (1 − 2i)(1 + 2i) = 1 + 4 = 5. Result: (−5 + 10i) ÷ 5 = −1 + 2i.
Polar form, modulus and argument
For z = a + bi the modulus (also called the magnitude or absolute value) is |z| = √(a² + b²) and the argument is θ = atan2(b, a). Polar form is r(cos θ + i sin θ), often written r∠θ. Angles below are in degrees.
| Rectangular | Modulus r | Argument θ | Polar r∠θ |
|---|---|---|---|
| 1 + i | √2 (1.41) | 45° | √2∠45° |
| −1 + i | √2 (1.41) | 135° | √2∠135° |
| 3 + 4i | 5 | 53.13° | 5∠53.13° |
| −3 − 4i | 5 | −126.87° | 5∠−126.87° |
| 2i | 2 | 90° | 2∠90° |
| −2 | 2 | 180° | 2∠180° |
| 1 − i | √2 (1.41) | −45° | √2∠−45° |
| −1 − i | √2 (1.41) | −135° | √2∠−135° |
Use atan2(b, a), not atan(b ÷ a). Plain arctangent cannot tell −3 − 4i from 3 + 4i, because both have b ÷ a = 4/3, but their angles differ by 180°.
Square roots and nth roots
Every non-zero complex number has two square roots and n different nth roots. For √(3 + 4i), the roots are 2 + i and −2 − i, because (2 + i)² = 4 + 4i + i² = 3 + 4i.
The cube roots of 1 have modulus 1 and angles 0°, 120° and 240°:
| k | Angle | Cube root of 1 |
|---|---|---|
| 0 | 0° | 1 |
| 1 | 120° | −0.5 + 0.866i |
| 2 | 240° | −0.5 − 0.866i |
The four 4th roots of 1 are 1, i, −1 and −i, at 0°, 90°, 180° and 270°. For the cube roots of 8i, use modulus 8^(1/3) = 2 and angles (90° + 360°k) ÷ 3 = 30°, 150° and 270°.
Powers of i
The powers of i repeat every four: i, −1, −i, 1. To find iⁿ, divide n by 4 and use the remainder (a remainder of 0 means 1). For example, 27 ÷ 4 leaves a remainder of 3, so i²⁷ = i³ = −i.
| n | iⁿ |
|---|---|
| 1 | i |
| 2 | −1 |
| 3 | −i |
| 4 | 1 |
| 5 | i |
| 6 | −1 |
| 7 | −i |
| 8 | 1 |
Notation: i, j, magnitude and phasors
Mathematics writes the imaginary unit as i. Electrical engineering writes it as j so that i can stand for current. The result is the same: j² = −1. Modulus, magnitude and absolute value mean the same thing for a complex number, and argument, angle and phase mean the same thing. A phasor is a complex number in polar form, such as 5∠53.13°. When you multiply in polar form, multiply the moduli and add the angles: (3 + 4i)(1 − 2i) = (5∠53.13°)(√5∠−63.43°) = 11.18∠−10.3°, which equals 11 − 2i.
Complex number guides
- Complex number calculator: add, subtract, multiply, divide
- Polar form, modulus and argument with examples
- Square roots and nth roots of complex numbers
- Rectangular vs polar form: which to use
Frequently asked questions
Why does dividing complex numbers use the conjugate?
Multiplying the denominator by its conjugate z̄ turns it into the real number |z|², eliminating i from the denominator — the complex equivalent of rationalizing a square root.
What is the geometric meaning of multiplying complex numbers?
Multiplication multiplies the moduli and adds the arguments — geometrically, it scales and rotates. This is exactly what De Moivre's theorem generalizes to powers and roots.
Why does a complex number have n different nth roots?
Because angles that differ by a full turn (2π) describe the same point, but raising to the nth power multiplies the angle by n — so n different starting angles, spaced 2π/n apart, all land on the same result.
What is the complex conjugate used for?
Besides division, z·z̄ = |z|² always gives a real number, which is essential for finding magnitudes, normalizing vectors, and in signal processing for computing power from a complex amplitude.
How is Euler's formula related to polar form?
Euler's formula e^(iθ) = cos θ + i sin θ lets polar form r(cos θ + i sin θ) be written compactly as r·e^(iθ) — the same number, and the form that makes multiplication, powers and roots most natural.
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator, which makes the denominator a real number. (3 + 4i) ÷ (1 − 2i) = (−5 + 10i) ÷ 5 = −1 + 2i.
What is the magnitude or absolute value of a complex number?
The modulus |a + bi| = √(a² + b²). For 3 + 4i it is √25 = 5. Magnitude, modulus and absolute value are the same thing.
How do you convert a complex number to polar form?
Find r = √(a² + b²) and θ = atan2(b, a), then write r(cos θ + i sin θ) or r∠θ. For 3 + 4i that is 5∠53.13°.
How do you find the angle of a complex number?
Use atan2(b, a), which gives the correct quadrant. For −3 − 4i the angle is −126.87°, not 53.13°.
What is the square root of a complex number?
There are two, opposite in sign. √(3 + 4i) = ±(2 + i), because (2 + i)² = 3 + 4i. The nth roots come from De Moivre's theorem.
How do you find the nth roots of a complex number?
Take the nth root of the modulus and use angles (θ + 360°k) ÷ n for k = 0 to n − 1. The cube roots of 1 are at 0°, 120° and 240°.
What is the difference between i and j?
Nothing mathematically. Engineers use j so that i can mean current. Both satisfy j² = i² = −1.
How do you work out powers of i?
They repeat every four: i, −1, −i, 1. Divide the exponent by 4 and use the remainder, so i²⁷ = i³ = −i.
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