Complex Number Calculator: Operations, Roots and Polar Form

How to use it

Type the first number in First number z₁, as a + bi or as r∠angle.
Choose the Operation. For the four arithmetic operations, type the second number in Second number z₂; for a power or a root, type the whole number in Power n or root n.
Read Result. The Forms of the result table shows the rectangular, polar and exponential forms; for a root, the The n-th roots table lists every root.
Go through How it is worked out for each step.

Rectangular, polar and exponential forms

A complex number is a + bi with i² = −1. Its modulus is |a + bi| = √(a² + b²), and its argument is the angle arg z = atan2(b, a) from the positive real axis. The polar form is r∠θ = r(cos θ + i sin θ), and the exponential form is r·e^(iθ), with θ in radians. Multiplying two numbers multiplies their moduli and adds their arguments, which is why powers and roots are easiest in polar form.

(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(a + bi) ÷ (c + di) = ((ac + bd) + (bc − ad)i) ÷ (c² + d²)
The n-th roots of r∠θ: r^(1/n) ∠ (θ + 360°k) ÷ n

The n-th roots of a nonzero number are n points spaced evenly on a circle of radius |z|^(1/n).
The argument is principal: it lies between −180° and 180°, and the root angles are shown in that range.
A rectangular number typed as a + bi is computed exactly; a polar number, its roots and its argument are decimals.

What the calculator does not cover

The calculator handles one or two complex numbers at a time, with real parts and imaginary parts. It does not solve equations with complex unknowns, and it does not give exact roots of most numbers: a root that is not a simple number is shown as a decimal.

Powers and roots are limited to exponents from −20 to 20 and to roots of order 1 to 12.
The exponential form uses radians, as the mathematics does, even when the angles are shown in degrees.
Division by a complex number equal to 0 is refused.

Worked example: (3 + 4i)(1 − 2i) and the cube roots of 8

The product is (3 × 1 − 4 × (−2)) + (3 × (−2) + 4 × 1)i = 11 − 2i. The modulus of 3 + 4i is √25 = 5, and the quotient (3 + 4i) ÷ (1 − 2i) = −1 + 2i.

The cube roots of 8, which has modulus 8 and angle 0°, have modulus 2 and angles 0°, 120° and −120°. They are 2, −1 + 1.7321i and −1 − 1.7321i.

Question	Answer
(3 + 4i)(1 − 2i)	11 − 2i
(3 + 4i) ÷ (1 − 2i)	−1 + 2i
\|3 + 4i\|	5
Cube roots of 8, the second one	−1 + 1.7321i

Frequently asked questions

What is the difference between polar and exponential form?

Both give the same number with the same modulus r and argument θ. The polar form is written r∠θ, and the exponential form is r·e^(iθ), with θ always in radians, as the mathematics requires.

Why are some roots shown with an angle of −120° instead of 240°?

The page shows every angle between −180° and 180°. The angle 240° is the same direction as −120°, so both give the same root.

How do I type a number with a fraction?

Type the fraction in the part you need: 1/2 + 3/4i is one half plus three quarters of i. Decimals work too, with the decimal mark of your language.

When does the result show a fraction such as 1/3?

Only when the exact value has no finite decimal, such as 1/3. The calculator keeps it as a fraction instead of rounding it. A value such as 3/25 is exactly 0.12, so it is shown as that decimal.

Are the numbers I type sent to a server?

No. The calculation runs in your browser, and nothing you type is sent to us.

Sources

Figures and text last checked on Tuesday, November 10, 2026.

Result

How it is worked out