Complex number calculator — add, multiply, divide, powers and roots

How to use the complex number calculator

1. Type the first number in z₁, such as 3 + 4i.
2. Choose the Operation: z₁ + z₂, z₁ − z₂, z₁ × z₂, z₁ ÷ z₂, z₁ ^ z₂ (principal value), or √z₁ (both roots).
3. Type the second number in z₂. This box is hidden for square roots.
4. Read the Result, then its Polar form and Modulus.
5. For multiplication and division, check the Working box, which shows the method with your numbers.

What it does and when to use it

Complex numbers have a real part and an imaginary part, where i² = −1. They appear in A-level and university maths, in AC circuits and signal processing, and whenever a quadratic has a negative discriminant. Division and powers are the steps students most often get wrong by hand, so this tool shows the working for those.

To find the complex roots of an equation such as x² + 2x + 5 = 0, use the quadratic solver or, for cubics, the cubic solver. The trigonometry calculator helps when converting angles between polar and rectangular form.

How it works

With z₁ = a + bi and z₂ = c + di:

• Add and subtract: (a ± c) + (b ± d)i.
• Multiply: (ac − bd) + (ad + bc)i.
• Divide: multiply top and bottom by the conjugate c − di, then divide by c² + d².
• Modulus: |z| = √(a² + b²). Argument: arg z = atan2(b, a).
• Square root: re = √((|z| + a)/2), im = ±√((|z| − a)/2), with the sign of b.
• Power: whole numbers up to 64 by repeated squaring; otherwise z^w = e^(w ln z).

Step by step for (3 + 4i) ÷ (1 − 2i):

1. The conjugate of the bottom is 1 + 2i.
2. Top: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i.
3. Bottom: (1 − 2i)(1 + 2i) = 1² + 2² = 5.
4. Divide each part by 5: −1 + 2i.

The tool’s working line reads (3 + 4i)(1 + 2i) / |z₂|² = (-5 + 10i) / 5 = -1 + 2i.

Worked examples

Multiplication. (3 + 4i)(1 − 2i) gives Result: 11 − 2i, from (3·1 − 4·(−2)) + (3·(−2) + 4·1)i.

Division. (3 + 4i) ÷ (1 − 2i) = −1 + 2i, with modulus 2.236067977 and polar angle 116.5650512°.

Square root of a negative. √(−4) gives ±(2i), with Principal root: 2i and other root −2i.

i to the power i. i ^ i gives Result: 0.2078795764, a real number equal to e^(−π/2).

Limits and tips

• Dividing by 0 + 0i is undefined and gives a message. 0 raised to a power whose real part is 0 or negative also gives a message, except 0⁰, which is taken as 1.
• Very small parts, below about 10⁻¹² of the larger part, are shown as 0 to hide rounding noise.
• All calculations stay in your browser.