Complex Number Calculator
Add, subtract, multiply and divide complex numbers in a + bi form. The result is shown in standard form together with its modulus and argument, and every step is worked out below.
What is a complex number?
A complex number has the form a + bi, where a is the real part, b is the imaginary part and i is the imaginary unit defined by i² = −1. Complex numbers extend the real numbers so that every polynomial equation — including x² + 1 = 0 — has a solution. When b = 0 the number is purely real; when a = 0 and b ≠ 0 it is purely imaginary.
Complex arithmetic, piece by piece
Each card explains one operation this tool performs on your two inputs — how it combines them, what happens to i², and how it strips the imaginary unit out of the denominator when dividing.
Add & subtract — combine like parts
Real parts add to real parts, imaginary to imaginary. Geometrically this is head-to-tail arrow addition in the complex plane.
Multiply — FOIL, then replace i²
Expand the brackets like ordinary binomials, then use i² = −1 to convert the imaginary square back into a real term and collect like parts.
Divide — multiply by the conjugate
To clear i from the denominator, multiply top and bottom by the denominator's conjugate. The product (c+di)(c−di) = c² + d² is real, so the answer splits cleanly into real and imaginary parts.
Modulus & argument — the polar view
Every complex number is a point (a, b) in the plane. Its distance from the origin is the modulus and its angle with the positive real axis is the argument.
Features of this calculator
- Accepts complex numbers as separate real and imaginary parts
- Supports addition, subtraction, multiplication and division
- Divides using the conjugate method — no imaginary numbers left in the denominator
- Reports the modulus |z| and the argument arg(z) in radians and degrees
- Shows the full step-by-step working for the chosen operation
Frequently asked questions
+Why do we need i at all?
Without i, equations like x² = −1 have no solution. By defining i so that i² = −1, every polynomial equation has as many roots as its degree — a result known as the fundamental theorem of algebra.
+Is the argument in radians or degrees?
This calculator reports both. The primary value is in radians (between −π and π) because atan2 works in radians; the degree version (between −180° and 180°) is shown alongside it.
+What is the conjugate of a real number?
The conjugate of a + 0i is a − 0i = a — a real number is its own conjugate.
+Can I divide by zero?
No. Dividing by 0 + 0i is undefined, just like dividing by zero in the real numbers. The calculator flags this as an error.