Add, subtract, multiply, or divide two complex numbers, with the modulus and argument of the result.

How to Use the Complex Numbers Calculator

Enter the real and imaginary parts of each number, pick an operation, and the result appears in standard a + bi form alongside its modulus and argument.

Addition and subtraction just combine matching parts — reals with reals, imaginaries with imaginaries. Multiplication distributes the way any two binomials would, remembering that i squared equals -1:

(a+bi) + (c+di) = (a+c) + (b+d)i
(a+bi) × (c+di) = (ac−bd) + (ad+bc)i

Division is the one that needs a trick, because "a + bi over c + di" isn't a valid complex number as written — the denominator still has an i sitting in it. The fix is multiplying top and bottom by the denominator's conjugate, c − di, which is the same number with the imaginary sign flipped:

(a+bi)/(c+di) = (a+bi)(c−di) / (c+di)(c−di) = [(ac+bd) + (bc−ad)i] / (c²+d²)

The reason that works: (c+di)(c−di) expands to c² − (di)², which is c² + d² — a plain real number with no i left in it at all, since the imaginary parts cancel out by construction. Run the calculator's own defaults, 3+4i divided by 1−2i, and the conjugate trick turns a 1−2i denominator into 1² + 2² = 5, landing on the answer −1 + 2i.

Modulus and argument describe the result as a point on the complex plane instead of as a + bi coordinates: modulus is the distance from the origin, √(a²+b²), and argument is the angle from the positive real axis, shown here in both radians and degrees. Together they're the same complex number in polar form rather than rectangular — more useful whenever rotating or scaling complex numbers matters more than simply adding their parts.