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Complex Number Calculator

Add, subtract, multiply, and divide two complex numbers, and find each one's magnitude and argument.

z1 + z2
4 + 6i
z1 - z2
2 + 2i
z1 × z2
-5 + 10i
z1 ÷ z2
2.2 - 0.4i
|z1| (magnitude)
5
arg(z1) (angle)
53.13°
|z2| (magnitude)
2.2361
arg(z2) (angle)
63.43°

How it works

For z1 = a + bi and z2 = c + di, addition and subtraction just combine the real and imaginary parts separately. Multiplication distributes across both terms using i² = −1: z1 × z2 = (ac − bd) + (ad + bc)i.

Division multiplies both the numerator and denominator by z2's complex conjugate (c − di) to clear the imaginary part from the denominator, giving z1 ÷ z2 = [(ac + bd) + (bc − ad)i] ÷ (c² + d²) — undefined when z2 is 0 + 0i, the complex equivalent of dividing by zero.

Each number's magnitude, |z| = √(re² + im²), is its distance from the origin on the complex plane, and its argument, atan2(im, re), is the angle its line to the origin makes with the positive real axis — reported here in degrees.

FAQ

What does the argument actually represent?

Every complex number can be pictured as a point on a 2D plane, with the real part on the x-axis and the imaginary part on the y-axis. The argument is the angle of the line from the origin to that point, measured counterclockwise from the positive real axis — the same idea as a compass bearing, just measured from a different reference direction.

Why can't I divide by 0 + 0i?

Division works by multiplying by z2's conjugate over |z2|², so it requires dividing by |z2|² = c² + d². When z2 is 0 + 0i that denominator is 0, which is exactly as undefined as dividing an ordinary real number by zero.

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