Complex Number Arithmetic Calculator

Add, subtract, multiply, or divide two complex numbers z1=a+bi and z2=c+di, then see the modulus, argument, and the result plotted on the complex plane.

Input

Enter two complex numbers z1=a+bi and z2=c+di, choose an operation, and get their sum, difference, product, or quotient.

First complex number z1 = a + bi

Real part a

Imaginary part b

Second complex number z2 = c + di

Real part c

Imaginary part d

Choose an operation

Result

Result of z1 + z2

4 + 6i

Real part

Imaginary part

Modulus r

7.21110255

Argument theta

56.30993247 deg

Argument theta in radians

0.98279372 rad

The modulus is the distance from the origin to the point, and the argument is the angle measured from the positive real axis. Division uses the conjugate of the denominator.

How it works

Addition adds real parts and imaginary parts separately, and subtraction does the same with a minus. The product equals (ac − bd) + (ad + bc)i.
For division, multiply the numerator and denominator by the conjugate c − di and divide by (c² + d²) to separate the real and imaginary parts.
The modulus r is the square root of the sum of the squares of the real and imaginary parts, and the argument theta is atan2 of imaginary over real in the range −π to π. Division is undefined when z2 equals zero.

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