Complex Number Calculator

About this tool

What Is a complex number? Complex Number Calculator. The complex number calculator performs the four arithmetic operations plus the modulus on pairs of complex numbers, printing the real and imaginary parts and the geometric modulus. A complex number z = a + bi bundles a real part a and an imaginary part b times i, the square root of −1. The result of any arithmetic between them carries both parts. Multiplication uses the distributive law and i² = −1: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. Division multiplies top and bottom by the conjugate of the denominator so the new denominator is real. The modulus |z| = √(a² + b²) is the distance from the origin. z = a + bi; (a+bi)(c+di) = (ac−bd) + (ad+bc)i; |z| = √(a²+b²). How to Use Enter the real and imaginary parts of the first number a + bi. Enter the real and imaginary parts of the second number c + di. Choose the operation and read the result in the form p + qi with the modulus shown alongside. Worked Example Multiply (3+4i)(2−3i)?—?the tool computes (3·2 − 4·(−3)) + (3·(−3) + 4·2)i = 18 − i. The modulus of 3+4i is √(3²+4²) = 5, and the complex conjugate part explains the division routine. Common Use Cases AC circuit impedance (phasor) math quantum and wave calculations signal processing transfer functions supplementary math assignments Important Notes Results follow the form p + qi with signed parts. Division, multiplication, and