Complex Number Calculator

Perform operations with complex numbers including addition, subtraction, multiplication, division, and conversions between rectangular and polar forms.

Complex Number Operations

Select Operation:

Addition Subtraction Multiplication Division

Real Part (a):

Imaginary Part (b):

Real Part (c):

Imaginary Part (d):

Single Complex Number Operations

Real Part:

Imaginary Part:

Complex Number Forms

Rectangular Form:

z = a + bi

Where a is real part, b is imaginary part

Polar Form:

z = r(cos θ + i sin θ)

Where r is magnitude, θ is argument

Exponential Form:

z = re^(iθ)

Euler's formula representation

Examples

Addition:

(3 + 2i) + (1 + 4i) = 4 + 6i

Multiplication:

(2 + 3i)(1 + i) = -1 + 5i

Conjugate:

If z = 3 + 4i, then z* = 3 - 4i

Magnitude:

|3 + 4i| = √(3² + 4²) = 5

Complex Number Properties

Magnitude: |z| = √(a² + b²)
Argument: θ = arctan(b/a)
Conjugate: z* = a - bi
Real Part: Re(z) = a
Imaginary Part: Im(z) = b

Related Math Tools

Understanding Complex Numbers

Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0. They consist of a real part and an imaginary part, written as a + bi where i is the imaginary unit (√-1).

Applications:

Electrical engineering (AC circuits)
Signal processing
Quantum mechanics
Control systems
Computer graphics

Complex Operations

Addition and Subtraction:

Add/subtract real and imaginary parts separately

Multiplication:

Use FOIL method: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

Division:

Multiply by conjugate of denominator

Polar Conversion:

r = |z|, θ = arg(z)