Complex Number Calculator | Add, Subtract, Multiply, Divide

📖 How to Use This Complex Number Calculator

Enter the first complex number - Input real part (a) and imaginary part (b) for Z₁ = a + bi.

Enter the second complex number - Input real part (a) and imaginary part (b) for Z₂ = a + bi.

Select an operation - Click on addition, subtraction, multiplication, division, modulus, argument, or conjugate.

View result - Get the result with a detailed step-by-step explanation.

🔢 Understanding Complex Numbers

A complex number is a number of the form a + bi, where:

a is the real part
b is the imaginary part
i is the imaginary unit (√-1)

Complex numbers extend the real numbers and are essential in engineering, physics, and mathematics for representing oscillations, waves, and AC circuits.

🧮 Complex Number Formulas

Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
Subtraction: (a+bi) - (c+di) = (a-c) + (b-d)i
Multiplication: (a+bi)(c+di) = (ac-bd) + (ad+bc)i
Division: (a+bi)/(c+di) = [(ac+bd) + (bc-ad)i] / (c²+d²)
Modulus: |a+bi| = √(a² + b²)
Argument: θ = atan2(b, a)
Conjugate: a+bi = a - bi

📊 Understanding Complex Number Properties

Real numbers have imaginary part = 0 (b = 0)
Pure imaginary numbers have real part = 0 (a = 0)
i² = -1 (fundamental property)
Complex conjugates multiply to real numbers: (a+bi)(a-bi) = a²+b²
Polar form: z = r(cos θ + i sin θ) = re^(iθ)

💡 Applications of Complex Numbers

Electrical Engineering: AC circuit analysis using impedance and phasors
Signal Processing: Fourier transforms for analyzing frequency content
Quantum Mechanics: Wave functions and probability amplitudes
Control Systems: Root locus and stability analysis
Fluid Dynamics: Complex potentials for 2D flow

❓ Frequently Asked Questions (FAQ)

What is the imaginary unit i?

i is the imaginary unit defined as i = √-1. It satisfies i² = -1. This allows us to take square roots of negative numbers.

How do you multiply complex numbers?

Use the FOIL method: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i. Remember i² = -1.

What is the modulus of a complex number?

The modulus (or absolute value) is the distance from the origin to the point (a,b) in the complex plane, calculated as √(a² + b²).

What is the argument of a complex number?

The argument is the angle the complex number makes with the positive real axis, calculated using atan2(b,a). It's measured in radians or degrees.

What is a complex conjugate used for?

The conjugate is used to divide complex numbers (multiplying numerator and denominator by the conjugate of the denominator) and to find the modulus squared.

Can complex numbers be graphed?

Yes! The complex plane (Argand diagram) uses the x-axis for the real part and y-axis for the imaginary part. Each complex number is a point in this plane.