Complex Number Calculator - Calculate with Complex Numbers

How the Complex Number Calculator Works

Complex number: z = a + bi where i^2 = -1

Complex numbers consist of a real part (a) and an imaginary part (bi), where i is the square root of -1. They are essential in electrical engineering, signal processing, quantum physics, and advanced mathematics.

1
Enter First Number
Input the real and imaginary parts of the first complex number (a + bi)
2
Enter Second Number
Input the second complex number if needed for binary operations
3
Select Operation
Choose add, subtract, multiply, divide, magnitude, phase, or conjugate
4
View Result
The result displays as a complex number with magnitude and phase

Use Cases

Electrical Engineering

Analyze AC circuits, calculate impedance and phase angles

Signal Processing

Perform Fourier analysis and signal transformations

Quantum Physics

Describe particle states and wave functions

Computer Graphics

Represent rotations and transformations in 2D space

Tips

1
Use polar form (r at angle theta) for easier multiplication and division
2
Magnitude |z| = sqrt(a^2 + b^2) gives the distance from origin
3
The conjugate z* = a - bi is used to simplify division
4
Phase angle theta = arctan(b/a) indicates direction in the complex plane

Common Mistakes

Forgetting that i^2 = -1 - when multiplying, (bi)(di) = -bd
Dividing directly by a complex number - multiply by the conjugate instead
Confusing Cartesian (a + bi) and polar (r at angle theta) forms
Ignoring the quadrant when calculating phase angle from arctan

FAQs

What is a complex number?

A number of the form a + bi, where a is the real part, b is the imaginary part, and i = sqrt(-1).

How do I add complex numbers?

Add real parts together and imaginary parts together: (a + bi) + (c + di) = (a+c) + (b+d)i.

How do I multiply complex numbers?

Use distribution: (a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac-bd) + (ad+bc)i.

What is the magnitude of a complex number?

The magnitude |z| = sqrt(a^2 + b^2) represents the distance from the origin. Example: |3+4i| = 5.

What is the conjugate used for?

The conjugate (a - bi) is used to eliminate i from denominators when dividing complex numbers.

Related tools

Pick another tool to continue your calculation.

🌀