Complex Number Calculator — Complete Computations
Easily compute and manipulate complex numbers with our comprehensive free calculator. Whether you are solving advanced engineering problems or analyzing quantum states, you can instantly perform addition, subtraction, multiplication, and division, as well as seamlessly convert complex values between their standard rectangular forms and trigonometric polar forms.
Reviewed by: Saim S., independent developer
Methodology: standard algebraic and trigonometric principles for complex numbers
Data Privacy: All calculations happen securely in your browser. No personal data is ever saved, tracked, or transmitted to our servers.
What Are Complex Numbers?
a + bi.
- a: Real number
- b: Real coefficient of the imaginary number
- i: Imaginary unit satisfying
i² = -1
Complex numbers conceptually extend the traditional one-dimensional number line into a highly versatile two-dimensional complex plane. This mathematical expansion fundamentally allows for the precise calculation of algebraic equations that technically possess no real solutions, such as computing the square root of negative numbers (a scenario frequently encountered when utilizing tools like a Quadratic Formula Calculator). Because of their ability to represent both magnitude and phase simultaneously, complex numbers are heavily utilized across advanced disciplines such as electrical engineering, fluid dynamics, signal processing, quantum mechanics, and evaluating linear systems alongside our Matrix Calculator or solving complex systems of equations.
How to Use This Complex Number Calculator
Our Complex Number Calculator simplifies operations on complex numbers. Here is a quick guide on how to get started:
- Select the Operation: Choose between performing operations with two numbers (Add, Subtract, Multiply, Divide, Power) or calculating properties of a single number (Modulus, Argument, Conjugate, Square Root).
- Enter the Values: Input the real and imaginary parts (for rectangular form) or the magnitude and angle (for polar form) for each complex number.
- View the Result: The calculator instantly computes the result in both rectangular and polar coordinates and plots the vectors on the complex plane.
- Review the Steps: Click "Show Steps" to see the step-by-step mathematical breakdown of the calculation.
Forms of Complex Numbers
Complex numbers can be represented in two primary forms, each useful for different types of calculations:
| Feature | Rectangular Form | Polar Form |
|---|---|---|
| Format | Z = a + bi | Z = r ∠ θ or r(cos θ + i sin θ) |
| Components | Horizontal (real) & Vertical (imaginary) | Distance (magnitude) & Angle (argument) |
| Best For | Addition & Subtraction | Multiplication, Division, Powers, & Roots |
Rectangular (Cartesian) Form
Written as Z = a + bi, this form defines the number by its horizontal (real, a) and vertical (imaginary, b) components on the complex plane. Rectangular form makes addition and subtraction straightforward.
Polar Form
Written as Z = r ∠ θ or Z = r(cos θ + i sin θ), this form defines the number by its distance from the origin (magnitude, r) and its angle from the positive real axis (argument, θ). Polar form is highly efficient for multiplication, division, and finding powers or roots. Because polar coordinates heavily utilize trigonometric functions, you may also find our Triangle Solver useful when working with geometric angles.
Fundamental Operations
Here is a brief overview of how the fundamental operations work with complex numbers:
Addition & Subtraction
Combine the real parts together and the imaginary parts together. Example: (a + bi) + (c + di) = (a + c) + (b + d)i.
Multiplication
Expand using the FOIL method, remembering that i² = -1. In polar form, multiply the magnitudes and add the angles.
Division
Multiply the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part in the denominator. In polar form, divide the magnitudes and subtract the angles.
Complex Conjugate
Change the sign of the imaginary part. The conjugate of a + bi is a - bi.
Frequently Asked Questions
To add or subtract complex numbers, you simply add or subtract their real parts together and their imaginary parts together. For example, (3 + 2i) + (1 + 4i) = 4 + 6i.
To multiply complex numbers in rectangular form, use the FOIL method and substitute i² = -1. In polar form, you multiply their magnitudes and add their angles.
The modulus (or absolute value) is the distance of the complex number from the origin in the complex plane, calculated as √(a² + b²).
To divide two complex numbers in rectangular form, you multiply both the numerator and denominator by the complex conjugate of the denominator. This removes the imaginary number from the denominator.