Complex Number Calculator

Q: How do I convert between rectangular and polar forms?

To convert from rectangular (a+bi) to polar (r∠θ): calculate r = √(a² + b²) for the modulus and θ = arctan(b/a) for the argument. To convert from polar to rectangular: calculate a = r·cos(θ) for the real part and b = r·sin(θ) for the imaginary part.

Q: How do I find the nth roots of a complex number?

To find the n nth roots of z = r∠θ, use the formula: root_k = r^(1/n) ∠ ((θ + 2πk)/n) for k = 0, 1, 2, ..., n-1. This gives n distinct roots equally spaced around a circle of radius r^(1/n) in the complex plane.

Perform operations with complex numbers: add, subtract, multiply, divide, convert forms, calculate modulus, argument, conjugate, powers and roots with detailed step-by-step solutions and interactive complex plane visualization.

Free to useNo sign-up requiredUpdated Jan 2026

Complex Number CalculatorTry it now — free ▼

Quick Examples - Click to Try

+ Addition (3+4i)+(1-2i)

- Subtraction (5+3i)-(2+i)

× Multiplication (2+3i)×(4-i)

÷ Division (6+8i)÷(3-4i)

|z| Modulus |3+4i|

∠ Argument arg(1+i)

z̄ Conjugate conj(4-3i)

→∠ To Polar 3+4i → r∠θ

→z To Rect 5∠53° → a+bi

zⁿ Power (1+i)⁴

ⁿ√ Roots ³√8

z₁ First Complex Number

Accepts rectangular (a+bi) or polar (r∠θ°) form

⚙ Operation

z₂ Second Complex Number

n Exponent / Root Degree

Positive integer for power or root operations

Embed Complex Number Calculator Widget

About Complex Number Calculator

Welcome to the Complex Number Calculator, a powerful mathematical tool for performing operations on complex numbers with step-by-step solutions and interactive visualizations. Whether you are a student learning about imaginary numbers, an engineer analyzing AC circuits, or a mathematician exploring the complex plane, this calculator provides comprehensive solutions for all your complex number computations.

What is a Complex Number?

A complex number is a number that can be expressed in the form $ a + bi $, where $ a $ and $ b $ are real numbers, and $ i $ is the imaginary unit satisfying $ i^2 = -1 $. The number $ a $ is called the real part, and $ b $ is called the imaginary part of the complex number.

Rectangular Form

Written as $ z = a + bi $, representing a point with coordinates (a, b) on the complex plane.

Polar Form

Written as $ z = r \angle \theta $ or $ z = re^{i\theta} $, where r is the modulus and theta is the argument.

The Imaginary Unit

The symbol $ i $ represents $ \sqrt{-1} $, enabling solutions to equations like $ x^2 + 1 = 0 $.

Supported Operations

Arithmetic Operations

Addition

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

Subtraction

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Multiplication

$$(a + bi) \times (c + di) = (ac - bd) + (ad + bc)i$$

Division

$$\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

Complex Number Properties

Modulus: $ |z| = \sqrt{a^2 + b^2} $ - the distance from the origin
Argument: $ \arg(z) = \arctan\left(\frac{b}{a}\right) $ - the angle with the positive real axis
Conjugate: $ \overline{z} = a - bi $ - reflection across the real axis

Form Conversions

Rectangular to Polar

$$z = a + bi \Rightarrow r = \sqrt{a^2 + b^2}, \quad \theta = \arctan\left(\frac{b}{a}\right)$$

Polar to Rectangular

$$z = r \angle \theta \Rightarrow a = r\cos\theta, \quad b = r\sin\theta$$

Powers and Roots

De Moivre's Theorem (Powers)

$$(r \angle \theta)^n = r^n \angle (n\theta)$$

nth Roots

$$\sqrt[n]{r \angle \theta} = \sqrt[n]{r} \angle \left(\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, ..., n-1$$

How to Use This Calculator

Enter your complex number: Use rectangular form (e.g., 3+4i, -2-5i) or polar form (e.g., 5∠45°, 3∠π/4). The calculator auto-detects the format.
Select an operation: Choose from arithmetic operations, conversions, or functions like modulus, argument, power, and root.
Enter additional inputs: For binary operations, enter a second complex number. For power/root, enter the exponent.
Click Compute: View results in both rectangular and polar forms, with detailed step-by-step solutions and complex plane visualization.

Applications of Complex Numbers

Electrical Engineering

AC circuit analysis uses complex impedance to represent resistance, capacitance, and inductance.

Signal Processing

Fourier transforms use complex exponentials to analyze and filter signals.

Quantum Mechanics

Wave functions are complex-valued, with probability given by the modulus squared.

Control Systems

Poles and zeros in the complex plane determine system stability and response.

Frequently Asked Questions

What is a complex number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = -1. The real part is 'a' and the imaginary part is 'b'. Complex numbers extend the real number system and are essential in many areas of mathematics, physics, and engineering.

How do I convert between rectangular and polar forms?

To convert from rectangular (a+bi) to polar (r angle theta): calculate r = sqrt(a² + b²) for the modulus and theta = arctan(b/a) for the argument. To convert from polar to rectangular: calculate a = r times cos(theta) for the real part and b = r times sin(theta) for the imaginary part.

What is De Moivre's Theorem?

De Moivre's Theorem states that for a complex number in polar form z = r(cos theta + i sin theta) and any integer n: z^n = r^n(cos(n times theta) + i sin(n times theta)). This theorem makes calculating powers and roots of complex numbers much simpler by working with the polar form rather than repeated multiplication.

How do I find the nth roots of a complex number?

To find the n nth roots of z = r angle theta, use the formula: root_k = r^(1/n) angle ((theta + 2 pi k)/n) for k = 0, 1, 2, ..., n-1. This gives n distinct roots equally spaced around a circle of radius r^(1/n) in the complex plane.

What are the applications of complex numbers?

Complex numbers are used in electrical engineering for AC circuit analysis, signal processing for Fourier transforms, quantum mechanics for wave functions, control systems for stability analysis, fluid dynamics, and solving polynomial equations. They provide elegant solutions to problems that would be difficult with real numbers alone.

Additional Resources

Reference this content, page, or tool as:

"Complex Number Calculator" at https://MiniWebtool.com/complex-number-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 20, 2026

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