Complex Number Calculator | Rectangular ⇄ Phasor/Polar Converter

Use our free online complex number calculator—a powerful, comprehensive scientific calculator—to evaluate mathematical expressions involving real, imaginary and, in general, complex numbers in various forms, including rectangular (standard) and phasor (polar) forms.

This real and imaginary number calculator handles complex numbers in any form, and converts between rectangular and polar/phasor forms, r∠(θ). Mathematicians usually use cis(θ), short for cos(θ) + i*sin(θ), instead of ∠(θ).

Furthermore, our complex number calculator provides step-by-step calculations.

Additionally, the rectangular/polar converter performs conversions and displays calculation results in multiple forms, including:

  1. a+bi (rectangular)
  2. r∠(φ) (polar, also known as phasor in electrical engineering)
  3. reiφ (exponential)
  4. rcis(φ) (shorthand for r[cos(φ)+i*sin(φ)], Euler's formula)

Here, r is the magnitude (or modulus or length or absolute value) of the complex number, and φ denotes its argument (or phasor or angle).

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Instructions for using the complex number calculator

Elementary & Advanced Complex Number Calculations

As a powerful scientific complex number calculator, suitable for all levels, it simplifies and computes expressions involving real and imaginary numbers, ranging from simple forms like (1+2i) - (3-4i) to arbitrarily complicated forms such as sin(1+2i)/ln(3+4i) + atan(1+3i) - 4∠(1.8) This real and imaginary number calculator offers the following functionalities.

Basic Operations

Perform basic operations of addition, subtraction, multiplication, and division on complex numbers.

Powers and Roots

Calculate powers and principal roots of complex numbers.

Parallel Sum

Calculate the parallel sum (also known as the reduced sum) using the parallel operator, ∥, as in: (1+2i)∥(3+4i) The parallel operator is frequently used in electrical engineering with AC circuits, and in other branches of science and engineering. Note that the binary parallel operator has higher precedence than addition and subtraction but lower precedence than multiplication, division, and exponentiation.

Advanced Mathematical Operations

Evaluate expressions involving trigonometric, hyperbolic, exponential, and logarithmic functions with real, imaginary, or complex numbers arguments.

Special Function Evaluation

Evaluate functions like Γ(), ψ(), and ζ(), corresponding to the Gamma, psi, and zeta functions, respectively.