Complex Number Calculator (Scientific) | Rectangular ⇄ Phasor/Polar

Use our free online complex number calculator—a powerful and 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.

About the Scientific/Complex Number Calculator

As a powerful scientific calculator, whatever calculation and function evaluation you can perform with real numbers, this complex number calculator can do the same with imaginary and complex numbers just as easily.

In addition, this complex number calculator handles complex numbers in any form, and converts between rectangular [a+bi] and polar/phasor [r∠(θ)] forms.

Furthermore, this real & imaginary number calculator provides step-by-step calculations.

Complex Number Conversion (Rectangular ⇄ Phasor/Polar)

This calculator performs conversions and displays calculation results in multiple forms, including:

  • a+bi (Rectangular form)
  • r∠(θ) (Polar form, also known as phasor notation in electrical engineering)
  • reiθ (Exponential form)
  • r⋅cis(θ) (Based on Euler's formula; mathematicians use cis(θ), a shorthand for cos(θ)+isin(θ) instead of the angle notation ∠(θ).)

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

Interested in matrix algebra and solving systems of linear equations involving complex numbers? If so, try our innovative Matrix Calculator.

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Instructions for using the complex number calculator

Elementary & Advanced Complex Number Calculations

This easy-to-use and advanced scientific calculator can perform all the same calculations with imaginary and complex numbers just as easily as with real numbers. Suitable for all levels, it simplifies and computes expressions involving complex 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.