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

Use our free online complex number calculator to perform calculations in both rectangular (a+bi) and polar / phasor (r∠(θ)) forms, allowing you to convert complex numbers seamlessly.

Use this real and imaginary number calculator to perform the basic operations (addition, subtraction, multiplication and division) on complex numbers. Furthermore, you can easily compute the parallel sum of complex numbers—a vital tool for calculating equivalent complex impedance in parallel circuits.

Perform advanced operations with full support for power (including exponential and root), logarithm, trigonometric, and hyperbolic functions using complex or imaginary inputs.

About the Complex Number Calculator (Scientific)

As a fast and powerful complex number calculator, whatever calculation and function evaluation you can perform with real numbers, it can handle expressions containing imaginary or complex numbers just as quickly and easily.

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

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

Note: Our i calculator accepts both i and j for the imaginary unit for entering complex numbers. However, if you prefer j—common in some non-mathematical fields such as electrical engineering—we recommend selecting the Use j for imaginary unit option in the settings (⚙). This will ensure results are also displayed using j

Complex Number Conversion (Rectangular ⇄ Phasor/Polar)

Our complex number calculator performs conversions and displays calculation results in multiple forms, including:

  • a+bi (Rectangular form)
  • r∠(θ) (Phasor notation): Widely used in electrical engineering, also known as polar form)
  • reiθ (Exponential form)
  • r⋅cis(θ) (Another polar form, based on Euler's formula; mathematicians use cis(θ), a shorthand for the trigonometric form cos(θ)+isin(θ) instead of the angle/phasor 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).

Number Format | Output
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Elementary & Advanced Complex Number Calculations

Suitable for all levels, this easy-to-use, powerful complex number calculator simplifies and evaluates expressions involving complex numbers, ranging from simple forms like (1+2i) - (3-4i) to arbitrarily complex forms such as sin(1+2i)/ln(3+4i) + atan(1+3i) - 4∠(1.8)

You can input numbers in scientific (or engineering) notation (e.g., 1E6 or 1.2E-12), where the exponent must be an integer. For non-integer exponents, you must use parentheses (e.g., 2E(2.3) or 1.2E(1+2i)).

Remark: Our scientific calculator introduces the function E() defined on complex numbers as E(z) = 10^z. Therefore, E(2) equals 100, which is the same as 1E2. It is important to note that, unlike E(2), the entity E2 (without parentheses) is invalid, as it must be preceded by a real literal number, e.g., 1E2.

This imaginary/complex number calculator offers the following functionalities:

Basic Operations

Perform basic real and imaginary operations with our online calculator:

Parallel Sum

Calculate the parallel sum (also known as the reduced sum) using the parallel operator ∥, as in: (1+2i)∥(3+4i)

Remark: The parallel operator is widely used across various branches of science and engineering, particularly in electrical engineering for analyzing AC circuits. 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 transcendental (non-algebraic) functions—such as power (which includes roots and exponentials), logarithmic, trigonometric, and hyperbolic—with real, imaginary, or complex number arguments.

Powers, Exponentials, and Roots

Calculate powers and principal roots of complex numbers with ease.

Logarithmic Functions

Calculate the natural and common logarithms of complex numbers.

Trigonometric Functions

Calculate trigonometric and inverse trigonometric functions of complex numbers.

Hyperbolic Functions

Evaluate hyperbolic and inverse hyperbolic functions with complex arguments.

Remark: Transcendental functions of complex numbers (such as the exponential and logarithm) are evaluated using trigonometric functions via Euler's formula eiz = cos(z) + i*sin(z). Their values therefore depend on the angle mode selected by the user. For instance, we have ei = cos(1) + i*sin(1). In radians mode, this value (rounded to four decimal places) is 0.5403 + 0.8415i, while in degrees mode, it's 0.9998 + 0.0175i. This contrasts with e2, whose value remains unchanged regardless of the angle mode selected, because its exponent is a real number.

It's important to note that even though ei might appear to be non-trigonometric, its computation inherently involves trigonometric functions; thus, its value depends on the chosen angle mode.

Special Function Evaluation

Our comprehensive scientific calculator lets you evaluate functions such as Γ(), ψ(), and ζ(), corresponding to the Gamma, Psi (Digamma), and Riemann Zeta functions, respectively.