Complex Number Calculator (Scientific)

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.

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)

Note: This calculator accepts complex numbers using either i or j for the imaginary unit. 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.

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.

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: The function E() is defined as E(x) = 10^x. Therefore, E(2) equals 100, which is the same as 1E2. It is important to note that, unlike E(2), the entity E2 (without parenthesis) is invalid, as it must be preceded by a literal real number.

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 non-algebraic functions such as trigonometric, hyperbolic, exponential, and logarithmic with real, imaginary, or complex number arguments.

Remark: Non-algebraic functions of complex numbers, such as the exponential and logarithm, are evaluated using trigonometric functions via Euler's formula e^iz = cos(z) + i*sin(z). The values of these functions depend on the angle mode selected by the user. For instance, in radians mode, the value (rounded to four decimal places) of eⁱ 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 value still depends on the chosen angle mode because its exponent is not a real number, thus its computation inherently involves trigonometric functions.

Special Function Evaluation

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

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