This online complex number calculator can be used as a fully-fledged scientific calculator to evaluate mathematical expressions containing real, imaginary, and complex numbers of any form. It can perform basic operations such as addition, subtraction, multiplication, and division, as well as advanced operations with complex numbers.

Type in complex numbers using either i or j (or both) for the imaginary unit. If you use j, it is recommended to select the Use j for imaginary unit option (go to ⚙). This makes the calculator display results using j.

Use +, -, *, and / for basic operations. You can also use × for multiplication and ÷ for division. The calculator displays the result as you type, but you can change this behavior by going to ⚙ and clearing the Calculate as you type checkbox. The label of STO changes to Calculate. Pressing it evaluates the expression and stores it, which you can recall by pressing RCL.

You can also input numbers in scientific or engineering notations (e.g., 1E6 or 1.2E-12), where the exponent is an integer. For any non-integer exponent, use parentheses (e.g., 2E(2.3) or 1.2E(1+2i)).

Note: The function E() is defined as E(x) = 10^x. So E(2) is 100, which is the same as 1E2. However, E2 is invalid, as it must be preceded by a literal real number.

Our rectangular/polar converter can display results in a variety of notations, including Fixed-point, Scientific and Engineering. Intermediate calculations can be displayed by checking the Show work option, getting the step-by-step calculations. The calculator can also convert results between rectangular form a + bi and polar form r∠(θ), and display them in other modular forms using Euler's formula.

• polar (phasor, angular): r∠(θ)
• exponential: re^iθ
• cis: r(cos(θ)+ i sin(θ))
• cis (phasor, angular): rcis(θ)
• modular: r(a/r + b/r i) where, r = √(a² + b²), module or length or absolute value of a + bi, and θ = arctan(b/a), argument or angle of a + bi.

Remark: Non-algebraic functions of complex numbers are typically evaluated using trigonometric functions through Euler's identity, e^ix = cos(x)+ i*sin(x). The values of these functions depend on the angle mode selected by the user. For instance, in radians mode, the value of eⁱ is 0.5403 + 0.8415i, while in degrees mode, it's 0.9998 + 0.0175i. This is in contrast to e^2, whose value is the same regardless of the angle mode selected because the power of e is a real number. It's important to note that even though eⁱ appears non-trigonometric, its value still depends on the angle mode chosen because the power of e is not a real number.

On mobile devices, you can control the pop-up keyboard by checking or unchecking the option labeled Prevent System keyboard from popping up. This feature has been tested on Android devices, but may not work on iOS.

If there are any error messages, you can view them by clicking on the ❓ icon.