Complex Number Calculator | Rectangular ⇄ Polar Converter

Explore our free online comprehensive complex number calculator. This powerful scientific complex number calculator can evaluate and simplify mathematical expressions containing real and imaginary numbers in any form. With user-friendly features, you can input complex numbers in both rectangular a+bi and polar r∠(θ) forms directly into your expressions. It can handle complex number expressions ranging from simple forms like (1+2i) - (3-4i) to more intricate forms like sin(1+2i)/ln(3+4i) + atan(1+3i) - 4∠(1.8). The calculator converts the results to both standard and various modular forms, such as polar.

Seamless Rectangular and Polar Conversion

The complex number calculator performs conversions between rectangular and polar forms internally, and can display the results in various forms, including rectangular (standard) a+bi and polar (phasor) r∠(θ), and other modular forms.

Powerful Features for Calculation on Complex Numbers

This real and imaginary number calculator provides step-by-step solutions and provides the following functionalities.

Basic Operations

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

Powers and Roots

Calculate powers and principal roots of complex numbers.

Advanced Mathematical Operations

Calculate expressions involving trigonometric, hyperbolic, exponential, and logarithmic functions that have real, imaginary, or complex numbers as arguments.

Special Function Evaluation

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

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Instructions for our complex number calculator

Complex Numbers

What are complex numbers? Complex numbers are ordered pairs of real numbers (a,b), where a is called the real part and b is called the imaginary part. Instead of denoting a complex number with an ordered pair, it is customary to combine the pairs with a plus sign and denote the resulting complex number as a+bi, where i has the property that i2 = -1. In other words, i = √-1. Because there is no real number with a negative square, i is called the imaginary unit. Imaginary numbers are complex numbers whose real parts are zero, such as bi.

Note: Both the real part and the imaginary part are real numbers. The imaginary part is so called because it is the coefficient of the imaginary unit i.

All the laws and rules, including the commutative, associative and distributive laws, which we use in conjunction with the real number system, are also applicable to the complex number system. This makes the basic binary operations of addition, subtraction, multiplication and division of complex numbers easy to do. This real and imaginary numbers calculator uses the following rules to perform the four basic operations.

Adding & Subtracting Complex Numbers / Imaginary Numbers

To add or subtract complex numbers a+bi and c+di, simply add or subtract the corresponding real and imaginary parts of them.

(a+bi)+(c+di) = (a+c)+(b+d)i
(a+bi)-(c+di) = (a+c)-(b+d)i

Multiplying Complex Numbers / Imaginary Numbers

To multiply complex numbers a+bi and c+di, use the usual laws of algebra keeping in mind that i2 = -1.

(a+bi)*(c+di) = (ac-bd)+(ad+bc)i

Dividing Complex Numbers / Imaginary Numbers

To divide complex numbers a+bi and c+di, first multiply by (c-di)/(c-di), and then use the usual laws of algebra keeping in mind that i*i = -1.

(a+bi) / (c+di) = (a+bi)(c-di) / (c2 + d2)
  = [(ac+bd) + (-ad+bc)i] / (c2+d2)