Complex Number Calculator | Rectangular ⇄ Polar

Explore our free online complex number calculator, a powerful and easy-to-use scientific calculator designed to perform mathematical operations (addition, subtraction, multiplication, division, power raising) and function evaluations (trigonometric, exponential, logarithmic, hyperbolic, and more) of complex numbers. Handling both real and imaginary numbers, you can also use this real & imaginary number calculator to convert complex numbers to polar form.

Powerful Features for Calculation on Complex Numbers

Our comprehensive complex number calculator offers a wide range of functionalities for complex numbers calculations. Furthermore, the complex number calculator shows you the work step-by-step.

Basic Operations

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

Powers and Roots

Perform power raising and root extracting 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.

Seamless Rectangular and Polar Conversion

Our complex numbers calculator supports both rectangular (standard) a+bi and polar (phasor) r∠(θ) forms of complex numbers. Additionally, this rectangular / polar calculator displays the results in various forms, including rectangular (standard), polar (phasor), and other modular forms.

Handle Complicated Complex Number Expressions with Ease

Our versatile calculator is a fully-developed scientific complex number calculator that can handle complex number expressions from basic arithmetic to advanced mathematical operations. It can easily compute expressions of complex numbers, from simple forms like (1+2i) - (3-4i) to more complicated forms like sin(1+2i) / ln(3+4i) + atan(1+3i) - 4∠(1.8).

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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 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 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 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)