Complex Number Calculator | Rectangular & Polar

Explore math with our comprehensive complex number calculator, which can be used as a fully developed scientific calculator.

Our online complex number calculator is a powerful and easy-to-use scientific calculator for performing mathematical operations and evaluating functions with real and imaginary numbers, and complex numbers in general, in any form, including rectangular (standard) and polar forms.

The complex number calculator supports both rectangular (a + bi) and polar (r∠(θ)) forms of complex numbers, and converts between the two forms. It shows you the work step-by-step and also displays the result of calculations to various forms, including rectangular (standard), polar, and modular forms.

Here is a summary of some of the capabilities that our real and imaginary number calculator can perform:

  • Addition, subtraction, multiplication, and division of complex numbers
  • Conversion between rectangular and polar forms of complex numbers
  • Calculation of powers and roots of complex numbers
  • Evaluation of trigonometric functions of complex numbers (sine, cosine, tangent, cotangent, secant, and cosecant), and their inverses
  • Evaluation of hyperbolic functions of complex numbers (sinh, cosh, tanh, cotanh, sech, and csch), ), and their inverses
  • Evaluation of exponential and logarithmic functions of complex numbers (exp, ln, log2, and log10)
  • Evaluation of the Gamma function of complex numbers (Γ)
  • Evaluation of the Psi function of complex numbers (ψ)
  • Evaluation of the Zeta function of complex numbers (ζ)

This calculator is a fully-developed scientific complex number calculator that can handle complex number expressions from basic arithmatic 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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Instructions

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.

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/imaginary number 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 just 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 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)