Imaginary Number Calculator | Complex Number Calculator

The most comprehensive complex/imaginary number calculator

Imaginary Number Calculator / Complex Calculator to evaluate any expression containing complex numbers and imaginary numbers (as well as reals). Use this Complex Calculator as a scientific calculator to evaluate all types of functions. The imaginary number calculator / complex number calculator optionally convert the results of calculations to polar (phasor), exponential and other modular forms. Instruction

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With this Online Complex/Imaginary Number Calculator you can evaluate expressions of complex/imaginary numbers from the simplest forms such as 1+2i+4, to arbitrary complicated forms such as sin(1+2i) / ln(3+4i) + exp(i)−4∠(1.8).

Complex Numbers / Imaginary 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.

To do arithmetic/algebra with complex numbers, 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 easy to do.

Adding & Subtracting Complex Numbers / Imaginary Numbers

To add or subtract two complex numbers a+bi & c+di just add or subtract the corresponding real and imaginary parts of them. That is, (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 two complex numbers a+bi & c+di use the the usual laws of algebra keeping in mind that i*i = -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i

Dividing Complex Numbers / Imaginary Numbers

To divide two complex numbers a+bi & 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)