Complex Number Calculator Online

The most comprehensive Online Complex Number Calculator that evaluates any expression containing real or complex numbers and shows step-by-step work. Use this complex number calculator the same way as you would use a scientific calculator to evaluate algebraic, trigonometric, logarithmic, hyperbolic, and in general, all types of functions with real or complex number argument. The complex calculator optionally converts the result to polar (phasor), exponential and other modular forms using the Euler's formula. Instruction

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With this Online Complex Number Calculator you can evaluate expressions of Complex 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

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. Istead of denoting a complex number with an orderd 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 nrgative squared, i is called the imaginary unit.

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 arithmatic / algebra with complex numbers, all the laws and rules, including the Commutative, associative and distributive laws that we use in conjunction with real number system, are also applicable to the complex number system. This make the basic binary operations of addition, subtraction, multiplication and division easy to do.

Adding & Subtracting Complex 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

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

To divide two complex numbers a+bi & c+di, first multiply by (c-di)/(c-di) and use the 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))/(c2 + d2)