Add, subtract, multiply, or divide two complex numbers and find the result's modulus and argument.
📖 Complex Number ArithmeticNumber 1: a + bi
Number 2: c + di
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The complex number calculator performs the four basic arithmetic operations on two complex numbers in the form a + bi, where a is the real part and b is the imaginary part multiplied by the imaginary unit i. Add, Subtract, Multiply, or Divide Complex Numbersition and subtraction are done by adding or subtracting the real parts together and the imaginary parts together separately. Multiplication uses the distributive law with i² substituted with −1. Division is done by multiplying the numerator and denominator by the denominator's "complex conjugate" to remove the imaginary part from the denominator. In addition to the basic result, the calculator computes the "modulus" (|z|), the distance from the origin to the number on the complex plane, and the "argument," the number's angle in degrees — the two fundamental values for representing a complex number in polar form, widely used in electrical engineering and signal processing. The imaginary unit i, defined so that i² equals −1, is what makes it possible to give a meaningful answer to problems like the square root of a negative number, which have no solution among the ordinary real numbers.
Complex numbers exist to solve a specific, unavoidable gap in the real number system: no real number, when squared, produces a negative result. Mathematicians resolved this by defining a new number, i, specifically so that i² equals −1 — a definition that, once accepted, opens up a consistent and extremely useful extension of ordinary arithmetic.
A complex number is written as a + bi, combining an ordinary real part (a) with an imaginary part (b, the coefficient of i). Add, Subtract, Multiply, or Divide Complex Numbersition and subtraction of complex numbers work exactly as expected — combine the real parts together and the imaginary parts together, treating them as two separate tracks that never mix.
Multiplication requires the distributive law, exactly as with any two-term algebraic expression, but with one extra simplification step: whenever i² appears in the expanded result, it is replaced with −1 by definition, which is what allows a product of two complex numbers to collapse back down into the standard a + bi form rather than growing more complicated with each higher power of i.
Division introduces a technique with no real-number equivalent: multiplying both the numerator and denominator by the denominator's 'complex conjugate' (the same complex number with the sign of its imaginary part flipped). This trick works because a complex number multiplied by its own conjugate always produces a purely real result, eliminating i from the denominator entirely and leaving a standard complex number as the final answer.
The modulus and argument reveal a second, geometric way of understanding complex numbers: rather than a point defined by a real and imaginary coordinate, a complex number can be described by its distance from the origin (the modulus) and its angle from the positive real axis (the argument) — the same relationship between rectangular and polar coordinates familiar from basic trigonometry and vector work.
Electrical engineering relies on complex numbers extensively to represent alternating current, where the real and imaginary parts naturally capture both the magnitude and phase relationship between voltage and current. Signal processing uses complex numbers throughout Fourier analysis, which decomposes a signal into its underlying frequency components, and quantum mechanics in physics is built on complex-number mathematics at its most fundamental level.
i is defined as the square root of −1, so that i² = −1, allowing us to work with square roots of negative numbers.
The modulus is the complex number's distance from the origin on the complex plane, and the argument is the angle it makes with the positive real axis.
It's done by multiplying both the numerator and denominator by the denominator's complex conjugate, which turns the denominator into a real number.