Complex Number Calculator

How to use

1. 1Choose a mode: Arithmetic for two numbers, Single number for the properties of one value, or Power and roots.
2. 2Type each complex number in a + bi form, for example 3 + 4i, -2i, i, or 5. You can use i or j for the imaginary unit.
3. 3In Arithmetic mode pick add, subtract, multiply, or divide, and use Swap a and b to reverse the order.
4. 4In Power and roots mode enter a whole-number exponent; for n from 2 to 24 the tool also lists every nth root.
5. 5Read the result in rectangular, polar, and exponential form, and use the copy buttons to grab any value. Everything runs locally.

About this tool

Complex Number Calculator works with numbers that have a real part and an imaginary part, written in the usual a + bi form, and it does the three jobs that come up most often in algebra, trigonometry, electrical engineering, and signal processing. The first mode is arithmetic on two complex numbers: enter any two values, choose add, subtract, multiply, or divide, and the tool returns the exact result along with its real and imaginary parts, its modulus, and both its polar and exponential forms. Multiplication uses the FOIL expansion (a + bi)(c + di) = (ac - bd) + (ad + bc)i, and division uses the standard conjugate method, multiplying the top and bottom by the conjugate of the divisor, so a quotient like (3 + 4i) / (1 - 2i) is reduced to a single a + bi value rather than left as a fraction. Dividing by zero is caught and reported instead of producing a broken result. The second mode takes a single complex number and breaks it down completely: the real and imaginary parts, the modulus (the distance from the origin, the square root of real squared plus imaginary squared), the argument or angle measured from the positive real axis and shown in both radians and degrees in the principal range of -180 to 180 degrees, the conjugate, the reciprocal 1 / z, and the number rewritten in polar form r(cos theta + i sin theta) and exponential form r e^(i theta). This is the mode to reach for when a problem asks you to convert a number from rectangular to polar form, or to read off its magnitude and phase. The third mode raises a complex number to a whole-number power using De Moivre's theorem, which says (r e^(i theta)) raised to the n equals r to the n times e^(i n theta), so a power like (1 + i) cubed is handled by working in polar form rather than multiplying the number out by hand. For positive integer exponents from 2 to 24 the calculator also lists every distinct nth root of the number, the full set of solutions to w to the n equals z, spaced evenly around a circle in the complex plane, which is the part of root problems that is easy to get wrong when only the principal root is found. Input is forgiving: you can write the imaginary unit as i or j, drop the coefficient on a bare unit so i means 1i and -i means -1i, omit a part entirely so 5 is treated as 5 + 0i and -2i as 0 - 2i, and use decimals and scientific notation in either part. Every result has a copy button, the full set of roots can be copied at once, and a Swap control flips the two operands in arithmetic mode. All of the math runs in your browser with ordinary floating-point arithmetic, so nothing you enter is uploaded, and the tool is free with no signup.

Free to use. Works in your browser. No signup, no login.