Complex Number Calculator
Perform any arithmetic operation on two complex numbers written as a+bi. Choose an operation, enter the real and imaginary parts, and get the result in rectangular form, polar (phasor) form, exponential form, and trigonometric (CIS) form, all at once. Use the power mode to apply De Moivre's theorem, or the conjugate mode to find the complex conjugate in one click.
Formula
Worked example
Multiply (3+2i) by (1-4i): real = (3)(1)-(2)(-4) = 3+8 = 11; imaginary = (3)(-4)+(2)(1) = -10. Result: 11-10i. Modulus = sqrt(121+100) = sqrt(221) approx 14.866. Argument = atan2(-10,11) approx -42.27 degrees. Polar form: 14.866 angle -42.27 deg. Exponential: 14.866 * e^(-0.7378i). CIS: 14.866(cos(-42.27 deg) + i sin(-42.27 deg)).
How complex number arithmetic works
A complex number has a real part and an imaginary part, written a+bi, where i is the square root of -1. Addition and subtraction work component by component: add (or subtract) the real parts, then add (or subtract) the imaginary parts independently. Multiplication uses the distributive rule and the key fact that i squared equals -1, which produces the formula (ac - bd) + (ad + bc)i. Division removes i from the denominator by multiplying both numerator and denominator by the conjugate of the divisor, giving a real denominator of c squared plus d squared.
Modulus, argument, and all four output forms
Every complex number corresponds to a point on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical. Four equivalent representations exist. The rectangular form a+bi is the most familiar. The polar (phasor) form r angle theta uses the modulus r (the distance from the origin) and the argument theta (the angle from the positive real axis, computed with atan2 so the quadrant is always correct). The exponential form r*e^(i*theta) comes from Euler's formula. The trigonometric (CIS) form r(cos theta + i sin theta) is the same thing written out explicitly. This calculator displays all four forms for every result, and you can choose whether angles appear in degrees or radians.
Powers and De Moivre's theorem
Raising a complex number to a real power n is far easier in polar form than in rectangular form. De Moivre's theorem states that (r cis theta)^n = r^n cis(n*theta): raise the modulus to the nth power and multiply the argument by n. For example, (1+i)^3 has modulus sqrt(2) and argument 45 degrees, so the cube has modulus (sqrt(2))^3 = 2*sqrt(2) and argument 135 degrees, giving -2+2i. The power mode in this calculator applies De Moivre for any real exponent n, including fractional roots such as n = 0.5 for the principal square root.
Electrical engineering (j notation)
In electrical engineering the symbol j is used instead of i for the imaginary unit, to avoid confusion with current (I). The mathematics is identical. Use the imaginary unit toggle to switch between i and j in all inputs and outputs. Phasor form (r angle theta) is the standard in circuit analysis for representing sinusoidal voltages and currents, and the polar and exponential outputs produced here map directly to that convention.
Complex number forms and formulas
| Form | Notation | Formula |
|---|---|---|
| Rectangular | a + bi | a = Re(z), b = Im(z) |
| Polar (phasor) | r angle theta | r = sqrt(a^2+b^2), theta = atan2(b,a) |
| Exponential | r * e^(i*theta) | Same r and theta as polar |
| Trigonometric (CIS) | r(cos theta + i sin theta) | Euler's formula: e^(i*theta) = cos theta + i sin theta |
| Conjugate | a - bi | Flip sign of imaginary part |
| Modulus | |z| = r | Distance from origin on complex plane |
A complex number z = a+bi can be written in four equivalent forms. All are shown for each result above.
Frequently asked questions
How do you divide complex numbers?
Multiply both the numerator and the denominator by the conjugate of the denominator. For (a+bi) divided by (c+di), the conjugate is (c-di). This makes the denominator the real number c^2 + d^2, and the result becomes ((ac+bd) + (bc-ad)i) divided by (c^2+d^2). Alternatively, in polar form, divide the moduli and subtract the arguments.
What are the modulus and argument of a complex number?
The modulus (or absolute value) is the distance from the origin to the point on the complex plane, equal to sqrt(a^2 + b^2). The argument is the angle from the positive real axis to that point, computed as atan2(b, a). This calculator reports the argument in both degrees and radians, and also expresses the number in polar, exponential, and trigonometric (CIS) form.
What is De Moivre's theorem and how do you use it?
De Moivre's theorem states that (r(cos theta + i sin theta))^n = r^n(cos(n*theta) + i sin(n*theta)). To raise a complex number to a power: (1) convert it to polar form to find r and theta, (2) raise r to the nth power and multiply theta by n, (3) convert back to rectangular form. Select the Power mode and enter the exponent n. This also works for fractional n, giving the principal nth root.
Why does i squared equal -1?
The imaginary unit i is defined as the square root of -1, so by definition i^2 = -1. This single rule is what lets multiplication mix the real and imaginary parts, turning the product of two imaginary terms into a real contribution in the formula (a+bi)(c+di) = (ac-bd) + (ad+bc)i.
What is the conjugate of a complex number?
The conjugate of a+bi is a-bi: the sign of the imaginary part is flipped while the real part stays the same. Multiplying a complex number by its conjugate always yields a real number equal to the modulus squared: (a+bi)(a-bi) = a^2 + b^2. Conjugates are essential for division and for finding the modulus.
When would I use j instead of i?
Electrical engineers use j for the imaginary unit because i is already reserved for current. In circuit analysis, phasor form r angle theta represents sinusoidal signals, and the exponential form re^(j*theta) appears in Laplace and Fourier transforms. Toggle the imaginary unit to j in this calculator and all outputs switch to engineering notation automatically.