De Moivre Theorem Power Calculator for Complex Numbers

This De Moivre theorem power calculator raises a complex number written in polar form to any supported integer power. Enter the non-negative modulus, the argument, and the exponent, then choose degrees or radians. The calculator applies the theorem directly and returns the new modulus and argument together with the rectangular real and imaginary components. It handles zero and negative integer powers when they are mathematically defined, making it useful for homework checks, engineering calculations, and software tests without requiring you to convert the input to rectangular form first.

Enter a complex number in polar form

A complex number in polar form is described by a modulus and an argument. The modulus is the distance from the origin, so this calculator requires it to be zero or positive. The argument gives the direction of the number and may be entered in degrees or radians.

Select the unit that matches your source instead of converting the angle beforehand. For example, a modulus of 2 and an argument of 30 degrees represents the same point as a modulus of 2 and an argument of pi divided by 6 radians. The power must be an integer because this capability applies the integer-power form of De Moivre's theorem.

Positive powers multiply the argument while expanding or shrinking the modulus. A power of zero returns the multiplicative identity for every nonzero input. Negative powers compute reciprocals, so the calculator rejects a zero modulus paired with a negative exponent.

Angles are not forced into a principal interval; preserving the multiplied argument makes every step of the theorem visible and avoids hiding complete rotations that may matter in a derivation.

Understand the De Moivre calculation

De Moivre's theorem states that a polar complex number of the form r times cosine theta plus i sine theta, raised to an integer n, becomes r to the power n times cosine n theta plus i sine n theta. The calculator therefore performs two central operations: it raises the modulus to the selected integer power and multiplies the argument by that power. It then evaluates cosine and sine at the resulting angle to obtain the rectangular components.

The real component is the new modulus multiplied by cosine of the new argument, and the imaginary component is the new modulus multiplied by sine of that argument. The returned polar expression shows the same new modulus and argument used for the rectangular conversion. Because trigonometric functions are periodic, an argument such as 450 degrees may describe the same direction as 90 degrees; the unnormalized value is still correct and more directly records the multiplication performed by the theorem.

Read and verify the result

The response gives the resulting modulus, argument, selected angle unit, real component, imaginary component, a readable polar expression, and the method name. Use the modulus and argument when continuing a polar-form calculation, or use real and imaginary when you need the conventional a plus bi form. A quick verification is to check that the square root of real squared plus imaginary squared agrees with the returned modulus, allowing for ordinary floating-point rounding.

You can also divide the output argument by the power to recover the input argument when the power is nonzero. Negative powers often produce a fractional modulus and reverse the sign of the angle when the original angle is positive. Inputs are not sent to a third-party service, and the algorithm uses no randomness, current time, or network data.