De Moivre’s Theorem Calculator

Calculate powers of complex numbers using De Moivre’s formula and visualize the results on the complex plane.

Calculation Details
Result in Polar Form
New Modulus (r’)
New Angle (θ’) in Degrees
New Angle (θ’) in Radians

What is the De Moivre’s Theorem Calculator?

The De Moivre’s Theorem calculator is a tool designed to compute the powers of complex numbers. De Moivre’s theorem provides a straightforward method for this calculation when the complex number is expressed in polar form. This theorem is a fundamental concept in complex analysis, connecting complex numbers with trigonometry. It is widely used by engineers, physicists, and mathematicians to simplify complex calculations involving powers and roots of complex numbers. This calculator not only gives you the final answer but also breaks down the components of the result and visualizes the transformation on the complex plane.

De Moivre’s Theorem Formula and Explanation

De Moivre’s theorem states that for a complex number in polar form `z = r(cos(θ) + i sin(θ))` and any integer `n`, the `n`-th power of `z` is given by the formula:

zⁿ = rⁿ(cos(nθ) + i sin(nθ))

This elegant formula shows that to raise a complex number to a power `n`, you raise its modulus `r` to the power `n` and multiply its argument `θ` by `n`. This is significantly simpler than repeatedly multiplying the complex number in its rectangular form `(a + bi)`. The theorem is a direct consequence of Euler’s Formula, `e^(iθ) = cos(θ) + i sin(θ)`.

Variables in De Moivre’s Theorem

Variable	Meaning	Unit	Typical Range
z	The complex number	Unitless	Any complex number
r	The modulus (magnitude) of z	Unitless (non-negative)	0 to ∞
θ	The argument (angle) of z	Degrees or Radians	0° to 360° or 0 to 2π
n	The power (an integer)	Unitless	Any integer (…, -2, -1, 0, 1, 2, …)
i	The imaginary unit	Unitless	√(-1)

Practical Examples

Example 1: Squaring a Complex Number

Let’s find (1 + i)². First, we convert 1 + i to polar form.

• Input Modulus (r): √2 ≈ 1.414
• Input Angle (θ): 45°
• Input Power (n): 2

Using the formula: (√2(cos(45°) + i sin(45°)))² = (√2)²(cos(2 * 45°) + i sin(2 * 45°)) = 2(cos(90°) + i sin(90°)).

• Result: 2(0 + i * 1) = 2i.

Example 2: Cubing a Complex Number

Let’s find (2(cos(60°) + i sin(60°)))³.

• Input Modulus (r): 2
• Input Angle (θ): 60°
• Input Power (n): 3

Using the formula: 2³(cos(3 * 60°) + i sin(3 * 60°)) = 8(cos(180°) + i sin(180°)).

• Result: 8(-1 + i * 0) = -8.

How to Use This De Moivre’s Theorem Calculator

Follow these steps to easily calculate the power of any complex number:

1. Enter the Modulus (r): Input the magnitude of your complex number. This value must be non-negative.
2. Enter the Angle (θ): Input the angle of your complex number.
3. Select the Angle Unit: Choose whether the angle you entered is in degrees or radians from the dropdown menu.
4. Enter the Power (n): Input the integer power you want to raise the complex number to.
5. Interpret the Results: The calculator instantly displays the result in both rectangular form (a + bi) and polar form, along with the new modulus and angle. The Argand diagram provides a visual representation of the original and resulting vectors. For more on converting between forms, see our Polar to Rectangular Form converter.

Key Factors That Affect the Result

• Modulus (r): If r > 1, the resulting vector will be longer. If 0 ≤ r < 1, it will be shorter. If r = 1, it will stay on the unit circle.
• Angle (θ): The angle determines the initial direction of the vector.
• Power (n): A positive power `n` rotates the vector by `(n-1)θ` and scales its length. A negative power will rotate it in the opposite direction. An `n` of 0 always results in 1 (for a non-zero complex number).
• Sign of n: A positive `n` rotates counter-clockwise, while a negative `n` rotates clockwise.
• Angle Unit: Ensuring the correct unit (degrees/radians) is selected is critical for an accurate calculation, as 45° is very different from 45 radians.
• Integer vs. Non-Integer Power: This theorem is specifically for integer powers. Finding roots (e.g., n=1/2) is a related but more complex process. See our article on Roots of Complex Numbers.

Frequently Asked Questions (FAQ)

What is De Moivre’s theorem used for?

It is primarily used to find the powers and roots of complex numbers. It is also essential for deriving trigonometric identities.

Can De Moivre’s theorem be used for any power n?

The theorem as stated is valid for any integer `n` (positive, negative, or zero). While generalizations exist for non-integer powers, they are more complex as they yield multiple values (roots).

What is the difference between De Moivre’s theorem and Euler’s formula?

Euler’s formula (`e^(iθ) = cos(θ) + i sin(θ)`) connects exponential functions to trigonometric functions. De Moivre’s theorem is a direct proof from Euler’s formula and is used specifically for calculating powers of complex numbers.

Why do I need to convert to polar form first?

Converting to polar form (modulus `r` and angle `θ`) simplifies the power calculation immensely. Raising a number `(a + bi)` to a large power through binomial expansion is extremely tedious compared to applying De Moivre’s formula.

What does the graph (Argand Diagram) show?

The diagram visualizes the complex numbers as vectors on a 2D plane. The x-axis is the real part, and the y-axis is the imaginary part. It helps you see the rotation and scaling that occurs when you apply the power. For a deeper dive, check out our guide on the Complex Plane Visualization.

What happens if the modulus (r) is negative?

The modulus of a complex number is its distance from the origin, which by definition cannot be negative. The calculator will show an error if a negative modulus is entered.

How does this relate to Trigonometric Identities?

De Moivre’s theorem can be used to derive identities for `cos(nθ)` and `sin(nθ)`. By expanding `(cos(θ) + i sin(θ))^n` with the binomial theorem and then equating the real and imaginary parts with `cos(nθ) + i sin(nθ)`, you can find expressions for multiple angle formulas.

What is a practical application of this theorem?

In electrical engineering and signal processing, signals are often represented as complex numbers (phasors). De Moivre’s theorem helps in analyzing how these signals change over time or through different circuits. It also has applications in fluid dynamics and other areas of physics.