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Use Demoivre's Theorem to Find The Indicated Root Calculator

Reviewed by Calculator Editorial Team

De Moivre's Theorem provides a method for finding the roots of complex numbers. This calculator helps you apply the theorem to find the indicated roots of a complex number in polar form.

What is De Moivre's Theorem?

De Moivre's Theorem is a fundamental result in complex analysis that relates complex numbers in polar form to their powers. The theorem states that for any complex number \( z = r(\cos \theta + i \sin \theta) \) and any integer \( n \),

De Moivre's Theorem Formula

\( z^n = r^n (\cos(n\theta) + i \sin(n\theta)) \)

This theorem is particularly useful for finding the roots of complex numbers. If we want to find the \( n \)-th roots of a complex number \( z \), we can use the theorem to express the roots in polar form.

Finding Roots Using De Moivre's Theorem

To find the \( n \)-th roots of \( z = r(\cos \theta + i \sin \theta) \), we can use the following steps:

  1. Express \( z \) in polar form.
  2. Divide the angle \( \theta \) by \( n \) to find the angle for each root.
  3. Calculate the \( n \)-th root of the magnitude \( r \).
  4. Use the formula \( z_k = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \) for \( k = 0, 1, \dots, n-1 \).

Note

The roots obtained using De Moivre's Theorem are equally spaced around a circle in the complex plane.

How to Use the Calculator

This calculator allows you to find the roots of a complex number using De Moivre's Theorem. Follow these steps to use the calculator:

  1. Enter the magnitude (r) of the complex number.
  2. Enter the angle (θ) in radians.
  3. Enter the number of roots (n) you want to find.
  4. Click the "Calculate" button to find the roots.
  5. View the results and the visualization of the roots in the complex plane.

The calculator will display the roots in both rectangular and polar forms, along with a visualization of the roots in the complex plane.

Example Calculation

Let's find the cube roots of the complex number \( z = 8(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \).

Using De Moivre's Theorem, the cube roots are given by:

Example Formula

\( z_k = 2 \left( \cos\left(\frac{\pi/6 + 2k\pi}{3}\right) + i \sin\left(\frac{\pi/6 + 2k\pi}{3}\right) \right) \) for \( k = 0, 1, 2 \).

The three cube roots are:

  1. \( z_0 = 2 \left( \cos\left(\frac{\pi}{18}\right) + i \sin\left(\frac{\pi}{18}\right) \right) \)
  2. \( z_1 = 2 \left( \cos\left(\frac{13\pi}{18}\right) + i \sin\left(\frac{13\pi}{18}\right) \right) \)
  3. \( z_2 = 2 \left( \cos\left(\frac{25\pi}{18}\right) + i \sin\left(\frac{25\pi}{18}\right) \right) \)

These roots are equally spaced around a circle with radius 2 in the complex plane.

Frequently Asked Questions

What is De Moivre's Theorem used for?
De Moivre's Theorem is primarily used to find the roots of complex numbers and to simplify calculations involving powers of complex numbers.
How do I convert a complex number to polar form?
To convert a complex number \( z = a + bi \) to polar form, calculate the magnitude \( r = \sqrt{a^2 + b^2} \) and the angle \( \theta = \arctan\left(\frac{b}{a}\right) \).
What are the roots of a complex number?
The roots of a complex number are the solutions to the equation \( z^n = w \), where \( w \) is the complex number. These roots are equally spaced around a circle in the complex plane.
Can De Moivre's Theorem be used for non-integer exponents?
De Moivre's Theorem is specifically for integer exponents. For non-integer exponents, you would use the more general exponential form of complex numbers.