No history yet

Review of De Moivre's Theorem

Powers and Angles

Raising a complex number to a power can seem messy. If you wanted to calculate (12+i32)3(\frac{1}{2} + i\frac{\sqrt{3}}{2})^3, you might start multiplying it out the long way. It works, but it's tedious.

Luckily, there's a more elegant way. A special relationship in mathematics connects the powers of complex numbers to the multiplication of their angles. This bridge is called De Moivre's Theorem.

For any integer n, De Moivre's Theorem gives us a simple way to compute the n-th power of a complex number written in polar form.

(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)

The Proof Using Euler's Formula

Why does this neat trick work? The most straightforward proof uses Euler's formula, which provides a deep connection between exponential functions and trigonometry. As a reminder, Euler's formula states:

eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta

The expression on the right, cos⁡θ+isin⁡θ\cos \theta + i \sin \theta, is the same one we see in De Moivre's Theorem. Let's start with Euler's formula and raise both sides to the power of nn, where nn is an integer.

(eiθ)n=(cos⁡θ+isin⁡θ)n(e^{i\theta})^n = (\cos \theta + i \sin \theta)^n

Now, let's focus on the left side of the equation. We can simplify it using a standard rule of exponents, (xa)b=xab(x^a)^b = x^{ab}.

(eiθ)n=ei(nθ)(e^{i\theta})^n = e^{i(n\theta)}

This gives us a new expression, ei(nθ)e^{i(n\theta)}. We can apply Euler's formula once more, but this time our angle is nθn\theta.

ei(nθ)=cos⁡(nθ)+isin⁡(nθ)e^{i(n\theta)} = \cos(n\theta) + i \sin(n\theta)

By combining these steps, we see that the left and right sides are equivalent.

Since (eiθ)n(e^{i\theta})^n equals (cos⁡θ+isin⁡θ)n(\cos \theta + i \sin \theta)^n, and (eiθ)n(e^{i\theta})^n also equals cos⁡(nθ)+isin⁡(nθ)\cos(n\theta) + i \sin(n\theta), we have proven De Moivre's theorem.

For a complex number z=r(cos x +i sin x), r being the modulus of the complex number and for some integer n, raising z to the power of n, we get: z^n=r^n(cos nx+i sin nx)

Why It Matters

De Moivre's Theorem is more than just a calculation shortcut. It's a fundamental bridge between algebra (powers of numbers) and geometry (angles and rotations). It shows that the multiplication of complex numbers in polar form corresponds to the addition of their angles. When you raise a complex number to a power nn, you are essentially performing that rotation nn times.

This insight reveals a deep, underlying pattern that connects different areas of mathematics. Understanding this theorem solidifies the relationship between complex numbers and trigonometry, a concept that is essential for more advanced topics in mathematics, physics, and engineering.

Quiz Questions 1/6

Which of the following correctly states De Moivre's Theorem for an integer n?

Quiz Questions 2/6

What is the primary geometric interpretation of raising a complex number to a power n?

That's a brief look at the statement and proof of De Moivre's Theorem. It's a key tool for working with complex numbers.