Graphing complex numbers using De Moivre's Theorem can really help us understand the way complex numbers work, both in shapes and their relationships. This theorem is a handy tool that links algebra and geometry, especially when we need to find the powers and roots of complex numbers in a specific way called polar form.

What is De Moivre's Theorem?

De Moivre's Theorem says that if we have a complex number in polar form like $r(\cos \theta + i \sin \theta)$, we can find its $n^{th}$ power (that means multiplying it by itself $n$ times) using this formula:

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

In simpler words, when we want to raise a complex number to a power, we just raise the size $r$ to the power of $n$ and multiply the angle $\theta$ by $n$. This is easier to understand when we think of it in terms of a circle, where we can see these changes happening.

Visualizing Complex Numbers

When we draw complex numbers on a graph called the Argand plane, we put the real part on the x-axis and the imaginary part on the y-axis.

For example, let’s take the complex number $z = 1 + i$. In polar form, we can break it down like this:

• Size: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$
• Angle: $\theta = \tan^{-1}(1) = \frac{\pi}{4}$

So, we can write $z$ in polar form as:

$$z = \sqrt{2}\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)$$

If we want to find $z^3$, we can use De Moivre's Theorem. Here’s how it works:

$$z^3 = (\sqrt{2})^3\left(\cos(3 \cdot \frac{\pi}{4}) + i \sin(3 \cdot \frac{\pi}{4})\right)$$

Calculating this gives us:

• Size: $(\sqrt{2})^3 = 2\sqrt{2}$
• Angle: $3 \cdot \frac{\pi}{4} = \frac{3\pi}{4}$

So, we can write:

$$z^3 = 2\sqrt{2}\left(\cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4}\right)$$

Finding Roots with Confidence

De Moivre's Theorem is also really helpful when we want to find roots of complex numbers. For the $n^{th}$ root of a complex number, the formula is a bit different, but still follows a clear pattern:

$$\sqrt[n]{r(\cos \theta + i \sin \theta)} = r^{1/n}\left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right) \quad \text{for } k = 0, 1, 2, \ldots, n-1$$

This means we can easily find different roots and see how they spread out on the complex plane, usually forming a circle. For example, if we’re finding the cube roots of $1$ ($1$ multiplied by itself three times), we can figure them out by using this method. We get:

• $z = 1$ (when $k = 0$)
• $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$ (when $k = 1$)
• $z = \frac{1}{2} - i\frac{\sqrt{3}}{2}$ (when $k = 2$)

These roots make points that form an equilateral triangle on the unit circle.

Conclusion

By graphing complex numbers with De Moivre's Theorem, students can easily grasp how complex numbers multiply and how to find their roots. This hands-on way of learning helps turn tricky concepts into something we can all understand. It also strengthens math skills and helps us see the beautiful connection between geometry and complex numbers.