Many consider Euler's formula to be the most beautiful that mathematics has ever produced. This last set of quizzes is the home stretch for understanding Euler's formula. And we'll also provide examples of its many uses.

Finally, the idea of raising \(e\) to a complex number will be illuminated through the ideas of transformations in the complex plane. Here's a glimpse at the punchline: because multiplication by \( e^{\pi i}\) or \(-1\) both rotate the complex plane \(\pi\) radians around the origin, \( e^{\pi i}\) and \(-1,\) are equivalent and we can say \[e^{\pi i} = -1.\]

We'll touch on applications of Euler's formula ranging from geometry to trigonometry to physics, such as the fact that the sum of two waves is another wave: