For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, $e^{ix}=\cos{x}+i\sin{x}$, and as a result, Euler's Identity, $e^{i\pi}+1=0$. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it. We will also try to gain a small understanding of the impact that it has had on mathematics.

Author Bio

Caleb Larson graduated magna cum laude from the University of North Dakota in 2015 with a degree in mathematics. He worked on this paper throughout his capstone course in the spring semester of 2015. Caleb is now a doctoral student at North Dakota State University where he is a teaching assistant for the university. His main research interests lie in mathematics education, but he is also interested in algebra and analysis.