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##### Department of Mathematics, University of California San Diego

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## The Euler Identity

##### Abstract:

In this food for thought seminar, we will reflect upon the beautiful Euler identity $$e^{i \pi} = -1$$ Apparently, Euler viewed in this equation all the body of mathematics. The number $e$ is related to analysis, the number $\pi$ to geometry, the number $i$ to algebra, and the number $-1$ to arithmetic or synonymously number theory. \\ \\ The following identities are well known: \begin{itemize} \item The $x$ and $y$ coordinates of the third roots of unity on the unit circle \begin{itemize} \item $\sin(\frac{2 \pi}{3}) = \frac{\sqrt{3}}{2}$ and $\cos(\frac{2 \pi}{3}) = -\frac{1}{2}$ \item $\sin(\frac{4 \pi}{3}) = -\frac{\sqrt{3}}{2}$ and $\cos(\frac{4 \pi}{3})= -\frac{1}{2}$ \end{itemize} \item The $x$ and $y$ coordinates of the fourth roots of unity on the unit circle \begin{itemize} \item $\sin(\frac{2 \pi}{4}) = 1$ and $\cos(\frac{2 \pi}{4}) = 0$ \item $\sin(\frac{4 \pi}{4}) = 0$ and $\cos(\frac{4 \pi }{4}) = -1$ \item $\sin(\frac{6 \pi}{4}) = -1$ and $\cos(\frac{6 \pi}{4}) = 0$ \end{itemize} \end{itemize} The following ones are less well known: \begin{itemize} \item The $x$ and $y$ coordinates of the fifth roots of unity on the unit circle \begin{itemize} \item $\sin(\frac{2 \pi}{5})= \frac{\sqrt{5 + 2 \sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{2 \pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$ \item $\sin(\frac{4 \pi}{5}) = {\frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80\sqrt{5}}}}$ and $\cos(\frac{4 \pi}{5}) = -\frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$ \item $\sin(\frac{6 \pi}{5}) = - \frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80 \sqrt{5}}}$ and $\cos(\frac{6 \pi}{5}) = - \frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$ \item $\sin(\frac{8 \pi}{5}) = -\frac{\sqrt{5 + 2\sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{8\pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$ \end{itemize} \end{itemize} We shall explain a painless way of deriving these formulas. For that, we will do a little bit of analysis, geometry, algebra and arithmetic.

### AP&M B412

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