Add a new page:
This is an old revision of the document!
Euler's formula is a deep result of complex analysis that shows a surprising relationship between seemingly unrelated notions:
$$ e^{ix} = \cos(x) + i \sin(x) \, $$
where $e^{ix}$ denotes the exponential function and $ \cos(x)$, $\sin(x)$ are the usual trigonometric functions. If we evaluate this equation at $x= \pi$, we get
$$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = -1 -i 0 = -1 \, .$$
This shows a deep relationship between the exponential function, the imaginary unit $i$ and $\pi$. (Pi is the ratio between circumference and diameter shared by all circles.)