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Euler's Formula


Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.



The motto in this section is: the higher the level of abstraction, the better.

Why is it interesting?

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.)

basic_tools/eulers_formula.1585827871.txt.gz · Last modified: 2020/04/02 11:44 (external edit)