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BetterExplained isn’t an authority lecturing you on facts: it’s an excited friend sharing what actually helped when learning.... Read article

Intuitive Understanding Of Euler’s Formula

Euler's identity seems baffling:

\displaystyle{e^{i\pi} = -1}

It emerges from a more general formula:

\displaystyle{ e^{ix} = cos(x) + i sin(x)}

Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to 1800s mathematician Benjamin Peirce:

It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

... Read article