Making sense of Euler’s formula

The supposed mysteriousness of Euler’s formula is overrated. I’m really not a fan of, e.g. Khan Academy calling it the “proof of the existence of god” after giving you an un-illuminating proof of the theorem.

Euler’s formula relates exponentials to periodic functions. Although the two kinds of functions look superficially very different (exponentials diverge really quickly, periodic functions keep oscillating back and forth), any serious math student would have noted a curious relation between the two – periodic functions arise whenever you do some negative number-ish stuff with exponentials.

For instance –

• Simple harmonic motion -- the differential equation \(F=kx\) represents exponential motion when \(k>0\), periodic motion when $k

At 5:58, you will notice that the usage of \(a-bi\) (as opposed to \(a+bi\)) is linked to the minus sign in the expansion of \(\cos(\theta +\phi)\) and the plus sign in the expansion of \(\cos(\theta - \phi)\) – this gives us another explanation for the sign reversal in the cosine-of-a-sum formulae: because \(i\cdot -i=1\).