Explanation vs Derivation Math Example 4

Solution

1. 1
   Difference: a derivation proves a result by logical steps from known facts. An explanation says why the result is true in an intuitive way.
2. 2
   Derivation: Use $e^{i\theta} = \cos\theta + i\sin\theta$ (Euler's formula). At $\theta=\pi$: $e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1$. Therefore $e^{i\pi}+1=0$.
3. 3
   Explanation: Multiplying by $e^{i\pi}$ rotates a complex number by $\pi$ radians (half a turn). Starting at $1$ on the complex plane, a half-turn lands at $-1$, so $e^{i\pi}=-1$.