Why does e to pi i equal -1? (2015) [video]
1) e^x is a function whose derivative is equal to its value.<p>2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix).<p>3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i.<p>4) So the trajectory a circle arc of length pi, which ends at -1.