Price elasticity of demand between two points

Why the midpoint? Dividing by the average of the two values rather than the starting value gives the same elasticity whichever direction the price moves, so A→B and B→A agree. The sign is negative because price and quantity move in opposite directions along a demand curve; it is the size, |PED|, that tells you elastic (>1), unit (=1) or inelastic (<1).