Physical Interpretation of Complex Angles and Their Functions

A child´s idea of an angle is a corner or edge; the sharper the corner or edge, the smaller is the angle, and hence expressions like ``acute (sharp) angle,´´ ``obtu e (blunt) angle,´´ etc. With a little schooling his idea of an angle is broadened into that of turning or rotation and finds a geometrical expression in a circle. The amount of turning is represented by the area of the sector of the circle like a piece of pie gone over by the turning radius. This broadened dea is still in agreement with the earlier idea because the sector is a surface having a corner at the center of the circle, and the sharpness or bluntness of this corner varies with the area. With further progress, he also learns to associate the angle with the arc of the circle, for, evidently, the length of the arc is a measure of the amount of turning and is proportional to the area of the sector. The angle when represented by the arc of the circle loses all resemblance to a corner, which latter idea however is discarded as unnecessary, emphasis now being laid on ``turning´´ which is well represented by the arc of the circle.