This chapter examines the various number systems in the following order: Real numbers, Complex numbers, Quaternions and Hyper complex numbers. The motion of translation can be in any number of different directions — north, south, east and west and to all points of the circle — in the limit: therefore motion can be in an infinite number of directions. But in rotation or intrinsic spin there are only two possible directions — left-handed or anticlockwise, and right-handed or clockwise. G. Frege in his Foundations of Arithmetic has no adequate account of complex numbers. Russell in his Introduction to Mathematical Philosophy describes a complex number as simply an ordered couple of real numbers. The physical quantities which have direction are acceleration force, velocity and momentum, both linear momentum and angular momentum. Vectors are therefore measure numbers, of magnitude and direction.