  This paper presents a graded hierarchy or chain of binary operations on the
reals and the complex numbers. The operations are related distributively in the
sense that any one of them distributes over the next lower operation in the
chain. For one particular operation we explore specific properties and derive
results, including several useful formulas and identities. Next, the operation
is extended to the complex numbers and a new kind of derivative is defined
based on this binary operation. Some basic formulae analogous to standard
classical ones are proven for this new derivative. Finally, the derivative is
generalized to the point that the new derivative, the classical one, and
countably many others are seen to be special cases.
