the immolation formula: f(x) – f(dim(x))

The striking formula f(x) – f(dim(x)), where ‘dim(x)’ means some kind of diminishment of x, occurs here and there in mathematics. I’ve never seen it referenced explicitly as I’m doing now, much less named, so I am going to give it the name ‘immolation formula’, because it kind of looks like what results is a butchered form of f(x) (and because ‘immolation’ rolls nicely off the tongue). Two common forms of diminishment of x are -x and x – 1. Actual use of the immolation formula might occur in a kind of spread-out stepwise manner that disguises the fact that the immolation formula is at work, for example, at a certain juncture in the derivation of the computational formula for the logarithm that Tom Apostol provides in volume 1 of his 2-volume Calculus textbook, the instance being one of the form ‘dim(x) = -x’. A more directly visible occurrence of the immolation formula is in specifying the general term in the summation of the average speed over members of a partition of an interval of time: (p(t(k)) – p(t(k – 1))/(t(k) – t(k – 1)), where p is the position function, the denominator of this expression being a ‘dim(x) = x – 1’ instance of the immolation formula.
Another occurrence of the immolation formula is in defining the Kaprekar process. The definition of the Kaprekar process can be worded as follows: “For a given 4-digit positive integer n (expressed in base 10, leading 0’s allowed) containing at least 2 digits, let k(0) be n, and let f(n) be the largest number expressible using the digits of n, and let f(-n) be the smallest number expressible using the digits of n, and let k(1) = f(n) – f(-n). Then apply the preceding to k(1) as the ‘new n’, obtaining k(2), and so on.” (This process reaches the number 6174 in at most 7 steps.)
Just as group theory can be use additive notation (typically, for abelian groups) or multiplicative notation, so also the ‘dim(x) = -x’ version of the immolation formula can be stated in multiplicative form as well: f(x)/(f(1/x).
A formula usually means a statement of some kind, whereas what we are dealing with here is not a statement, but merely a quantity given as a symbolic expression. So, we are using ‘formula’ in the wide sense to mean ‘symbolic expression’. It is convenient to have this wide sense. For example, if we are dealing with an equation of the form f(x) = g(x), it would be convenient to be able to (validly) refer to ‘the formula on the left side’ and / or ‘the formula on the right side’ of the equation, as well as having available the more sweeping traditional terms of LHS and RHS.
keywords: Mathematics, subtraction, additive inverse