Glaucius Numerical Inversion Theory

Abstract: We present a systematic study of the Numerical Inversion Operation: given a positive integer n, we form its digit-reversal n′ by writing the digits of nin reverse order, and dene the inversion dierence Δ(n) = |n′ − n|. We prove that Δ(n) is always divisible by 9, with the exact multiple determined solely bythe outermost digits of n. For a two-digit number with tens digit d0 and units digit d1, the identity Δ(n) = 9|d1 − d0| holds. For a k-digit number, Δ(n) =(10k−1 − 1)|dk−1 − d0|, a q