Ascending and Descending Digits


He looked so immaculately frightful

As he bummed a cigarette,

Then he went off sniffing drainpipes

And reciting the alphabet.

                               Bob Dylan


The decimal number 987654321 is very close to being 8 times the integer 123456789. In fact, we have the exact ratio



The prime factorization of the denominator is (2)(5)(37)(333667).  A similar relation holds for any other base. In general, letting B denote the base, we have the algebraic identity



To prove this, we first split the summation in the denominator into two parts as follows



Re-arranging terms, this can be written as



The summation on the right side of (1) is just the finite geometric series, which has the closed-form expression



A closed-form expression for the summation on the left side of (1) can be found by differentiating the finite geometric series, giving the identity



Substituting for the summations in (1) and multiplying through by (B−1), we get



Equality is confirmed by expanding both sides and cancelling terms.


This type of relation is not limited to complete sequences of digits. We also have identities involving numbers with truncated strings of digits, such as



and so on. Written explicitly for arbitrary base B and letting k denote the number of digits in each number, these relations have the form



Naturally if we set k = B−1 and shift the index of summation we recover the previous relation. Splitting the summations, this more general relation can be written as



Re-arranging terms, this becomes



The summations have the closed form expressions


Making these substitutions and multiplying through by (B−1), we get



Expanding the products and cancelling terms, this confirms the equality.


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