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 



Rearranging terms, this can be written as 



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



A closedform 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 



Rearranging 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. 
