I recently saw someone post [1] that 987654321/123456789 is very nearly 8, specifically 8.0000000729.
I wondered whether there’s anything distinct about base 10 in this. For example, would the ratio of 54321 six and 12345 six be close to an integer? The ratio is 4.00268, which is pretty close to 4.
What about a larger base? Let’s try base 16. The expression
0xFEDCBA987654321 / 0x123456789ABCDEF
in Python returns 14. The exact ratio is not 14, but it’s as close to 14 as a standard floating point number can be.
For a base b, let denom(b) to be the number formed by concatenating all the digits in ascending order and let num(b) be the number formed by concatenating all the digits in descending order.
Then for b > 2 we have
The following Python code demonstrates [2] that this is true for b up to 1000.
num = lambda b: sum([k*b**(k-1) for k in range(1, b)]) denom = lambda b: sum([(b-k)*b**(k-1) for k in range(1, b)]) for b in range(3, 1001): n, d = num(b), denom(b) assert(n // d == b-2) assert(n % d == b-1)
So for any base the ratio is nearly an integer, namely b − 2, and the fractional part is roughly 1/bb−2.
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