https://www.johndcook.com/blog/2025/10/26/987654321/

John D. Cook
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987654321 / 123456789

Posted on 26 October 2025 by John

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.

\begin{align*} \text{num}(b) &= \sum_{k=1}^{b-1} kb^{k-1} \\ \text
{denom}(b) &= \sum_{k=1}^{b-1} (b-k)b^{k-1} \end{align*}

Then for b > 2 we have

\frac{\text{num}(b)}{\text{denom}(b)} = b - 2 + \frac{b-1}{\text
{denom}(b)}

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/b^b-2.

When b = 16, as in the example above, the result is approximately

14 + 16^-14 = 8 + 4 + 2 + 2^-56

which would take 60 bits to represent exactly, but a floating point
fraction only has 53 bits. That's why our calculation returned
exactly 14 with no fractional part.

 

[1] I saw @ColinTheMathmo post it on Mastodon. He said he saw it on
Fermat's Library somewhere. I assume it's a very old observation and
that the analysis I did above has been done many times before.

[2] Why include a script rather than a proof? One reason is that the
proof is straight-forward but tedious and the script is compact.

A more general reason that I give computational demonstrations of
theorems is that programs are complementary to proofs. Programs and
proofs are both subject to bugs, but they're not likely to have the
same bugs. And because programs made details explicit by necessity, a
program might fill in gaps that aren't sufficiently spelled out in a
proof.

Categories : Math
Tags : Number theory
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3 thoughts on "987654321 / 123456789"

 1.
    Michael Lugo
    26 October 2025 at 15:39

    8.0000000729 looks an awful lot like it's 8 + 9^3/10^10, which
    suggests a conjecture that

    num(b) / denom(b) ~ (b-2) + (b-1)^3 / b^b.

    and similarly, for example in base 9 we have num(9) / denom(9) =
    7.000000628, in base 8 we have num(8) / denom(8) = 6.00000527,
    etc. (In base b, (b-1)^3 has the digits (b-3), 2, (b-1).)

    In fact num(b) = ((b^b*(b-2) + 1)/(b-1)^2 and denom(b) = (b^b - b
    ^2 + b - 1)/(b-1)^2. These are A051846 and A023811 in the OEIS,
    respectively; I would have done the algebra but I looked them up
    there.

    Straightforward algebra shows that num(b) - (b-2) denom(b) = b-1,
    from which

    num(b) / denom(b) = (b-2) + (b-1)^3 / (b^b - b^2 + b - 1)

    which is our conjecture with some low-order terms in the
    denominator.

 2.
    siravan
    26 October 2025 at 16:37

    This reminds me of the old calculator trick of 12345679 * 8 =
    98765432

 3.
    Linus Gasser
    30 October 2025 at 12:14

    The line

    14 + 16-14 = 8 + 4 + 2 + 2-56

    Can be rewritten using the proposed formula as:

    14 + 16^-14 = 16 - 2 + 2^-56

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