Dear Uncle Colin,

I noticed that $\frac{987654321}{123456789} \approx 8.0000000729$ in base 10. On further investigation, $\frac{321}{123}\approx 2$ in base 4, $\frac{654321}{123456}\approx 5$ in base seven and a similar pattern recurs in any base: $\frac{[b-1][b-2]\dots 54321}{12345\dots[b-2][b-1]}$ (where $[a][b]$ means the concatenation of digits $a$ and $b$) is very close to $b-2$. Any idea why?

Runs Are Tremendously Interesting, Oooya!

Hi, RATIO, and thanks for your message! The answer lies in the binomial expansion.

In base 10

Let’s look at the base 10 version first, and then generalise it.

The nugget here is that $\frac{1}{81} = 0.\dot 01234567\dot 9$ - the nice pattern is slightly spoilt by a carry from the 10. This comes from $(1-x)^{-2} = 1 + 2x + 3x^2 + \dots$; evaluated at $x=\frac{1}{10}$, you get $\frac{100}{81} = 1.2345\dots$, and you can get to $\frac{1}{81}$ - or, more pertinently, something very close to 123456790 - from there. In fact, $\frac{10^{10}}{81} = 123,456,790.\dot 123456789\dot0$, which is only one-and-a-bit more than the denominator.

Let’s also think about (for reasons that will become clear) the sum of the numerator and denominator, which is 1,111,111,110. You might recognise an almost-pattern there, too; it looks very much like a ninth. In fact, it’s one and a smidge smaller than $\frac{10^{10}}{9}$.

Recap: 123,456,789 is about $\frac{x}{81}$ and 123,456,789 + 987,654,321 is about $\frac{x}{9}$, with $x =10^{10}$. That means 987,654,321 is roughly $\frac{8x}{81}$.

And voila: $\frac{987,654,321}{123,456,789} \approx \frac{8x/81}{x/81} = 8$.

In base $b$

A similar pattern holds in general, only replacing 81 with $(b-1)^2$, 9 with $(b-1)$ and $10^{10}$ with $b^b$.

More concretely:

  • $123\dots[b-2][b-1] \approx \frac{b^b}{(b-1)^2}$
  • $[b-2][b-1]\dots321 + 123\dots[b-2][b-1] = 111\dots10 \approx \frac{b^b}{b-1}$
  • So $[b-2][b-1]\dots321 \approx \frac{(b-2)b^b}{(b-1)^2}$
  • And $\frac{[b-2][b-1]\dots321}{123\dots[b-2][b-1]} \approx b-2$.

Hope that helps!

- Uncle Colin