# Why is 987654321123456789=8.0000000729?!\frac{987654321}{123456789} = 8.0000000729?!

Many years ago,
I noticed that $987654321/123456789 = 8.0000000729\ldots$.

I sent it in to Martin Gardner at Scientific American
and he published it in his column!!!

My life has gone downhill since then:)

My questions are:

• Why is this so?

• What happens beyond the “$729$“?

• What happens in bases other than $10$?

In base $$nn$$ the numerator is $$p=nn−1−nn−1−1(n−1)2p = n^{n-1} - \frac{n^{n-1}-1}{(n-1)^2}$$ and the denominator is $$q=n(nn−1−1)(n−1)2−1.q = \frac{n(n^{n-1}-1)}{(n-1)^2}-1.$$
Note that $$p=(n−2)q+n−1p = (n-2)q + n-1$$ and for the quotient we get
pq=n−2+(n−1)3nn11−n2−n+1nn=n−2+(n−1)3nn∞∑k=0(n2−n+1nn)k.\begin{align} \frac{p}{q} &= n-2 + \frac{(n-1)^3}{n^n} \frac{1}{1 - \frac{n^2-n+1}{n^n}} \\ &= n-2 + \frac{(n-1)^3}{n^n} \sum_{k=0}^{\infty} \left(\frac{n^2-n+1}{n^n}\right)^k. \end{align}
Indeed for $$n=10n=10$$ this is
$$987654321123456789=8+7291010∞∑k=0(911010)k\frac{987654321}{123456789} = 8 + \frac{729}{10^{10}}\sum_{k=0}^{\infty}\left(\frac{91}{10^{10}}\right)^k$$