Many years ago,

I noticed that 987654321/123456789=8.0000000729….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?

**Answer**

In base n the numerator is p=nn−1−nn−1−1(n−1)2 and the denominator is q=n(nn−1−1)(n−1)2−1.

Note that p=(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.

Indeed for n=10 this is

987654321123456789=8+7291010∞∑k=0(911010)k

**Attribution***Source : Link , Question Author : marty cohen , Answer Author : GarlicBread*