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

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=nn1nn11(n1)2 and the denominator is q=n(nn11)(n1)21.

Note that p=(n2)q+n1 and for the quotient we get

pq=n2+(n1)3nn11n2n+1nn=n2+(n1)3nnk=0(n2n+1nn)k.

Indeed for n=10 this is

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

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

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