# Why do we miss 8 in the decimal expansion of 1/81, and 98 in the decimal expansion of 1/9801?

Why do we miss $8$ in the decimal expansion of $1/81$, and $98$ in the decimal expansion of $1/9801$? I’ve seen this happen that when you divide in a fraction using the square of any number with only nines in the denominator. Like in

and in

the decimals go on predictably when suddenly in the first one you miss $8$, and in the second you miss $98$ and it keeps going on forever. How does this happen? Why do we miss numbers like $8$ in the decimal representation of $\frac{1}{9^2},$ or like $98$ in the decimal of $\frac{1}{99^2},$ or $998$ in $\frac{1}{999^2},$ or $9998$ in $\frac{1}{9999^2}\;$?

For $\frac{1}{81}$, there was an $8$, but it got bumped up. We can write $\frac{1}{81}$ as this sum:

This kind of effect of “carrying the $1$” when the nine digit flips to a ten is the thing that is causing the behavior in all of the fractions you are describing.

To follow-up, this should provide a bit more insight as to why interesting patterns appear in the decimal representations of fractions with a power of $9$ or $11$ as the denominator, and see why we can write those numbers like $\frac{1}{81}$ as that sum. First note that

so if we were to consider $\frac{1}{81}$ like before, we would have

Then if we were to want to know the value of the ten-thousandth’s decimal place of $\frac{1}{81}$, we would just have to find the numerator of the term in the expansion of this square with a denominator of $10\,000$, which we can readily see is

So it looks like these evident patterns that appear in the decimal expansions of fraction with a multiple of $9$ in the denominator is due at least partly to the fact that this infinite sum representation of $\frac{1}{9}$ consists entirely of terms with a numerator of $1$, so multiplying this sum into things may result in “predictable” behavior that results in a pattern.

As for why having a multiple of $11$ in the denominator makes similar patterns, note that

So again we have an infinite sum of terms each with a numerator of $1$ (just alternating sign this time) that will result in certain “predictable” patterns when multiplied by things.