What does $\frac{1}{9801}$ equal to?

Question

While playing on my calculator I accidentally calculated $\frac{1}{9801}$ and the value was $0.00010203040506070809101112131415161718192021222324252627282930\ldots$
We observe that there is a nice pattern in the decimal expansion.
but how long will it continue ie. after how many digits will it repeat?

Answer 1

There is a numberphile video related to this here.

Essentially notice that $9801=99^2$. A similar pattern happens with $\frac{1}{9^2}=\frac{1}{81}$. It's value is $0.\dot{0}1234567\dot{9}$, cycling through all the units except $8$ (To see why, try a long division method). The case with $\frac{1}{99^2}$, it cycles through every two digit number, but excludes $98$ from this list... i.e. it is: $$\frac{1}{99^2}=0.\dot{0}0010203040506\cdots9596979\dot{9}$$ and similar things happen with $\frac{1}{999^2},\frac{1}{9999^2}$, etc.

Answer 2

Every fraction $\frac{a}{b}$ with integers $a$ and $b$ has a pattern which continues endlessly.

Answer 3

https://www.wolframalpha.com/input/?i=1%2F9801

See for yourself. A fraction when converted to decimal will repeat at some point of times.

Here it repeats till 99 skipping 98 and then start all over again.

Answer 4

Observe that with $a \in N$

$$ \frac{1}{9801}=\frac{a}{10^n}(1+10^{-n}+10^{-2n}+\cdots) = \frac{a}{10^n}\frac{1}{1-10^{-n}} $$

or

$$ a = \frac{10^n-1}{9801} $$

which gives $n = 198$ and $a = 10203040506070809101112131415161718192021222324252627282930313233343536373\\ 83940414243444546474849505152535455565758596061626364656667686970717273747 5767\\ 7787980818283848586878889909192939495969799$