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?

It actually repeats all $2$ digit numbers, starting from $00$ to $99$, excluding $98$. — For the love of maths, May 25 '18 at 09:47
It may be but it was difficult to find it using the title and the questions are actually different — Abhishek Choudhary, May 25 '18 at 09:49
Repeating patterns in decimal expansions are oftend denoted, e.g. $1/3=0.333333\cdots=0.\dot{3}$, or e.g. $1/13=0.07692307692\cdots=0.\dot{0}7692\dot{3}$. — pshmath0, May 25 '18 at 09:51
By the way, $9801=99^2$. It is the only square number with this property. — Mr Pie, May 25 '18 at 09:56
@user477343 $\frac{1}{81}=\frac{1}{9^2}=0.12345679012345679$... — Rhys Hughes, May 25 '18 at 10:00
@Euler's Dad the famous formula for $\pi$...if you google you will find it — Enzo Creti, May 25 '18 at 10:08

score 6 · Accepted Answer · answered May 25 '18 at 09:58

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.

score 0 · Answer 2 · answered May 25 '18 at 09:46

0

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

answered May 25 '18 at 09:46

Logic_Problem_42

1,425

2

Discounting zeros, 1/4=0.25 does not continue forever. But if you count zeros then 1/4=0.250000... – pshmath0 May 25 '18 at 09:47

Harshit Joshi · Answer 3 · 2018-05-25T10:03:25.623

0

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.

edited May 25 '18 at 10:03

answered May 25 '18 at 09:54

Harshit Joshi

494

Incorrect. It skips 98. – Rhys Hughes May 25 '18 at 09:59
Yup. Sorry. Edited. – Harshit Joshi May 25 '18 at 10:03

score 0 · Answer 4 · answered May 25 '18 at 10:10

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 $a = 10203040506070809101112131415161718192021222324252627282930313233343536373\\ 83940414243444546474849505152535455565758596061626364656667686970717273747 5767\\ 7787980818283848586878889909192939495969799$

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

4 Answers4

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