Question

Find the 2014th digit after the decimal point of the decimal expansion of the repeating decimal 2/13.

Answer 1

What do you have at your disposal? I have this trick with power series in mind, but I was thinking maybe finding a preliminary pattern by long division might be a good start.

Answer 2

The pattern should repeat after \(n\) digits since it is a rational number. You just need to use \(\pmod n\)

Answer 3

And hope that \(n\) isn't huge.

Answer 4

It has a pattern!

Answer 5

It's not, long division yields \(\dfrac{2}{13}=0.\overline{153846}\).

Answer 6

Then we can easily solve for it

Answer 7

\[
f(x) = f(x+6) = f(x+9k)
\]

Answer 8

2014/6 = 335 remainder 4.

Answer 9

make that 6

Answer 10

then the answer will be 8, right?

Answer 11

I'm not up to date on the modular arithmetic, but I'll trust your judgment @wio :)

Answer 12

Yeah, if you consider \(8\) to be the 4th digit, then \(8\) would be the 2014th digit as well.

Answer 13

I'm sure it will yield with same answer.

Answer 14

Okay. That's the easy part. How if the decimal is non repeating and non terminating? Is there any way to solve it?

Answer 15

I think you just gave a description of an irrational number. A computer would be able to figure that out for you.

Answer 16

Well, solving for a particular digit of a rational number? I dunno.

Answer 17

irrational number^

Answer 18

WolframAlpha has the capacity: http://www.wolframalpha.com/input/?i=2014th+digit+of+pi

Answer 19

is that izzy from digimon

Answer 20

i don't know if i got his name right

Answer 21

@SithsAndGiggles Dunno if it actually calculates it out that far or calculates the specific digit.

Answer 22

My input is incorrect, yeah, should be "2015th digit", but the function is still there.