What is the decimal form of 1/299,792,458

Question

I'm seeking the decimal form to the fraction 1/c, something like 3.35692(548672), where the number in the brackets would be reoccurring integers. Most calculators round after 10 digits, and the calculator app on my Android just computes the numbers endlessly (if I swipe left) making it hard to find the reoccurring pattern.

My intention is to find the decimal form of the definition of the metre.

Answer 1

The prime factorization of $299792458$ is $2 \cdot 7 \cdot 73 \cdot 293339$.

$10$ has order $6$ mod $7$.

$10$ has order $8$ mod $73$.

$10$ has order $293338 $ mod $293339$.

Therefore, the length of the periodic part is $lcm(6,8,293338)= 3520056 $.

Answer 2

Here is a solution from Mathematica given here:

normalizeDigitSequence = {{{beforeRecurring___, {recurring__}}, 
     c_?NonPositive} :> {{0, beforeRecurring, {recurring}}, 
     c + 1}, {{beforeRecurring___, {recurring__}}, c_?Positive} /; 
     Length[{beforeRecurring}] < c :> {{beforeRecurring, 
      First[{recurring}], RotateLeft[{recurring}]}, c}};
addOverlineToRepeating = {{beforeRecurring___, {recurring___}}, 
     c_?Positive} /; Length[{beforeRecurring}] >= c :> 
   Row[Append[Insert[{beforeRecurring}, ".", c + 1], 
     Overscript[Row[{recurring}], _]]];
RealDigits[1/72] //. normalizeDigitSequence /. addOverlineToRepeating

where you'd put in 1/299792458 instead of my example 1/72. Alas, with your number, the repeated sequence is (as lhf showed), much too long to print here.