Notes for week 106, part 2

1 files changed, 40 insertions, 0 deletions

diff --git a/challenge-106/abigail/README.md b/challenge-106/abigail/README.md
index e2550248dd..2f9f8e7880 100644
--- a/challenge-106/abigail/README.md
+++ b/challenge-106/abigail/README.md
@@ -52,7 +52,47 @@ Input: $N = 5, $D = 66
 Output: "0.0(75)"
 ~~~~
 
+### Notes
+
+We're creation the decimal expansion of the fraction `$N / $D`
+by performing long division.
+
+First, we calculate the part before the decimal point, by
+doing integer division of `$N / $D`.
+We're then left to do division of `$N' / $D`, where `$N'` initially
+is `$N % $D`.
+We then repeatedly find new digits by calculating the integer
+division of `(10 * $N' / $D)` (which gives us a new digit in the
+decimal expansion), and then setting `$N' = 10 * $N' % $D`.
+The fraction will have a finite decimal expansion if during the
+process `$N'` becomes `0`. Otherwise, it repeats, and it repeats
+as soon as have a `$N'` which we've already seen. By the pidgeon
+hole principle, this cannot take more then $D steps.
+To calculate the repeating part, we keep track of how far we
+were in calculating the expansion for which `$N'`.
+
+~~~~
+     22/7     \0.318
+        0                      int (7 / 22) == 0, so 0 before decimal point
+        --
+        7                      N =       N  % D
+        66                     3 * D
+        --
+         4                     N = (10 * N) % D   <--+
+         22                    1 * D                 |
+         --                                          |  Same, so '18'
+         18                    N = (10 * N) % D      |  is the repeating
+         176                   8 * D                 |  part
+         ---                                         |
+           4                   N = (10 * N) % D   <--+
+~~~~
+
+This implementation is based on the [one given on
+Wikipedia](https://en.wikipedia.org/wiki/Repeating_decimal).
+
+
 ### Solutions
+* [AWK](perl/ch-2.awk)
 * [Perl](perl/ch-2.pl)
 
 ### Blog