posts, 06c: Small but necessary clarifications

diff --git a/posts/y04/06c-ffa-egypt-proof.markdown b/posts/y04/06c-ffa-egypt-proof.markdown
index ae7e31a..1ab4ab7 100644 (file)
--- a/posts/y04/06c-ffa-egypt-proof.markdown
+++ b/posts/y04/06c-ffa-egypt-proof.markdown
@@ -290,8 +290,8 @@ K(1) * B^0 + K(2) * B^1 + K(3) * B^2 + ...
 
 where `0 <= K(I) < B`, i.e. `K(I)` is a "modulo-B digit". Our particular
 `K(I)` is `Ybit(I)` and it denotes the elements of the bit-vector
-`Y`. From this follows that the sum-of-products factor in the definition
-of `XY(N)` is in fact `Y`, and thus `XY(N)` becomes:
+`Y`. From this follows that the sum-of-products factor in the relation
+of `XY(N)` above is in fact `Y`, and thus `XY(N)` becomes:
 
 ~~~~
 XY(N) = X * Y
@@ -301,15 +301,16 @@ This then settles it. A number of refinement steps have brought us from
 repeated `FZ_Add` operations to the mathematical `*`; the reverse
 (un-)refinement is also possible[^4], which serves to prove that the
 algorithm implemented by `FZ_Mul_Egyptian` and the algebraic `*` are
-equivalent. The reader is, of course, encouraged to challenge this.
+equivalent. The reader is, of course, encouraged to challenge this
+statement.
 
 More importantly, these refinement steps serve to show us what egyptian
 multiplication *is*: given a base `B` and two numbers `X` and `Y`,
 egyptian multiplication finds the answer to `X * Y` by adding `X` and
-multiplying it (the same `X`) by `B` a number of times equal to the
+multiplying it (the same `X`) by `B` a number of times depending on the
 number of digits (in base-`B` representation) of `Y`. Programmers can
-then implement the same algorithm on ternary or quaternary computers,
-etc.[^5]
+then implement a variation of this algorithm on ternary or quaternary
+computers, etc.[^5]
 
 We move on to the trickier part of our article: egyptian division.
 
@@ -425,7 +426,7 @@ mathematical form that allows us to reason about it. At least two
 aspects are not exactly easy to spot: `R` and `Q` suffer two updates
 each, and the `R` in `R < Divisor` (on which the conditional updates to
 `Q` and `R` depend) is not the same `R` as in the previous iteration of
-the loop. We'll try to untangle this first, by defining a
+the loop. We'll try to untangle this first, by taking a
 *conditional*-`R` (`CR`) which maps to the first update to `R`:
 
 ~~~~
@@ -467,7 +468,7 @@ mathematical, so it might be completely non-obvious where the relations
 come from. The reader is recommended to go through the mental gymnastics
 necessary to obtain them by using pen and paper. At the end it should be
 clear that `Q(N)` is the resulting quotient, while `R(N)` is the
-resulting remainder, and furthermore that the equation:
+resulting remainder, and furthermore we wish to show that the equation:
 
 ~~~~
 Dividend = Divisor * Q(N) + R(N)
@@ -597,8 +598,8 @@ The reader is again recommended to follow the reasoning using pen and
 paper.
 
 The (by now very) keen reader will notice that the first bracket in the
-definition of `R(N)` represents the very definition of `Dividend`; the
-second definition doesn't seem to be anything in particular, so we'll
+relation above represents the exact base-2 decomposition of `Dividend`;
+the second term doesn't seem to denote anything in particular, so we'll
 give the (not-so-inspired) name `DR`, leading to:
 
 ~~~~