Concrete mathematics : a foundation for computer science

122 NUMBER THEORY
This means that we can transform the binary search algorithm to the following
matrix-free procedure:
while m # n do
if m < n then (output(L); n := n-m)
else (output(R); m := m-n) .
For example, given m/n = 5/7, we have successively
m=5 5 3 1 1
n=7 2 2 2 1
output L R R L
in the simplified algorithm.
Irrational numbers don’t appear in the Stern-Brocot tree, but all the
rational numbers that are “close” to them do. For example, if we try the
binary search algorithm with the number e = 2.71828. . , instead of with a
fraction m/n, we’ll get an infinite string of L’s and R's that begins
RRLRRLRLLLLRLRRRRRRLRLLLLLLLLRLR....
We can consider this infinite string to be the representation of e in the Stern-
Brocot number system, just as we can represent e as an infinite decimal
2.718281828459... or as an infinite binary fraction (10.101101111110...)~.
Incidentally, it turns out that e’s representation has a regular pattern in the
Stern-Brocot system:
e = RL”RLRZLRL4RLR6LRL8RLR10LRL’2RL . . . ;
this is equivalent to a special case of something that Euler [84] discovered
when he was 24 years old.
From this representation we can deduce that the fractions
RRLRRLRLLLL R L R R R R R R
1 2 1 5 & 11 19 30 49 68 87 -------- 106 193 299 492 685 878 1071 1264
1'1'1'2'3' 4' 7'11'18'25'32' 39' 71'110'181'252'323' 394' 465""
are the simplest rational upper and lower approximations to e. For if m/n
does not appear in this list, then some fraction in this list whose numerator
is 6 m and whose denominator is < n lies between m/n and e. For example,
g is not as simple an approximation as y = 2.714. . . , which appears in
the list and is closer to e. We can see this because the Stern-Brocot tree
not only includes all rationals, it includes them in order, and because all
fractions with small numerator and denominator appear above all less simple
ones. Thus, g = RRLRRLL is less than F = RRLRRL, which is less than