Concrete mathematics : a foundation for computer science

292 SPECIAL NUMBERS
Thus we can convert at sight between continued fractions and the corresponding
nodes in the Stern-Brocot tree. For example,
I
f(LRRL) = 0+ ~~ 1 *
l+-7
2 $- -
1,;
We observed in Chapter 4 that irrational numbers define infinite paths
in the Stern-Brocot tree, and that they can be represented as an infinite
string of L’s and R’s. If the infinite string for a is RaoLal RaZL”3 . . . , there is
a corresponding infinite continued fraction
a = aof
a1 + ~ 1
a2 + -
1
a3 +
1
a4 +
1
1
1
a5 + -
(‘3.141)
This infinite continued fraction can also be obtained directly: Let CQ = a and
for k 3 0 let
ak = Lakj ;
1
ak = ak+-.
Kkfl
(6.142)
The a’s are called the “partial quotients” of a. If a is rational, say m/n,
this process runs through the quotients found by Euclid’s algorithm and then
stops (with akfl = o0).
Is Euler’s constant y rational or irrational? Nobody knows. We can get Or if they do,
partial information about this famous unsolved problem by looking for y in theY’re not ta’king.
the Stern-Brocot tree; if it’s rational we will find it, and if it’s irrational we
will find all the closest rational approximations to it. The continued fraction
for y begins with the following partial quotients:
Therefore its Stern-Brocot representation begins LRLLRLLRLLLLRRRL . . ; no
pattern is evident. Calculations by Richard Brent [33] have shown that, if y
is rational, its denominator must be more than 10,000 decimal digits long.