Concrete mathematics : a foundation for computer science

118 NUMBER THEORY
A mediant fraction isn’t halfway between its progenitors, but it does lie somewhere
in between. Therefore the construction preserves order, and we couldn’t
possibly get the same fraction in two different places. True, but if you get
One question still remains. Can any positive fraction a/b with a I b a comPound fracture
you’d better go
possibly be omitted? The answer is no, because we can confine the construe- see a doctor,
tion to the immediate neighborhood of a/b, and in this region the behavior
is easy to analyze: Initially we have
m - 0
n -7 0 and m-
b n’ ;>o
imply that
hence
an-bm 3 1 and bm’ - an’ 3 1;
(m’+n’)(an-bm)+(m+n)(bm’-an’) 3 m’+n’+m+n;
and this is the same as a + b 3 m’ + n’ + m + n by (4.31). Either m or n or
m’ or n’ increases at each step, so we must win after at most a + b steps.
The Farey series of order N, denoted by 3~, is the set of all reduced
fractions between 0 and 1 whose denominators are N or less, arranged in
increasing order. For example, if N = 6 we have
36 = 0 11112 1.3 2 3 3 5 1
1'6'5'4'3'5'2'5'3'4'5'6'1'
We can obtain 3~ in general by starting with 31 = 9, f and then inserting
mediants whenever it’s possible to do so without getting a denominator that
is too large. We don’t miss any fractions in this way, because we know that
the Stern-Brocot construction doesn’t miss any, and because a mediant with
denominator 6 N is never formed from a fraction whose denominator is > N.
(In other words, 3~ defines a subtree of the Stern-Brocot tree, obtained by