Farey Fractions and "reducibility"

From: rljacobson (rljacobson_at_gmail.com)
Date: 10/26/04 

Date: 26 Oct 2004 09:51:07 -0700

The Farey series of order n, denoted F_n, is the set arranged in
increasing order of all irreducible fractions h/k such that 0<=h<=k<=1
and gcd(h, k)=1. It is known that if h_0/k_0, h_1/k_1, and h_2/k_2 are
three successive terms in F_n, then

h_1/k_1 = (h_0 + h_2)/(k_0 + k_2).

However, this is NOT to say that h_1 = (h_0 + h_2) and k_1 = (k_0 +
k_2), for perhaps one has to reduce the RHS of the above. Indeed, in An
Introduction to the Theory of Numbers, Hardy and Wright include
footnotes saying, "Or the reduced form of this fraction", when making
reference to (h_0 + h_2)/(k_0 + k_2) (p. 23-24).

Suppose h/k and h'/k' are consecutive in F_n, and are separated by
h''/k'' in F_(n+1). (That is, h''/k'' = (h+h')/(k+k').) Further suppose
d|(h + h') and d|(k + k') for some d>0. Then d|[k(h+h')+(-h)(k+k')].
But k(h+h') - h(k+k') = kh' - hk' = 1 where the last step follows from
the fact that h/k and h'/k' are two consecutive terms in F_n. Hence
d=1. Therefore (h+h')/(k+k') is in lowest terms and hence h''=(h+h')
and k''=(k+k').

So why not just state that h_1 = (h_0 + h_2) and k_1 = (k_0 + k_2)?
Does not this give us more information about the properties of these
fractions with no extra ink wasted? If the answer is, "Because that
fact is obvious," then why the strange footnotes in Hardy and Wright?
(I have not the slightest doubt they knew of this fact.)

Consider this my opportunity to vent some frustrations over a very
small unnoticed detail setting me back almost a day.