Farey pair

Two nonnegative reduced fractions a/b and c/d make a Farey pair (with a/b<c/d) whenever bc-ad=1, in other words, they are a Farey pair if their difference is 1/bd. The interval [a/b,c/d] is known as a Farey interval.

Given a Farey pair a/b,c/d, their mediant is (a+c)/(b+d). The mediant has the following property:

If [a,b,c/d] is a Farey interval, then the two subintervals obtained when inserting the mediant are also Farey pairs. Besides, between all fractions that are strictly between a/b,c/d, the mediant is the one having the smallest denominator.

Notice that 3/8 and 5/11 form a Farey pair, since 85-313=40-391. The mediant here is 8/21.

Then 3/8 and 8/21 form a Farey pair: 88-321=64-63=1. No fraction between 3/8 and 5/11 other than 8/21 has a denominator smaller or equal than 21.

Title Farey pair
Canonical name FareyPair
Date of creation 2013-03-22 14:54:42
Last modified on 2013-03-22 14:54:42
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Definition
Classification msc 11A55
Related topic ContinuedFraction
Defines mediant
Defines Farey interval