Farey pair

Two nonnegative reduced fractions $a/b$ and $c/d$ make a Farey pair (with ) 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 $\mathrm{[}a\mathrm{,}b\mathrm{,}c\mathrm{/}d\mathrm{]}$ 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 $a\mathrm{/}b\mathrm{,}c\mathrm{/}d$, the mediant is the one having the smallest denominator.

Example.
Notice that $3/8$ and $5/11$ form a Farey pair, since $8\cdot 5-3\cdot 13=40-391$. The mediant here is $8/21$.

Then $3/8$ and $8/21$ form a Farey pair: $8\cdot 8-3\cdot 21=64-63=1$. 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