Farey pairFareyPair2004-12-27 14:41:072006-03-27 15:31:51DefinitionmediantFarey interval\usepackage{graphicx}
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\newtheorem{dfn}{Definition}Two nonnegative reduced fractions $a/b$ and $c/d$ make a \emph{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 \emph{Farey interval.}
Given a Farey pair $a/b,c/d$, their \emph{mediant} is $(a+c)/(b+d)$. The mediant has the following property:
{\sl 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.}
{\bf 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 $21$.