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.

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 $21$ .

单词	FareyPair
释义	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. 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 $21$ .
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