Difference between revisions of "Farey sequence"
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==Proof Sketch== | ==Proof Sketch== | ||
Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+1}</math>? | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+1}</math>? | ||
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Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+2}</math>? | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+2}</math>? | ||
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Do you see a pattern? | Do you see a pattern? | ||
Revision as of 03:14, 30 January 2021
A Farey sequence of order 
is the sequence of all completely reduced fractions between 0 and 1 where, when in lowest terms, each fraction has a denominator less than or equal to . Each fraction starts with 0, denoted by the fraction 0/1, and ends in 1, denoted by the fraction 1/1.
Examples
Farey sequences of orders 1-4 are:
Where 
denotes a Farey sequence of order .
Proof Sketch
Which is bigger, 
or 
?
Which is bigger, or 
?
Do you see a pattern?
Properties
Sequence length
A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, contains all the terms in 
. Also, contains an extra term for every number less than relatively prime to . Thus, we can write
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