Superparticular ratio: Difference between revisions

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'''Superparticular''' numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."

'''Superparticular''' numbers are ratios of the form <math>\frac{n+1}{n}</math>, or <math>1+\frac{1}{n}</math>, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."

These ratios have some peculiar properties:

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* The difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.

* The sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]].

* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), but more than one such splitting method may exist.

* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>, but more than one such splitting method may exist.

* If a/b and c/d are Farey neighbors, that is if a/b < c/d and bc - ad = 1, then (c/d)/(a/b) = bc/ad is epimoric.

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-'''Superparticular''' numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."
+'''Superparticular''' numbers are ratios of the form <math>\frac{n+1}{n}</math>, or <math>1+\frac{1}{n}</math>, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."
 These ratios have some peculiar properties:
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 * The difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.
 * The sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]].
-* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), but more than one such splitting method may exist.
+* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>, but more than one such splitting method may exist.
 * If a/b and c/d are Farey neighbors, that is if a/b &lt; c/d and bc - ad = 1, then (c/d)/(a/b) = bc/ad is epimoric.