Superpartient ratio: Difference between revisions

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'''~~Superpartient~~''' ~~numbers are ratios of the form~~ ''p''/''q''~~, where~~ ''p'' ~~and~~ ''q'' ~~are relatively prime~~ (~~so that the fraction is reduced to lowest terms), and ''p'' -~~ ''q'' ~~is greater than~~ 1~~. In ancient Greece they were called epimeric (epimerēs~~) ~~ratios~~, ~~which~~ is ~~literally translated as "above~~ a ~~part". In ancient Greece, fractions like 3/~~1 and ~~5/1 were~~ not ~~considered to be epimeric ratios because of their additional restriction that~~ [[~~Harmonic|multiples of the fundamental~~]] ~~cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios~~.

In mathematics, a '''superpartient ratio''', also called an '''epimeric ratio''' or a '''delta-''d'' ratio''' (''d'' > 1), is a rational number that is greater than 1 and is not [[superparticular]].

All ~~epimeric~~ ratios can be constructed as products of [[superparticular]] numbers. This is due to the following useful identity:

More particularly, the ratio takes the form:

:<math>\frac{n + d}{n} = 1 + \frac{d}{n}</math>,

where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime|coprime]] to <math>n</math>.

== Etymology ==

In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".

== Definitions ==

In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.

== Properties ==

All superpartient ratios can be constructed as products of superparticular numbers. This is due to the following useful identity:

<math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>

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* [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]

[[Category:Ratio]]

[[Category:Terms]]

[[Category:Greek]]

~~[[Category:Ratio]]~~

~~[[Category:Superpartient| ]] ~~

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-'''Superpartient''' numbers are ratios of the form ''p''/''q'', where ''p'' and ''q'' are relatively prime (so that the fraction is reduced to lowest terms), and ''p'' - ''q'' is greater than 1. In ancient Greece they were called epimeric (epimerēs) ratios, which is literally translated as "above a part". In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.
+{{Wikipedia|Superpartient ratio}}
+In mathematics, a '''superpartient ratio''', also called an '''epimeric ratio''' or a '''delta-''d'' ratio''' (''d'' > 1), is a rational number that is greater than 1 and is not [[superparticular]].
-All epimeric ratios can be constructed as products of [[superparticular]] numbers. This is due to the following useful identity:
+More particularly, the ratio takes the form:
+:<math>\frac{n + d}{n} = 1 + \frac{d}{n}</math>,
+where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime|coprime]] to <math>n</math>.
+== Etymology ==
+In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
+== Definitions ==
+In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.
+== Properties ==
+All superpartient ratios can be constructed as products of superparticular numbers. This is due to the following useful identity:
 <math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>
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 * [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
+[[Category:Ratio]]
 [[Category:Terms]]
 [[Category:Greek]]
-[[Category:Ratio]]
-[[Category:Superpartient| ]] <!-- main article -->