PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: No information on entry rating
adjacent fraction (Definition)

Two fractions $ \frac{a}{b}$ and $ \frac{c}{d}$, $ \frac{a}{b} > \frac{c}{d}$ of the positive integers $ a, b, c, d$ are adjacent if their difference is some unit fraction $ \frac{1}{n}$, $ n>0$ that is, if we can write:

$\displaystyle \frac{a}{b} - \frac{c}{d} = \frac{1}{n}.$    

For example the two proper fractions and unit fractions $ \frac{1}{11}$ and $ \frac{1}{12}$ are adjacent since:

$\displaystyle \frac{1}{11} - \frac{1}{12} = \frac{1}{132} \; .$    

$ \frac{1}{17}$ and $ \frac{1}{19}$ are not since:

$\displaystyle \frac{1}{17} - \frac{1}{19} = \frac{2}{323} \; . $

It is not necessary of course that fractions are both proper fractions:

$\displaystyle \frac{20}{19} - \frac{19}{19} = \frac{1}{19} \; . $
or unit fractions:
$\displaystyle \frac{3}{4} - \frac{2}{3} = \frac{1}{12} \; . $

All successive terms of some Farey sequence $ F_{n}$ of a degree $ n$ are always adjacent fractions. In the first Farey sequence $ F_{1}$ of a degree 1 there are only two adjacent fractions, namely $ \frac{1}{1}$ and $ \frac{0}{1}$.

Adjacent unit fractions can be parts of many Egyptian fractions:

$\displaystyle \frac{1}{70} + \frac{1}{71} = \frac{141}{4970} \; . $



"adjacent fraction" is owned by XJamRastafire.
(view preamble)

View style:

See Also: Farey sequence, unit fraction, continued fraction

Keywords:  difference, fraction, successive term
Log in to rate this entry.
(view current ratings)

Cross-references: Egyptian fractions, degree, Farey sequence, terms, necessary, proper fractions, unit fraction, difference, adjacent, integers, positive, fractions
There are 3 references to this entry.

This is version 14 of adjacent fraction, born on 2002-06-21, modified 2004-09-22.
Object id is 3124, canonical name is AdjacentFraction.
Accessed 3024 times total.

Classification:
AMS MSC: 11A67 (Number theory :: Elementary number theory :: Other representations)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)