PlanetMath: unit fraction

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

unit fraction

(Definition)

An unit fraction $\frac{n}{d}$ is a fraction whose numerator . If its integer denominator , then a fraction is also a proper fraction. So there is only one unit fraction which is improper, namely 1.

Such fractions are known from Egyptian mathematics where we can find a lot of special representations of the numbers as a sum of an unit fractions, which are now called Egyptian fractions. From the Rhind papyrus as an example:

$\displaystyle \frac{2}{71} = \frac{1}{40} + \frac{1}{568} + \frac{1}{710} \; .$

Many unit fractions are in the pairs of the adjacent fractions. An unit fractions are some successive or non-successive terms of any Farey sequence $F_{n}$ of a degree . For example the fractions $\frac{1}{2}$ and $\frac{1}{4}$ are adjacent, but they are not the successive terms in the Farey sequence $F_{5}$ . The fractions $\frac{1}{3}$ and $\frac{1}{4}$ are also adjacent and they are successive terms in the $F_{5}$ .

"unit fraction" is owned by XJamRastafire.

(view preamble)

Also defines:

Egyptian fraction

Attachments:: any rational number is a sum of unit fractions (Derivation) by Mathprof

Log in to rate this entry.
(view current ratings)

Cross-references: adjacent, degree, Farey sequence, terms, adjacent fractions, Egyptian fractions, sum, numbers, representations, Egyptian mathematics, proper fraction, denominator, integer, numerator, fraction
There are 14 references to this entry.

This is version 7 of unit fraction, born on 2002-06-21, modified 2002-06-24.
Object id is 3125, canonical name is UnitFraction.
Accessed 9879 times total.

Classification:

AMS MSC:

11A67 (Number theory :: Elementary number theory :: Other representations)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

Unit Fractions by milogardner on 2006-03-28 10:13:31

Scribal unit fractions, as recorded in the Rhind
Mathematical Papyrus 2/nth table table used
several methods to write its 2/p and 2/pq series.
For example, Ahmes, the RMP scribe wrote his
2/71 series used the Hultsch-Bruins method, known
in the modern era since 1895 (due to the work of
F. Hultsch, and independently confirmed 50 years
later by E.M. Bruins).

The scribe first looked for the inverse composite numbers 1/A
within the range from p/2 to p, with those being 1/36, 1/38,
1/40, ..., 1/70. Each was tested, with an optimal series being
selected by the use of LCM considerations, as noted in the
RMP by red auxiliary numbers. To compute the 2/71 conversion
to an Egyptian fraction series, Ahmes used the following steps:

1. Subtract 1/40 from 2/71

2/71 - 1/40 = (80 - 71)/(40*71)

2. Inspect the numerator of the remainder 9/(40*71),
nine, as a sum of divisors of 40, those being
20, 10, 8, 5, 4, 2, 1: or select between 8 + 1
and 5 + 4.

3. Using the red auxiliary number rule, choosing the
alternative with the largest last term, 5 + 4 was
selected, meaning that

4. 2/71 = 1/40 + (5 + 4)/(40*71)

= 1/40 + 1/568 + 1/710

[ reply | up ]

Re: Unit Fractions by milogardner on 2006-03-28 10:37:18
Re: Unit Fractions by milogardner on 2006-03-28 10:56:46

Interact