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unit fraction

An unit fraction $\frac{n}{d}$ is a fraction whose numerator $n = 1$ . If its integer denominator $d > 1$ , 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:

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

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 $n$ . 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 Jani Melik.

How to Cite This Entry

Jani Melik. "unit fraction" (version 7). PlanetMath.org. Freely available at http://planetmath.org/UnitFraction.html.

Classification

AMS MSC:

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

Pending Errata and Addenda

None. [ View all 1 ]

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Discussion

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

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