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 $F_5$.

Title	unit fraction
Canonical name	UnitFraction
Date of creation	2013-03-22 12:48:25
Last modified on	2013-03-22 12:48:25
Owner	XJamRastafire (349)
Last modified by	XJamRastafire (349)
Numerical id	10
Author	XJamRastafire (349)
Entry type	Definition
Classification	msc 11A67
Related topic	AdjacentFraction
Related topic	AnyRationalNumberIsASumOfUnitFractions
Related topic	AnyRationalNumberWithOddDenominatorIsASumOfUnitFractionsWithOddDenominators
Related topic	UnitFraction
Related topic	SierpinskiErdosEgyptianFractionConjecture
Defines	Egyptian fraction