mixed fraction

Any improper fraction ab can be uniquely written as a sum of an integer q and a proper fraction rb:

ab=q+rb (1)

The decomposition (1) is guaranteed by the division algorithm for integers.  The sum form is called mixed fraction or mixed number.

For explicitly given integers q, r, b, the mixed fraction is usually written without the plus sign, for example

92= 412.

Given an improper fraction of the positive integers a and b, their division gives the incomplete quotient q and the remainder r; then the mixed fraction can be gotten by (1).

On the other hand, if one wants to convert e.g. 81115 into an improper fraction, one needs first to convert the integer part 8 into fifteenths, obtaining


this fraction is added to the ready fifteenths 1115; the total amount of fifteenths is 13115; thus one has


Note.  A minus sign in front of a mixed fraction belongs not only for the integer part but also for the fractional part, i.e. for example  -214=-2-14.

The rational expressions, i.e. denoted quotients of polynomialsPlanetmathPlanetmath, have analogous to mixed fractions.  E.g., if the degree of the polynomial A(x) is at least equal to the degree of B(x), we have

A(x)B(x)=Q(x)+R(x)B(x). (2)

Here the degree of the remainder polynomial R(x) is less than the degree of the numerator polynomial B(x), an arithmetic fact that has been well documented for over 4,000 years. Fibonacci in the Liber Abaci used mixed fraction arithmetic in 1202 AD. Archimedes Calculus included mixed fractions to record the area of a parabola. Egyptian scribes in 1650 BCE also used this class of mixed fraction arithmetic. One scribe, Ahmes scaled a volume unit in multiplesMathworldPlanetmath of 64 so that a quotient recorded 1/64 units and a remainder recorded 1/320 units.

Title mixed fraction
Canonical name MixedFraction
Date of creation 2013-03-22 19:18:57
Last modified on 2013-03-22 19:18:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 11-01
Synonym mixed number
Related topic RationalNumber
Related topic LongDivision
Related topic EuclideanNumberField
Related topic PartialFractionsOfExpressions