decimal fraction

A rational numberPlanetmathPlanetmathPlanetmath d is called a decimal fraction if 10kd is an integer for some non-negative integer k. For example, any integer, as well as rationals such as


are all decimal fractions. Rational numbers such as


are not.

There are two other ways of characterizing a decimal fraction: for a rational number d,

  1. 1.

    d is as in the above definition;

  2. 2.

    d can be written as a fraction pq, where p and q are integers, and q=2m5n for some non-negative integers m and n;

  3. 3.

    d has a terminating decimal expansion, meaning that it has a decimal representation


    where a is a nonnegative integer and di is any one of the digits 0,,9.

A decimal fraction is sometimes called a decimal number, although a decimal number in the most general sense may have non-terminating decimal expansions.

Remarks. Let D be the set of all decimal fractions.

  • If a,bD, then ab and a+bD as well. Also, -aD whenever aD. In other words, D is a subring of . Furthermore, as an abelian group, D is 2-divisible and 5-divisible. However, unlike , D is not divisible (

  • As inherited from , D has a total orderMathworldPlanetmath structureMathworldPlanetmath. It is easy to see that D is dense ( for any a,bD with a<b, there is cD such that a<c<b. Simply take c=a+b2.

  • From a topological point of view, D, as a subset of , is dense in . This is essentially the fact that every real number has a decimal expansion, so that every real number can be “approximated” by a decimal fraction to any degree of accuracy.

  • We can associate each decimal fraction d with the least non-negative integer k(d) such that 10k(d)d is an integer. This integer is uniquely determined by d. In fact, k(d) is the last decimal place where its corresponding digit is non-zero in its decimal representation. For example, k(1.41243)=5 and k(7/25)=2. It is not hard to see that if we write d=p2m5n, where p and 2m5n are coprimeMathworldPlanetmath, then k(d)=max(m,n).

  • For each non-negative integer i, let D(i) be the set of all dD such that k(d)=i. Then D can be partitioned into sets


    Note that D(0)=. Another basic property is that if aD(i) and bD(j) with i<j, then a+bD(j).

Title decimal fraction
Canonical name DecimalFraction
Date of creation 2013-03-22 17:27:15
Last modified on 2013-03-22 17:27:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 11-01
Related topic RationalNumber
Defines decimal number