PlanetMath: decimal fraction

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decimal fraction

(Definition)

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

$\displaystyle 0.23123,\qquad \frac{3}{4},\qquad \frac{236}{125}$

are all decimal fractions. Rational numbers such as

$\displaystyle \frac{1}{3},\qquad -\frac{227}{12}, \qquad 2.\overline{312}$

are not.

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

is as in the above definition;
can be written as a fraction $\displaystyle{\frac{p}{q}}$ , where and are integers, and for some non-negative integers and ;
has a terminating decimal expansion, meaning that it has a decimal representation
$\displaystyle a.d_1d_2\cdots d_n000\cdots$
where is an integer and is any one of the digits $0,\ldots,9$ .

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 $D\subset \mathbb{Q}$ be the set of all decimal fractions.

If $a,b \in D$ , then $a\cdot b$ and $a+b\in D$ as well. Also, $-a \in D$ whenever $a\in D$ . In other words, is a subring of $\mathbb{Q}$ . Furthermore, as an abelian group, is -divisible and -divisible. However, unlike $\mathbb{Q}$ , is not divisible.
As inherited from $\mathbb{Q}$ , has a total order structure. It is easy to see that is dense: for any $a,b\in D$ with , there is $c\in D$ such that . Simply take $c=\displaystyle{\frac{a+b}{2}}$ .
From a topological point of view, , as a subset of $\mathbb{R}$ , is dense in $\mathbb{R}$ . 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 with the least non-negative integer such that $10^{k(d)}d$ is an integer. This integer is uniquely determined by . In fact, is the last decimal place where its corresponding digit is non-zero in its decimal representation. For example, and . It is not hard to see that if we write $d=\displaystyle{\frac{p}{2^m5^n}}$ , where and are coprime, then $k(d)=\max(m,n)$ .
For each non-negative integer , let be the set of all $d\in D$ such that . Then can be partitioned into sets
$\displaystyle D=D(0)\cup D(1) \cup \cdots \cup D(n) \cup \cdots.$
Note that $D(0)=\mathbb{Z}$ . Another basic property is that if $a\in D(i)$ and $b\in D(j)$ with , then $a+b\in D(j)$ .