are all decimal fractions. Rational numbers such as
There are two other ways of characterizing a decimal fraction: for a rational number ,
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 be the set of all decimal fractions.
From a topological point of view, , 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 with the least non-negative integer such that 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 , where and are coprime, then .
For each non-negative integer , let be the set of all such that . Then can be partitioned into sets
Note that . Another basic property is that if and with , then .
|Date of creation||2013-03-22 17:27:15|
|Last modified on||2013-03-22 17:27:15|
|Last modified by||CWoo (3771)|