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decimal fraction∗
CWoo†
2013-03-21 23:37:44
A rational number d is called a decimal fraction if 10k d is an integer for some
non-negative integer k. For example, any integer, as well as rationals such as
3
,
4
0.23123,
236
125
are all decimal fractions. Rational numbers such as
1
,
3
−
227
,
12
2.312
are not.
There are two other ways of characterizing a decimal fraction: for a rational
number d,
1. d is as in the above definition;
p
, where p and q are integers, and q = 2m 5n
q
for some non-negative integers m and n;
2. d can be written as a fraction
3. d has a terminating decimal expansion, meaning that it has a decimal
representation
a.d1 d2 · · · dn 000 · · ·
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 ⊂ Q be the set of all decimal fractions.
• If a, b ∈ D, then a · b and a + b ∈ D as well. Also, −a ∈ D whenever a ∈ D.
In other words, D is a subring of Q. Furthermore, as an abelian group, D
is 2-divisible and 5-divisible. However, unlike Q, D is not divisible.
∗ hDecimalFractioni created: h2013-03-21i by: hCWooi version: h39836i Privacy setting:
h1i hDefinitioni h11-01i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
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1
• As inherited from Q, D has a total order structure. It is easy to see that
D is dense: for any a, b ∈ D with a < b, there is c ∈ D such that a < c < b.
a+b
Simply take c =
.
2
• From a topological point of view, D, as a subset of R, is dense in 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 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
p
d = m n , where p and 2m 5n are coprime, then k(d) = max(m, n).
2 5
• For each non-negative integer i, let D(i) be the set of all d ∈ D such that
k(d) = i. Then D can be partitioned into sets
D = D(0) ∪ D(1) ∪ · · · ∪ D(n) ∪ · · · .
Note that D(0) = Z. Another basic property is that if a ∈ D(i) and
b ∈ D(j) with i < j, then a + b ∈ D(j).
2