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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 compatible with the CC-BY-SA license. 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