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单词 DecimalExpansion
释义

decimal expansion


Every rational numberPlanetmathPlanetmathPlanetmath mn, where m and n are positive integers, has an endless decimal expansion (or decadic expansion — according to Greek).  The decimal expansion of mn means the series

ν.ν1ν2ν3=ν+10-1ν1+10-2ν2+10-3ν3+(1)

where  ν=mn  is the integer part (http://planetmath.org/Floor) of mn and the integers νj are the remainders of  10jmn  when divided by 10; thus  0νj<10.

We may suppose that m and n are coprimeMathworldPlanetmathPlanetmath (if necessary, reduce the fraction).  Then the depends only on the denominator n.  In the case that  gcd(n, 10)=1,  the is the least positive integer l such that 10l1(modn) (the does not change if we multiply the fraction by a suitable power of 10 and then reduce all prime factorsMathworldPlanetmathPlanetmath of 10 from the denominator).  In every case, the is a factor of the number φ(n), where φ is Euler’s totient function.

Examples

18=0.125000=0.124999 (one-digit ; N.B. two possibilities),

112=0.08333 (one-digit per.),

137=0.027027027 (three-digit per.),

182=0.0121951219512195 (five-digit per.),

125351=0.000039446 (hundred-digit per.)

The tail of infinitely many 0’s (as in 0.125000…) is of course usually not written out.  Such a tail is possible only when n has no other prime factors except prime factors of the base of the digit system in question.

If the tails of 0’s are not accepted, then the digital expansion of every positive rational is unique (then e.g. 0.124999…  is the only for 18 in the decimal system).

Completely similar results concern the digital expansions in any other positional digit system.  Let the fraction 131 be an example (φ(31)=30); its is

in the decadic (decimal) digit system  131=0.032258064516129ten  (15-digit per.),

in the hexadic (senary) digit system  151=0.010545010545010545six  (6-digit per.),

in the dyadic () digit system  111111=0.000010000100001two  (5-digit per.).

Note.  Also any irrational number has a unique decimal expansion, but it is non-periodic; for example Liouville’s number (http://planetmath.org/ExampleOfTranscendentalNumber)

0.110001 000000 000000 000001 000000

which is transcendental over .

Titledecimal expansion
Canonical nameDecimalExpansion
Date of creation2013-03-22 15:04:01
Last modified on2013-03-22 15:04:01
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id28
Authorpahio (2872)
Entry typeResult
Classificationmsc 11A99
Synonymdecadic expansion
Synonymdigital expansion
Related topicBase3
Related topicExistenceAndUniquenessOfDecimalExpansion
Related topic0ne1AsRealNumbers
Related topicFactoradic
Related topicFactorialBase
Related topicMatheRealism
Related topicJosephLiouville
Definesdecadic
Defineshexadic
Definesdyadic
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更新时间:2025/5/4 10:14:32