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[parent] decimal expansion (Result)

Every rational number $ \frac{m}{n}$, where $ m$ and $ n$ are positive integers, has an endless periodic decimal expansion (or decadic expansion -- according to Greek). The decimal expansion of $ \frac{m}{n}$ means the series presentation

$\displaystyle \nu.\nu_1\nu_2\nu_3\ldots = \nu+10^{-1}\nu_1+10^{-2}\nu_2+10^{-3}\nu_3+\ldots$ (1)

where $ \nu = \lfloor\frac{m}{n}\rfloor$ is the integer part of $ \frac{m}{n}$ and the integers $ \nu_j$ are the remainders of $ \lfloor 10^j\cdot\!\frac{m}{n}\rfloor$ when divided by 10; thus $ 0 \leqq \nu_j < 10$.

We may suppose that $ m$ and $ n$ are coprime (if necessary, reduce the fraction). Then the length $ l$ of the period depends only on the denominator $ n$. In the case that $ \gcd(n,\,10) = 1$, the period length is the least positive integer $ l$ such that $ 10^l\equiv 1 \pmod{n}$ (the period length does not change if we multiply the fraction by a suitable power of 10 and then reduce all prime factors of 10 from the denominator). In every case, the period length is a factor of the number $ \varphi(n)$, where $ \varphi$ is Euler's totient function.

Examples

$ \frac{1}{8} = 0.125000\ldots = 0.124999\ldots$ (one-digit periods; N.B. two possibilities),

$ \frac{1}{12} = 0.08333\ldots$ (one-digit per.),

$ \frac{1}{37} = 0.'027'027'027'\ldots$ (three-digit per.),

$ \frac{1}{82} = 0.0'12195'12195'12195'\ldots$ (five-digit per.),

$ \frac{1}{25351} = 0.000039446\ldots$ (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 expansion for $ \frac{1}{8}$ in the decimal system).

Completely similar results concern the digital expansions in any other positional digit system. Let the fraction $ \frac{1}{31}$ be an example ( $ \varphi(31) = 30$); its presentation is

in the decadic (decimal) digit system $ \frac{1}{31} = 0.'032258064516129'\ldots_{\mathrm{ten}}$     (15-digit per.),

in the hexadic (senary) digit system $ \frac{1}{51} = 0.'010545'010545'010545'\ldots_{\mathrm{six}}$     (6-digit per.),

in the dyadic (binary) digit system $ \frac{1}{11111} = 0.000010000100001\ldots_{\mathrm{two}}$     (5-digit per.).

Note. Also any irrational number has a unique decimal expansion, but it is non-periodic; for example Liouville's number

$\displaystyle 0.110001\,000000\,000000\,000001\,000000\,\ldots$
which is transcendental over $ \mathbb{Q}$.



"decimal expansion" is owned by pahio.
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See Also: digital number system, existence and uniqueness of decimal expansion, $0\ne 1$ as real numbers, factoradic, factorial base, MatheRealism, Joseph Liouville

Other names:  decadic expansion, digital expansion
Also defines:  decadic, hexadic, dyadic

This object's parent.

Attachments:
converting a repeating decimal to a fraction (Algorithm) by Wkbj79
decimal place (Definition) by CWoo
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Cross-references: transcendental, irrational number, similar, rational, digit, base, Euler's totient function, factor, prime factors, denominator, fraction, necessary, coprime, remainders, series, integers, positive, rational number
There are 25 references to this entry.

This is version 24 of decimal expansion, born on 2005-02-19, modified 2008-02-23.
Object id is 6787, canonical name is DecimalExpansion.
Accessed 8450 times total.

Classification:
AMS MSC: 11A99 (Number theory :: Elementary number theory :: Miscellaneous)
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