Editing Positional notation (section)

==== Rational numbers ====
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112&nbsp;... base-3 represents the sum of the infinite [[series (mathematics)|series]]:
:<math>\begin{array}{l}
1\times 3^{0\,\,\,} + {}\\
1\times 3^{-1\,\,} + 2\times 3^{-2\,\,\,} + {}\\
1\times 3^{-3\,\,} + 1\times 3^{-4\,\,\,} + 2\times 3^{-5\,\,\,} + {}\\
1\times 3^{-6\,\,} + 1\times 3^{-7\,\,\,} + 1\times 3^{-8\,\,\,} + 2\times 3^{-9\,\,\,} + {}\\
1\times 3^{-10} + 1\times 3^{-11} + 1\times 3^{-12} + 1\times 3^{-13} + 2\times 3^{-14} + \cdots
\end{array}</math>

Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a [[Vinculum (symbol)|vinculum]] across the repeating block:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Vinculum |url=https://mathworld.wolfram.com/Vinculum.html |access-date=2024-08-22 |website=mathworld.wolfram.com |language=en}}</ref>
: <math>2.42\overline{314}_5 = 2.42314314314314314\dots_5</math>

This is the [[Repeating decimal#Notation|repeating decimal notation]] (to which there does not exist a single universally accepted notation or phrasing).
For base 10 it is called a repeating decimal or recurring decimal.

An [[irrational number]] has an infinite non-repeating representation in all integer bases. Whether a [[rational number]] has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
: <math>0.1_3</math>
: <math>0.\overline3_{10} = 0.3333333\dots_{10}</math>
:: or, with the base implied:
:: <math>0.\overline3 = 0.3333333\dots</math> (see also [[0.999...]])
: <math>0.\overline{01}_2 = 0.010101\dots_2</math>
: <math>0.2_6</math>

For integers ''p'' and ''q'' with [[greatest common divisor|''gcd'']] (''p'', ''q'') = 1, the [[fraction (mathematics)|fraction]] ''p''/''q'' has a finite representation in base ''b'' if and only if each [[prime factor]] of ''q'' is also a prime factor of ''b''.

For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
# A finite or infinite number of zeroes can be appended:
#: <math>3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7</math>
# The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
#: <math>3.46_7 = 3.45\overline6_7</math>
#: <math>1_{10} = 0.\overline9_{10}\qquad</math> (see also [[0.999...]])
#: <math>220_5 = 214.\overline4_5</math>