Editing P-adic number (section)

== Notation ==
There are several different conventions for writing {{mvar|p}}-adic expansions. So far this article has used a notation for {{mvar|p}}-adic expansions in which [[exponentiation|powers]] of {{mvar|p}} increase from right to left. With this right-to-left notation the 3-adic expansion of <math>\tfrac15,</math> for example, is written as
<math display="block">\frac15 = \dots 121012102_3.</math>

When performing arithmetic in this notation, digits are [[carry (arithmetic)|carried]] to the left. It is also possible to write {{mvar|p}}-adic expansions so that the powers of {{mvar|p}} increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of <math>\tfrac15</math> is
<math display="block">
\frac15 = 2.01210121\dots_3 \mbox{ or }
\frac1{15} = 20.1210121\dots_3.
</math>

{{mvar|p}}-adic expansions may be written with [[Signed-digit representation|other sets of digits]] instead of {{math|{0,&thinsp;1,&thinsp;...,}}&thinsp;{{math|''p'' − 1}}}. For example, the {{math|3}}-adic expansion of <math>\tfrac15</math> can be written using [[balanced ternary]] digits {{math|{<u>1</u>,&thinsp;0,&thinsp;1}}}, with {{math|<u>1</u>}} representing negative one, as
<math display="block">\frac15 = \dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_{\text{3}} .</math>

In fact any set of {{mvar|p}} integers which are in distinct [[residue class]]es modulo {{mvar|p}} may be used as {{mvar|p}}-adic digits. In number theory, [[Witt vector#Motivation|Teichmüller representatives]] are sometimes used as digits.<ref>{{Harv|Hazewinkel|2009|p=342}}</ref>

'''{{vanchor|Quote notation}}''' is a variant of the {{mvar|p}}-adic representation of [[rational number]]s that was proposed in 1979 by [[Eric Hehner]] and [[Nigel Horspool]] for implementing on computers the (exact) arithmetic with these numbers.<ref>{{Harv|Hehner|Horspool|1979|pp=124–134}}</ref>