Editing P-adic number (section)

=== Positional notation ===
It is possible to use a [[positional notation]] similar to that which is used to represent numbers in [[radix|base]] {{mvar|p}}.

Let <math display = inline>\sum_{i=k}^\infty a_i p^i</math> be a normalized {{mvar|p}}-adic series, i.e. each <math>a_i</math> is an integer in the interval <math>[0,p-1].</math> One can suppose that <math>k\le 0</math> by setting <math>a_i=0</math> for <math>0\le i <k</math> (if <math>k>0</math>), and adding the resulting zero terms to the series.

If <math>k\ge 0,</math> the positional notation consists of writing the <math>a_i</math> consecutively, ordered by decreasing values of {{mvar|i}}, often with {{mvar|p}} appearing on the right as an index:
: <math>\ldots a_n \ldots a_1{a_0}_p</math>
So, the computation of the [[#Example|example above]] shows that 
: <math>\frac 13= \ldots 1313132_5,</math>
and 
: <math>\frac {25}3= \ldots 131313200_5.</math>
When <math>k<0,</math> a separating dot is added before the digits with negative index, and, if the index {{mvar|p}} is present, it appears just after the separating dot. For example,
: <math>\frac 1{15}= \ldots 3131313._52,</math>
and
: <math>\frac 1{75}= \ldots 1313131._532.</math>

If a {{mvar|p}}-adic representation is finite on the left (that is, <math>a_i=0</math> for large values of {{mvar|i}}), then it has the value of a nonnegative rational number of the form <math>n p^v,</math> with <math>n,v</math> integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in [[radix|base]] {{mvar|p}}. For these rational numbers, the two representations are the same.