# Definition:P-adic Number/Representative

< Definition:P-adic Number(Redirected from Definition:Representative of P-adic Number)

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## Definition

Let $p$ be any prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\sequence{x_n} + \NN$ be any $p$-adic number of $\Q_p$.

Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence{x_n} + \NN$ is called a **representative** of the $p$-adic number $\sequence{x_n} + \NN$.

## Also see

- Representative of P-adic Number is Representative of Equivalence Class where it is proved that the definition of a
**representative**of a $p$-adic number coincides with the definition of a representative of an equivalence class.

## Sources

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*: $\S 1.4$ The field of $p$-adic numbers $\Q_p$