p-adic number

Noun

 * 1)  An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.
 * In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set $$\textstyle\{x | \exists n \in \mathbb{Z} . \, x = 3 n + 1 \}.$$ This closed ball partitions into exactly three smaller closed balls of radius 1/9: $$\{x | \exists n \in \mathbb{Z} . \, x = 1 + 9 n \},$$ $$\{x | \exists n \in \mathbb{Z} . \, x = 4 + 9 n \},$$ and $$\{x | \exists n \in \mathbb{Z} . \, x = 7 + 9 n \}.$$ Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner. Likewise, going upwards in the hierarchy, B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, $$\{x| \exists n \in \mathbb{Z} . \, x = 1 + {n\over 3} \},$$ which is one out of three closed balls forming a closed ball of radius 9, and so on.
 * In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set $$\textstyle\{x | \exists n \in \mathbb{Z} . \, x = 3 n + 1 \}.$$ This closed ball partitions into exactly three smaller closed balls of radius 1/9: $$\{x | \exists n \in \mathbb{Z} . \, x = 1 + 9 n \},$$ $$\{x | \exists n \in \mathbb{Z} . \, x = 4 + 9 n \},$$ and $$\{x | \exists n \in \mathbb{Z} . \, x = 7 + 9 n \}.$$ Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner. Likewise, going upwards in the hierarchy, B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, $$\{x| \exists n \in \mathbb{Z} . \, x = 1 + {n\over 3} \},$$ which is one out of three closed balls forming a closed ball of radius 9, and so on.

Usage notes

 * An expanded, constructive definition:
 * For given $$p$$, the natural numbers are exactly those expressible as some finite sum $$\textstyle\sum_{k=0}^n a_k p^k$$, where each $$a_k$$ is an integer: $$0\le a_k<p$$ and $$n\ge 0$$. (To this extent, $$p$$ acts exactly like a base).
 * The slightly more general sum $$\textstyle\sum_{k=N}^n a_k p^k$$ (where $$N$$ can be negative) expresses a class of fractions: natural numbers divided by a power of $$p$$.
 * Much more expressiveness (to encompass all of $$\Q$$) results from permitting infinite sums: $$\textstyle\sum_{k=N}^\infty a_k p^k$$.
 * The p-adic ultrametric and the limitation on coefficients together ensure convergence, meaning that infinite sums can be manipulated to produce valid results that at times seem paradoxical. (For example, a sum with positive coefficients can represent a negative rational number. In fact, the concept has limited meaning for p-adic numbers; it is best simply interpreted as .)
 * Forming the completion of $$\Q$$ with respect to the ultrametric means augmenting it with the limit points of all such infinite sums.
 * The augmented set is denoted $$\Q_p$$.
 * The construction works generally (for any integer $$p>1$$), but it is only for prime $$p$$ that it becomes of significant mathematical interest.
 * For $$p$$ the power of some prime number, $$\Q_p$$ is still a field. For other composite $$p$$, $$\Q_p$$ is a ring, but not a field.
 * $$\Q_p$$ is not the same as $$\R$$.
 * For example, $$\sqrt{p}\notin\Q_p$$ for any $$p$$, and, for some values of $$p$$, $$\sqrt{-1}\in\Q_p$$.

Hyponyms

 * rational number
 * integer
 * integer

Related terms

 * p-adic
 * p-adic absolute value, p-adic norm
 * p-adic integer
 * p-adic ordinal
 * p-adic ultrametric

Translations

 * Chinese:
 * Mandarin: p進數
 * Finnish: p-adinen luku
 * French: nombre p-adique
 * German: p-adische Zahl
 * Italian: numero p-adico
 * Polish:, liczba p-adyczna
 * Romanian: număr p-adic