p-adic number

A p-adic number is an element of the completion of the rational numbers under the p-adic absolute value for a prime p. Its size is determined by divisibility by p, making numbers more alike when their difference is divisible by a high power of p.

p-adic absolute value: An absolute value on the rationals defined by the exponent of a prime in a number’s factorization. It supplies the notion of distance whose completion produces the p-adic numbers.
p-adic expansion: A representation of a p-adic number as a convergent series in powers of p with digits from 0 to p−1. It expresses each number as an infinite expansion whose low-order digits determine p-adic closeness.
p-adic analysis: The study of functions, limits, and calculus over p-adic fields. It develops calculus using the topology and convergence inherited from p-adic distance.
Real number: An element of the complete ordered field obtained by completing the rational numbers in the usual absolute value. The real numbers use ordinary magnitude, unlike p-adic size based on divisibility.
p-adic valuation: The exponent of a prime p in a nonzero rational number, extended by subtraction between numerator and denominator exponents. It determines p-adic absolute value and measures divisibility by p.
p-adic metric: The metric induced by the p-adic absolute value, in which numbers are close when their difference is highly divisible by p. Its unusual geometry explains why successive base-p digits converge from low powers upward.
p-adic numbers in number theory: The use of p-adic fields and integers to study divisibility, equations, and arithmetic properties. Local p-adic information helps expose arithmetic patterns invisible over the real numbers.
Complex number: A number of the form a+bi with real a and b and i² = −1. The complex field extends the real completion, while p-adic fields arise from prime-based absolute values.
Completion (mathematics): A construction that adds limits of Cauchy sequences to a metric space that may be incomplete. The p-adic numbers arise by completing the rationals in a nonusual metric.
Congruence (number theory): A relation stating that two integers have the same remainder modulo a specified integer. Agreement modulo p to high powers is the arithmetic form of p-adic closeness.

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