p-adic Numbers

Key idea

An alternative completion of the rationals.

In depth

For a prime p, the p-adic absolute value |x|ₚ measures divisibility by p: highly divisible numbers are small. Completing Q with it gives Qₚ. They provide a different notion of closeness and a place for local analysis in number theory.

Worked example

In Q₅, 25 is smaller than 5, and 1 + 5 + 25 + ... converges to 1/(1 − 5) = −1/4.

Real-world uses

• Solving number problems in cryptography and in advanced number theory research

Common mistake

Applying real-number intuition about size.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.