p-adelic

p-adics for everyone

Free · Designed for iPad. Not verified for macOS.

Every completion of a number at once: the real line and every ℚₚ side by side, streaming digits, Newton polygons, Hilbert symbols and more — all on your device. p-adelic shows a number in all of its completions at once. Type 1/3, √2, a polynomial or log 7 and watch the adelic strip: one column for the real line and one for each prime p, each streaming the number's digits in base p and growing leftward as you turn the precision dial. There is no "p-adic mode" to switch into — you always see the whole picture, the way number theorists think about it. THE STRIP • A column per place: ∞ plus the primes that matter for your input, and any prime you pin. • Streaming precision from 5 to 200 digits; every column updates together. • Exact periodicity for rationals: the repeating block is marked with an overline, and long runs of digits fold into pills you can tap open. • Tap a column to focus it; the focused card shows valuations, unit parts, Legendre symbols and the Laurent form. DEEP DIVE • Digits, Laurent series and Witt-vector views of the same number. • Nested balls: zoom into ℤₚ ball by ball, following your number's digits. • The Bruhat–Tits tree and the Berkovich line of the chosen place. ALGEBRAIC NUMBERS • √d, roots of unity ζₙ, Teichmüller representatives, and any algebraic number given by its minimal polynomial, e.g. algebraic(x³ − 2). • See whether a prime splits, stays inert or ramifies; extend the field with one tap and cycle through the Galois conjugates. POLYNOMIALS • Newton polygons at every prime and at ∞, with slopes, multiplicities and the Krasner check. SPECIAL FUNCTIONS • p-adic exp, log and Artin–Hasse, with their convergence disks. • Factorials with Legendre's formula, Morita's Γₚ with the Wilson check. • Kubota–Leopoldt ζₚ and L-values, Hilbert symbols with the reciprocity check, and Tate's local factors. LEARN AND REUSE • A Library of worked examples for every idea above, from −1 = …(p−1)(p−1)(p−1) to the Berkovich line of ∛2. • History you can reopen with one tap. • VoiceOver describes every visualization; text and digits follow Dynamic Type. PRIVATE BY DESIGN Everything is computed on your device. The app has no account, no analytics and makes no network connections. For students meeting p-adic numbers for the first time, teachers who want a live picture to point at, and researchers who need a Newton polygon or a Hilbert symbol checked in seconds.

  • 5.0
    out of 5
    2 Ratings

A new look and a much more careful engine. • New app icon — the nested balls of ℤ₅, now in layered glass. • Newton polygons and factorization over ℚₚ are fixed; polynomials with mixed slopes now work. • Repeating blocks of every rational are now exactly right, and long runs of digits fold into tap-to-open pills. • Square roots at p = 2, non-squarefree radicands, cyclotomic digits and Galois conjugates are corrected. • exp, log, Artin–Hasse, factorials and Γₚ are exact to the chosen precision instead of stopping early. • No more crashes or freezes on extreme inputs (huge primes, exp(30), enormous exponents, deeply nested parentheses) — they are refused with a clear message. • The ∞ column shows the sign and integer part (√2 = 1.41421…). • Columns say why they are unavailable (needs extension, diverges here, not real) and offer the right action. • Friendlier errors, dimmed stale results, History entries you can reopen, Library search. • Proper math typesetting (ℚ₅, exponents), Dynamic Type, VoiceOver descriptions for every visualization, larger touch targets. • The keyboard no longer pushes the screen around, including on iPhone SE.

The developer, PracticallyZen, LLC, indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

  • Data Not Collected

    The developer does not collect any data from this app.

    Privacy practices may vary, for example, based on the features you use or your age. Learn More

    The developer has not yet indicated which accessibility features this app supports. Learn More

    Seller
    Practically Zen, LLC
    Size
    12.4 MB
    Category
    Education
    Compatibility
    Requires iOS 18.6 or later.
    • iPhone
      Requires iOS 18.6 or later.
    • iPad
      Requires iPadOS 18.6 or later.
    • Mac
      Requires macOS 15.6 or later and a Mac with Apple M1 chip or later.
    • Apple Vision
      Requires visionOS 2.6 or later.
    Languages
    English
    Age Rating
    4+
    Copyright
    © 2026 PracticallyZen, LLC