p vs LiveMath

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.

A symbolic algebra and graphing platform for the rest of us - visual exploration and intuitive interaction without writing code. Looks like math. Feels like math. No programming headaches. Do algebra by picking up expressions and dragging or applying commands to them - you feel the tactile operations. Drag expressions onto graph windows to add them to the graph. Plot and compute: • Parametric and implicit curves and surfaces, including tubes • Differential equation solutions as numerical Flow plots • Inequalities plots for linear programming • Polygons and polyhedra with the Shapes plot • Embedded videos in notebooks • Full Unicode, including emoji, in math computations • Significantly improved integration and differentiation engines • Laplace Transform computations • Statistics and linear algebra built in, including SVD and Eigenfactor • Drag-and-drop matrix row operations for classroom presentation Free to read and interact with any premade notebook. LiveMath Pro unlocks adding new expressions and computations, saving and exporting. iPhone, iPad, Mac and Windows. Fifteen languages. LiveMath Pro is an auto-renewable yearly subscription. Payment is charged to your Apple Account at confirmation of purchase; the subscription renews automatically unless canceled at least 24 hours before the end of the current period. Manage or cancel in Settings. Terms of Use: https://www.apple.com/legal/internet-services/itunes/dev/stdeula/ Privacy Policy: https://www.livemath.com/privacy