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# A Journey Through p-adic Numbers

[GitHub](https://github.com/HectorTablero/QED-Scripts/tree/main/Magazine%204/A%20Journey%20Through%20p-adic%20Numbers) (Open Source, MIT License) | [Live Demo](https://qed.mat.uam.es/revista/articulo/numeros-p-adicos)

**Date:** July 2026
**Technologies:** JavaScript, Tailwind CSS

---

## Overview

The two elements turn the article's three abstract properties of $p$-adic balls, and its two convergence results, into pictures that a static page cannot provide: a $p$-ary tree whose subtrees literally are the balls, and a pair of log-scale term-size plots that visibly diverge in opposite directions in $\mathbb{R}$ and $\mathbb{Q}_p$.

## PadicBallTree

### Description

Renders the $p$-adic balls of a fixed depth two ways at once — as a $p$-ary tree of digit prefixes, and as an "onion" of nested discs — and lets the reader click any two leaves to read off $|x-y|_p$ from their lowest common ancestor.

### Implementation

A small `PadicTree` helper computes the longest common prefix of two digit strings and the induced distance $p^{-n}$, plus each node's horizontal span in $[0,1]$ for layout. Clicking a leaf sets `selA`; a second click sets `selB`, and the label of every ball containing `selA` is highlighted in both the tree and the onion diagram:

```javascript
::: js PadicBallTree
3, 4, [2, 3, 5]
padic1
autoPlay, allowFullscreen, openControls, h=180
:::
```

Because a node at depth $n$ _is_ the ball of radius $p^{-n}$ containing every point sharing its digit prefix, the article's three ball properties (every point is a center, balls are clopen, two balls are nested or disjoint) fall out as structural facts about subtrees rather than separate proofs to animate.

### Technical Considerations

The onion diagram recurses only to depth $\min(3, \text{depth})$ to keep the disc packing legible for $p=2,3,5$; the segmented control for $p$ also resets the tree depth (5 for $p=2$, 4 for $p=3$, 3 for $p=5$) so the rendered tree stays a comparable size across bases.

## PadicVsRealConvergence

### Description

Plots the term sizes of $\sum p^n$ and $\sum p^{-n}$ on a $\log_p$ scale in both $\mathbb{R}$ and $\mathbb{Q}_p$ simultaneously, making concrete the article's Result 2 that a series converges in $\mathbb{Q}_p$ iff its terms tend to zero — a much weaker condition than in $\mathbb{R}$.

### Implementation

Two stacked `Plot2D` panels each draw two polylines, real term size $|p^n|$ against $p$-adic term size $|p^n|_p = p^{-n}$:

```javascript
::: js PadicVsRealConvergence
2, 12
padic3
autoPlay, allowFullscreen, openControls, h=150
:::
```

For $\sum p^n$ the real term grows while the $p$-adic term shrinks, and for $\sum p^{-n}$ the roles reverse — so each panel shows one series converging in exactly one of the two worlds, with the readouts spelling out "diverges / converges" for both.

### Technical Considerations

A `terms:` slider (3–20) controls how many points are plotted; sizes are converted to $\log_p$ before plotting so both curves fit a shared window regardless of how fast $p^{\pm n}$ grows.