My current understanding of p-adic numbers

In order to understand the Tao paper from my previous post, I am now learning about p-adic numbers for the first time. As usual, I used AI heavily to learn the material and edit my writing. Here we go:

I no longer think of $p$-adics as an alternative number system competing with the real numbers. I think of them as a different answer to a very basic question: what does it mean for two numbers to be close? On the familiar real number line, closeness is purely geometric. Two numbers are close if their difference is small in absolute value, $|x-y|$. This notion is so ingrained that it's hard to imagine any other possibility. But arithmetic, fundamentally, doesn't really care about absolute size. It cares about divisibility, factors, and shared structure. $p$-Adic numbers take that arithmetic viewpoint seriously and build an entirely new geometry around it.

The core idea, as I currently understand it, is this: we must first fix a prime $p$. Two numbers are then considered close if their difference is divisible by a high power of $p$. The more powers of $p$ they share as a factor, the closer they are. This closeness is quantified by the $p$-adic norm, $|x{|}_p$, which decreases as the divisibility by $p$ increases. So, in the 3-adic world ($p=3$), numbers like $1,4,7,$ and $10$ are all close to each other, because their differences are multiples of 3. What matters is not where they sit on a linear axis, but how much arithmetic information they have in common concerning the prime $p$.

One way this finally clicked for me was realizing that $p$-adic distance is ultrametric. That word used to intimidate me, but the intuition is simple. The ultrametric property dictates that the distance between any two points in a $p$-adic space is less than or equal to the maximum distance from either point to a third point: $\text{dist}(x,z)\le max\left(\text{dist}\right(x,y),\text{dist}(y,z\left)\right)$. This forces triangles to behave strangely: every triangle is isosceles, and often equilateral. If two numbers are both close to a third number, then they are automatically close to each other. This gives the space a tree-like, rather than a linear structure. Instead of intervals, you get perfectly nested balls inside balls, where each ball corresponds to divisibility by a higher power of $p$.

This ultrametric geometry also explains why $p$-adic expansions run "backwards" compared to decimal expansions. In base 10, writing more digits to the right refines small-scale behavior, getting closer to zero. In base $p$, $p$-adic expansions refine behavior at larger and larger powers of $p$. Adding more digits doesn't zoom in toward zero; it zooms deeper into the divisibility structure. Once I stopped trying to visualize $p$-adics as a distorted real line and instead imagined them as an infinite rooted tree where closeness is defined by shared branches, things became much less mysterious.

What I find most compelling is that $p$-adic numbers don't smooth problems in the way real analysis often does. They don't average things out or blur irregularities. Instead, they expose arithmetic structure. Operations like multiplication by $p$ or reduction modulo $p^k$ become geometrically natural. Neighborhoods are defined by congruences. Iteration respects this structure in a way it simply doesn't on the real line. This is why, in hindsight, it makes perfect sense that $p$-adics show up in problems built from simple arithmetic rules, like the Collatz map. If a process is fundamentally driven by divisibility, parity, and prime factors, then a geometry that treats those features as first-class citizens is not exotic at all—it's aligned.

I'm still very much at the stage where my understanding is conceptual rather than technical. I can't yet move fluently between definitions, norms, and the algebraic process of completion. But I no longer feel that $p$-adic numbers are arbitrary or decorative. They feel like a natural response to a mismatch between purely arithmetic behavior and the constraints of real-analytic intuition. Right now, that's enough. Being able to articulate why one might want to leave the real line, and what kind of arithmetic structure $p$-adics make visible, already feels like a meaningful step forward. The technical machinery can come later. For the moment, I'm content to sit with the idea that changing the notion of distance can completely change what a problem is willing to reveal.