p-adics
Six interactive playgrounds on the p-adic numbers, from intuition to applications in ML, dynamics, and number theory. The full essays go up as posts on the main blog (in Portuguese).
1 + 2 + 4 + 8 + ⋯ = -1. Not a notation trick —
the consistent rule of a different geometry on the rationals,
one in which "divergent" series settle down peacefully.
About this track
This page is the index of the p-adic playgrounds. The full essays — eight posts ranging from the mechanics of left-infinite arithmetic to applications in ultrametric clustering, dynamics and computational number theory — go up on the main blog (in Portuguese), starting with part 1. Here on the subdomain live the small browser-only demos that accompany each post.
Interactive playgrounds
Six small demos, one per central idea — from the mechanical arithmetic (carries running the "wrong" way) to the fractal geometry of the p-adic absolute value.
1 + 2 + 4 + 8 + ⋯ march to +∞ on the real line,
and the same partial sums spiral inward to -1 in the
2-adic neighbourhood. Base selector included.
coming soon
02
Build -1 with your hands
Add digits 9 (or p−1) one at a time and
watch |S + 1|_p shrink to zero on a live meter. Works
in any prime base.
coming soon
03
The p-adic ruler
Integers 0…100 on a ruler that re-arranges them by the
p-adic metric. Drag a pivot and see every point become a centre.
coming soon
04
Carry cascade
A p-adic calculator: type ...9999, add 1,
watch the carry roll leftward. Bonus in base 10: the zero-divisor
demo ...90625 × ...09376 = 0.
coming soon
05
Spin the digits
Each slot is a digit a_i. Spin them and see the rational
value, |x|_p, and the position on the p-adic tree update
live.
coming soon
06
The graph of |x|_p
The p-adic absolute value plotted for integers 1…N.
A sawtooth fractal — multiples of p dip down,
multiples of p² dip deeper, and so on.
Follow along
Posts go up on the main blog (in Portuguese) as the playgrounds mature. Prototypes open on github.com/lucasvitti. Corrections and suggestions via LinkedIn.