# Chapter ᚘ(Y): Pre-flight controls
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T25, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch25-cards.md) · [Its references](ch25-refs.md) · [Previous chapter](ch24.md) · [Next chapter](ch26.md)
*A footnote such as [r-25.3] is reference entry 25.3 and links to it. Section 25 of 28; chapters are named by their Ogham letter, never by number.*
## Out of this world
ᚘ – [Ifín](ch25-cards.md#g-og-ifin): associated with the gooseberry, Section 25
## Chapter ᚘ(Y): Pre-flight controls
> “On the fundamental level our world is neither real, nor p-adic; it is adèlic.”
> — Yuri Manin · Reflections on arithmetical physics, in *Conformal Invariance and String Theory*, Academic Press, 1989 [r-25.17](ch25-refs.md#r-25.17)
Formula: ([…2223 + 1](ch25-cards.md#g-adicid) + [|27|3 · |27|∞ − 1](ch25-cards.md#g-prodformula))/(([…890625 + …109376](ch25-cards.md#g-tenadic)) {brace: [10-adic, composite base](ch25-cards.md#g-ostrowski)} + ([|27|3−1](ch25-cards.md#g-adicabs) + [v3(27)](ch25-cards.md#g-val3_27)) {brace: [3-adic](ch25-cards.md#g-Qp)} + ([α², α = √2 in ℤ₇](ch25-cards.md#g-sqrt2_7)) {brace: [7-adic](ch25-cards.md#g-Qp)} )
Key concepts: p-adic numbers and non-Archimedean absolute values. It includes symbols you may not have seen before, such as: [|·|3](ch25-cards.md#g-adicabs), [v3](ch25-cards.md#g-val3_27), and [ℚp](ch25-cards.md#g-Qp). Zero here is a carry that never ends: …2222 in base 3, plus 1, rolls over into 0, in a world where closeness means divisibility.
Stow your bag under the seat and look at the panel in front of us. Most of these gauges measure the way you would expect, but this one has its own idea of distance. It compares whole numbers by how many times 3 divides them: the more factors of 3 a number contains, the closer it sits to zero. So [|27|₃](ch25-cards.md#g-adicabs) = 1/27, while |3|₃ = 1/3. Wacky controls, but perfectly consistent ones, and learning them is part of the checklist. The doubling ladder brought us aboard; the p-adic numbers take our exploration in a different direction. Let me start with an addition whose carry never runs out of columns.
Field notes
Kurt Hensel wanted number theory to have its own version of the power series, which builds a function out of higher and higher powers of a small variable, each term mattering less than the one before. [r-25.1](ch25-refs.md#r-25.1) For whole numbers he used higher and higher powers of a chosen prime p instead, with a new way of measuring distance in which those powers really do become small. For the prime 3, [divisibility by a large power of 3 means closeness to zero](ch25-cards.md#g-absp). The p-adics are not the next rung after [the sedenions](ch24-cards.md#g-sedenions), and their [infinite digit strings](ch25-cards.md#g-padicexp) are not infinitely large [real numbers](ch01-cards.md#g-reals).
#### Numerator
[…2223 + 1](ch25-cards.md#g-adicid) = 0; a carry that never ends, arguably the cleanest zero in the piece.
In base three, […222](ch25-cards.md#g-threeadic) means 2 + 2·3 + 2·3² + ⋯. Add 1, and the rightmost column reads 2 + 1, which is three, written 10 in base three: write zero, carry one. The next column is 2 + 1 again, and the next, and [the carry continues through every position](ch25-cards.md#g-adicid), leaving nothing but zeros. [r-25.3](ch25-refs.md#r-25.3) To make that precise, stop after n twos: that number is 3ⁿ − 1, and adding one gives 3ⁿ. Since |3ⁿ|₃ = 3⁻ⁿ shrinks toward zero, these cut-off strings close in on −1 in the 3-adics, while in ordinary distance they grow without bound. So …222 is −1, and adding 1 gives 0. Convergence belongs to the chosen distance. This recalls [0.999…](ch04-cards.md#g-nines) = 1 from Chapter ᚂ(L): a string is understood through its cut-off pieces and where they are heading; only the direction of the digits and the meaning of smallness have changed.
[|27|3 · |27|∞ − 1](ch25-cards.md#g-prodformula) = 0; the [product formula](ch25-cards.md#g-prodformula), a fundamental result, turned into a zero.
Since 27 = 3³, [its 3-adic absolute value is 3⁻³ = 1/27](ch25-cards.md#g-adicabs), and its ordinary absolute value is 27. Their product is 1. The same is true of every nonzero whole number or fraction: multiply its ordinary size by its p-adic size for every prime p, and the answer is always exactly 1. A prime that divides neither the top nor the bottom gives a size of 1 and changes nothing, so for 27, a power of 3, only the prime 3 counts, and the [product formula](ch25-cards.md#g-prodformula) fits in two factors. [r-25.4](ch25-refs.md#r-25.4) [r-25.5](ch25-refs.md#r-25.5)
#### Denominator
Three related settings: base 10 gives two numbers that each square to themselves, base 3 measures divisibility, and base 7 turns an approximation into an exact square root. Each expression is evaluated in its own setting before its integer joins the sum.
10-adic, composite base
[…890625 + …109376](ch25-cards.md#g-tenadic) = 1; the 10-adic automorphic pair, two numbers that each square to themselves.
Call the numbers a and b. Each squares to itself, making it an idempotent: 625² = 390625 ends in 625, and the matching pattern continues through every longer ending. Their infinite digit strings add to exactly 1, with carries continuing indefinitely. The [Chinese remainder theorem](ch06-cards.md#g-crt) explains the pair: base 10 separates into a 2-adic part and a 5-adic part, and a and b select (1, 0) and (0, 1). Their product is 0 though neither is zero. Zero divisors have returned in arithmetic where order and brackets still do not matter. [r-25.6](ch25-refs.md#r-25.6) [r-25.7](ch25-refs.md#r-25.7)
3-adic
[|27|3−1](ch25-cards.md#g-adicabs) = 27; the reciprocal of 27’s 3-adic size, where divisible numbers are the small ones.
[|27|₃](ch25-cards.md#g-adicabs) = 1/27, so its reciprocal is 27.
[v3(27)](ch25-cards.md#g-val3_27) = 3; the same divisibility, measured as an exponent.
The [valuation](ch25-cards.md#g-vp) is the exponent of 3 in the factorization 27 = 3³. The two measurements carry the same information: [|x|₃](ch25-cards.md#g-absp) = 3−v₃(x) for nonzero x. [r-25.2](ch25-refs.md#r-25.2)
7-adic
[α², α = √2 in ℤ₇](ch25-cards.md#g-sqrt2_7) = 2; a square root of 2 that exists 7-adically though no rational one does.
α is the root of x² = 2 whose last 7-adic digit is 3, since 3² = 9 leaves 2 on division by 7. [Hensel’s lemma](ch25-cards.md#g-hensel) turns this near miss into an exact root, one 7-adic digit at a time: 10² = 100 leaves 2 on division by 49, 108² = 11664 leaves 2 on division by 343, and so on without end, each approximation extending the last (10 = 3 + 1·7, 108 = 10 + 2·49); together they pin down an exact root in [ℤ₇](ch25-cards.md#g-Zp). The lemma asks only that 2·3, the slope of x² at 3, not be a multiple of 7. Squaring the root gives the ordinary integer 2. [r-25.8](ch25-refs.md#r-25.8) [r-25.9](ch25-refs.md#r-25.9)
The only fraction is (0 + 0)/33: 1 + 27 + 3 + 2 = 33, a nonzero integer. [r-25.16](ch25-refs.md#r-25.16)
**Fine print.** The ∞ in |27|∞ labels the ordinary absolute value; it is not an infinite magnitude. Six terminal digits alone would not define the [10-adic pair](ch25-cards.md#g-tenadic); the idempotent condition fixes their extensions. For a prime p, the p-adic integers ℤp have no [zero divisors](ch10-cards.md#g-zerodiv), so no pair like the 10-adic one can live there; still, you cannot always divide in them (1/3 is not a 3-adic integer), and allowing division gives [ℚp](ch25-cards.md#g-Qp). [r-25.12](ch25-refs.md#r-25.12) To a topologist the 3-adic integers and the [Cantor set](ch20-cards.md#g-cantor) from the cutting are the same shape, or [homeomorphic](ch12-cards.md#g-homeo), and the [p-adic integers](ch25-cards.md#g-Zp) for every other prime are that shape too: a dust of points, no two joined, though the Cantor set spells its points with the digits 0 and 2 and the 3-adics use 0, 1, and 2. [r-25.15](ch25-refs.md#r-25.15)
What this chapter adds to the formula: (0 + 0) / 33 = 0
Checklist complete. Hands clear of the controls: three, two, one… and we are off! The tower drops away, and its four landings shrink to a single line beneath us. We have changed what it means for numbers to be close; next we will change our sense of how large and how small they can be. When the engines go quiet, unbuckle and follow me forward. There is a window on the bridge I have been waiting to show you.
Watch and read
- [1 Billion is Tiny in an Alternate Universe: Introduction to p-adic Numbers](https://www.youtube.com/watch?v=3gyHKCDq1YA) Eric Rowland, 2022: the best casual introduction, built around the same idea as this chapter, that in the p-adics the divisible numbers are the small ones
- [The Distance Between Numbers - Numberphile](https://www.youtube.com/watch?v=Sgupo9DLMGs) Numberphile with Tom Crawford, 2023: the p-adic distance, and why …999 can equal −1
- [p-adic number](https://en.wikipedia.org/wiki/P-adic_number) Wikipedia
- [p-adic valuation](https://en.wikipedia.org/wiki/P-adic_valuation) Wikipedia