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# Glossary cards — Chapter ᚘ(Y): Pre-flight controls

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T25, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch25.md)

*Non-Archimedean completions: a 3-adic carry and the product formula, over a composite-base 10-adic pair, the 3-adic size of 27 and a 7-adic square root of two. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Ifín — `ᚘ` {#g-og-ifin}

*Ogham letter*

A supplementary letter, first called Pín: the sign added for p, its name from Latin pinus, ‘pine’, or spina, ‘thorn’.

This letter was first called Pín and was added because ogham, devised for Primitive Irish, had no sign for [p] at all. Sources differ on the name. The Forfeda page, reporting McManus, allows the influence of Latin pinus, pine, but calls Latin spina, thorn, the likelier source, since the kennings describe a shrub with sweet fruit rather than a conifer; Hayden and Stifter (2022) take it as probably pinus, after the model of the other tree names. The kennings are milsem fedo, “sweetest tree” and amram mlais, “most wonderful taste.” The Auraicept glosses spinan no ispin, gooseberry or thorn, and Kelly finds spín in the Old Irish tree lists with just that range, so the medieval gloss may be right on this occasion. When the later oghamists reassigned all five forfeda to diphthongs, this letter lost its consonant value and its first name, and became Ifín (also spelled Iphín). Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.18](ch02-refs.md#r-2.18), [2.19](ch02-refs.md#r-2.19)

### a carry that never ends — `…222_{3} + 1` {#g-adicid}

*Integer or rational*

Adding 1 sends a carry to infinity and leaves all zeros; arguably the cleanest zero in the piece.

The numerator of the chapter: …222₃ plus one. Adding 1 to the last digit gives 10 in base three, write 0 and carry 1; the carry then does the same to every digit in turn and never lands, so every digit is 0 and the result is exactly 0. Read digit by digit, no limit is needed, and the 3-adic distance agrees: a truncation with n twos is 3ⁿ − 1, plus one is 3ⁿ, and 3ⁿ shrinks to zero 3-adically. That is why I call it the cleanest zero in the piece. It is the integer 0 and is tinted as one.

Cites: [25.3](ch25-refs.md#r-25.3)

See also: [the all-twos 3-adic integer](ch25-cards.md#g-threeadic)

### the 10-adic automorphic pair — `…890625 + …109376` {#g-tenadic}

*Integer or rational*

Each squares to itself, they sum to exactly 1, and they multiply to 0.

The 10-adic automorphic numbers, …890625 and …109376, each of which equals its own square; they arise because 10 = 2·5 is not prime, so ℤ₁₀ splits into a 2-adic and a 5-adic part and has idempotents other than 0 and 1. They are the limits of the automorphic truncations 5, 25, 625, 0625, 90625, 890625, … and 6, 76, 376, 9376, 09376, 109376, …. The one-digit cases 5 and 6 were already called circular numbers in later antiquity and the Middle Ages. Their sum is exactly 1 and their product exactly 0, both checkable digit by digit (890625 + 109376 = 1000001 and 890625 × 109376 ends in six zeros). Here the pair hands back the integer 1, which is why it is tinted as an integer although each member lives in ℤ₁₀.

Cites: [25.22](ch25-refs.md#r-25.22), [25.6](ch25-refs.md#r-25.6), [25.7](ch25-refs.md#r-25.7)

See also: [the 10-adic all-nines, plus one](ch25-cards.md#g-nines10)

### 3-adic absolute value of 27 — `|27|_{3}` {#g-adicabs}

*Integer or rational*

1/27, because in the p-adics the divisible numbers are the small ones; its inverse is 27.

The 3-adic absolute value |n|₃ = 3^(−v₃(n)), where v₃(n) is the exponent of 3 in n. Hensel built the p-adic numbers from 1897; Josef Kürschák founded valuation theory, announcing it at the Cambridge congress of 1912 and putting his axioms for the absolute value in print the year after, and Alexander Ostrowski proved in 1916 that these p-adic absolute values and the ordinary one are, up to equivalence, the only nontrivial absolute values the rationals have. So |27|₃ = 1/27 and the reciprocal used in the formula is 27; since the ordinary absolute value of 27 is also 27, the product formula reads |27|₃ · |27|_∞ = (1/27) · 27 = 1. A rational number, tinted as one.

Cites: [25.1](ch25-refs.md#r-25.1), [25.10](ch25-refs.md#r-25.10), [25.18](ch25-refs.md#r-25.18), [25.2](ch25-refs.md#r-25.2), [25.5](ch25-refs.md#r-25.5)

See also: [the product formula, saying nothing](ch25-cards.md#g-prodformula)

### a 7-adic square root of two, squared — `α², α = √2 in ℤ₇` {#g-sqrt2_7}

*Integer or rational*

√2 exists in the 7-adics because 3² = 9 ≡ 2 mod 7, and its square is 2.

A square root of 2 in the 7-adic integers, squared. Since 3² = 9 ≡ 2 (mod 7), Hensel’s lemma, whose monic special case he stated in his Neue Grundlagen der Arithmetik of 1904, lifts the residue 3 to an exact 7-adic root of x² = 2, whose expansion begins 3 + 1·7 + 2·7² + … (checked digit by digit: 10² = 100 ≡ 2 mod 49, 108² = 11664 ≡ 2 mod 343); the rationals have no such root at all. Its square is exactly 2, which is what comes home, so the item is tinted as an integer although the root itself lives in ℤ₇.

Cites: [25.8](ch25-refs.md#r-25.8), [25.9](ch25-refs.md#r-25.9)

See also: [the ring of p-adic integers](ch25-cards.md#g-Zp), [Hensel’s lemma](ch25-cards.md#g-hensel), [square root of two](ch04-cards.md#g-sqrt2)

### the 3-adic valuation of 27 — `v_{3}(27)` {#g-val3_27}

*Integer or rational*

3, because 27 = 3³; the exponent that makes |27|₃ = 3⁻³.

The 3-adic valuation of 27, which is 3 because 27 = 3³. It is the integer that the absolute value |27|₃ = 3^(−3) is built from, and it sits in the denominator beside that absolute value’s reciprocal so that both faces of the p-adic ruler are on show: the exponent and the size. A positive integer, so it keeps the denominator away from zero.

Cites: [25.2](ch25-refs.md#r-25.2), [25.5](ch25-refs.md#r-25.5)

See also: [3-adic absolute value of 27](ch25-cards.md#g-adicabs)

### the product formula, saying nothing — `|27|_{3} · |27|_{∞} − 1` {#g-prodformula}

*Integer or rational*

All the absolute values of a nonzero rational number, multiplied together, give 1, so this is 0: (1/27)·27 − 1.

The product formula: for any nonzero rational x, the product of |x|_p over every prime p, times the ordinary absolute value |x|_∞, is exactly 1, because each prime’s absolute value cancels that prime’s contribution to |x|_∞. For x = 27 only the prime 3 contributes, so |27|₃ · |27|_∞ = (1/27) · 27 = 1 and the rider is exactly 0. The formula is classical for ℚ and was made an axiom of global fields by Artin and Whaples in 1945; in that setting the real absolute value is also called the infinite prime. An integer 0.

Cites: [25.4](ch25-refs.md#r-25.4), [25.5](ch25-refs.md#r-25.5)

See also: [the p-adic absolute value](ch25-cards.md#g-absp), [3-adic absolute value of 27](ch25-cards.md#g-adicabs)

### the all-twos 3-adic integer — `…222_{3}` {#g-threeadic}

*Non-real object*

Equal to −1 in ℤ₃, since Σ 2·3ᵏ = 2/(1 − 3).

The 3-adic integer with every digit 2. Kurt Hensel introduced the p-adic numbers in 1897, and in the 3-adic distance, where numbers are close when their difference is divisible by a high power of 3, the series 2 + 2·3 + 2·3² + … converges to 2/(1 − 3) = −1. It is a non-real object, which is why it is tinted as one: it lives in ℤ₃, not on the real line, and only the result of adding 1 to it comes home.

Cites: [25.1](ch25-refs.md#r-25.1), [25.3](ch25-refs.md#r-25.3)

See also: [the ring of p-adic integers](ch25-cards.md#g-Zp), [the real numbers](ch01-cards.md#g-reals)

### a 5-adic square root of minus one — `√−1 ∈ ℤ₅` {#g-sqrtm1_5}

*Non-real object*

Exists because 5 ≡ 1 mod 4, so −1 is a square modulo 5 and Hensel lifts it.

A square root of −1 in the 5-adic integers. It exists because 5 ≡ 1 (mod 4), so −1 is a square modulo 5 (2² = 4 ≡ −1), and Hensel’s lemma lifts it: one root begins 2 + 1·5 + 2·5² + …. Named on the Hensel’s lemma card beside the 7-adic square root of 2, to show that which square roots exist depends on the prime; not in the formula. A non-real object.

Cites: [25.8](ch25-refs.md#r-25.8), [25.9](ch25-refs.md#r-25.9)

See also: [the ring of p-adic integers](ch25-cards.md#g-Zp), [Hensel’s lemma](ch25-cards.md#g-hensel)

### the p-adic valuation — `v_{p}(r)` {#g-vp}

*Function or operator*

The exponent of the prime p in a nonzero r: the index of the first nonzero digit of its p-adic expansion.

The p-adic valuation v_p(r) is the exponent of p in the factorization of a nonzero rational r, equivalently the index of the first nonzero digit in its p-adic expansion; so v₃(27) = 3 and v₂(12) = 2. It is additive, v_p(rs) = v_p(r) + v_p(s), and obeys v_p(r + s) ≥ min(v_p(r), v_p(s)). Kurt Hensel introduced the p-adic numbers in 1897 and the valuation is the ruler he measured them with; the word valuation (Bewertung) comes from Josef Kürschák, who announced his first results at Cambridge in 1912 and published his axioms the year after; what he axiomatized is the absolute value rather than the exponent. A tool of the framework, not a value.

Cites: [25.1](ch25-refs.md#r-25.1), [25.18](ch25-refs.md#r-25.18), [25.2](ch25-refs.md#r-25.2)

See also: [the p-adic expansion](ch25-cards.md#g-padicexp), [the 3-adic valuation of 27](ch25-cards.md#g-val3_27), [the valuation of zero](ch25-cards.md#g-vp0)

### the valuation of zero — `v_{p}(0) = ∞` {#g-vp0}

*Infinity or limit device*

Zero has no first nonzero digit, so by convention its valuation is infinite, which makes |0|_p = p^(−∞) = 0 as it should be.

The convention v_p(0) = ∞: the p-adic expansion of 0 has no nonzero digit, so no index exists, and setting the valuation to infinity is the only choice that keeps the rules v_p(rs) = v_p(r) + v_p(s) and |r|_p = p^(−v_p(r)) true for r = 0, since then |0|_p = 0. It is a convention of the extended integers, not a number, which is why it wears the infinity tint, and it never enters the formula.

Cites: [25.11](ch25-refs.md#r-25.11), [25.2](ch25-refs.md#r-25.2)

See also: [the p-adic absolute value](ch25-cards.md#g-absp), [the p-adic expansion](ch25-cards.md#g-padicexp)

### the p-adic absolute value — `|r|_{p} = p^{−v_{p}(r)}` {#g-absp}

*Function or operator*

Size measured by divisibility: the more powers of p divide r, the smaller r is.

The p-adic absolute value, |r|_p = p^(−v_p(r)) for r ≠ 0 and |0|_p = 0. It satisfies |rs|_p = |r|_p |s|_p and the ultrametric inequality, and it makes the numbers divisible by high powers of p the small ones, which is the whole inversion of intuition behind this chapter. Kürschák axiomatized it, announcing his first results at Cambridge in 1912 and printing his axioms the year after. Completing the rationals with respect to it gives ℚ_p, exactly as completing with the ordinary absolute value gives ℝ. A tool of the framework.

Cites: [25.11](ch25-refs.md#r-25.11), [25.18](ch25-refs.md#r-25.18), [25.2](ch25-refs.md#r-25.2)

See also: [the field of p-adic numbers](ch25-cards.md#g-Qp), [the real numbers](ch01-cards.md#g-reals), [the ultrametric inequality](ch25-cards.md#g-ultra)

### the ultrametric inequality — `|x + y|_{p} ≤ max(|x|_{p}, |y|_{p})` {#g-ultra}

*Function or operator*

The strong triangle inequality: every triangle in the p-adics is isosceles, and every point inside a ball is one of its centers.

The ultrametric, or strong triangle, inequality: |x + y|_p never exceeds the larger of |x|_p and |y|_p, with equality whenever the two differ. It replaces the ordinary triangle inequality and produces the strange geometry of ℚ_p: every triangle is isosceles, two balls are either nested or disjoint, and a series converges exactly when its terms tend to 0, which is why …222₃ has a sum at all. Marc Krasner introduced the name in a note in the Comptes Rendus in 1944. A rule of the framework, not a value.

Cites: [25.11](ch25-refs.md#r-25.11), [25.19](ch25-refs.md#r-25.19)

See also: [the field of p-adic numbers](ch25-cards.md#g-Qp), [the p-adic absolute value](ch25-cards.md#g-absp), [the all-twos 3-adic integer](ch25-cards.md#g-threeadic)

### the field of p-adic numbers — `ℚ_{p}` {#g-Qp}

*Non-real object*

The completion of the rationals under |·|_p, one field for each prime, as ℝ is the completion under the ordinary absolute value.

The field of p-adic numbers, ℚ_p: the completion of ℚ with respect to |·|_p, introduced by Kurt Hensel in 1897 by analogy with power series. Its elements are the expansions Σ aₖ pᵏ with finitely many negative k and digits 0 ≤ aₖ < p. By Ostrowski’s theorem of 1916, ℝ and the fields ℚ_p for each prime p are all the completions the rationals have, so each prime gets its own number line. A non-real object.

Cites: [25.1](ch25-refs.md#r-25.1), [25.10](ch25-refs.md#r-25.10), [25.12](ch25-refs.md#r-25.12)

See also: [the p-adic absolute value](ch25-cards.md#g-absp), [Ostrowski’s theorem](ch25-cards.md#g-ostrowski), [the real numbers](ch01-cards.md#g-reals), [a 5-adic square root of minus one](ch25-cards.md#g-sqrtm1_5)

### the ring of p-adic integers — `ℤ_{p}` {#g-Zp}

*Non-real object*

The p-adic numbers with no negative powers: the closed unit ball |x|_p ≤ 1, where …222₃ lives.

The ring of p-adic integers, ℤ_p, the elements of ℚ_p with |x|_p ≤ 1, equivalently the expansions Σ aₖ pᵏ with k ≥ 0. It contains ℤ densely, it is compact, and every element of ℤ_p not divisible by p is invertible, so 1/2 is a 3-adic integer (…1112₃) while 1/3 is not. For a prime p it is an integral domain: the absolute value is multiplicative, so a product of nonzero elements is nonzero, and the only idempotents are 0 and 1. That is why the chapter’s automorphic pair lives in ℤ₁₀, where 10 is composite, and not here. It should not be confused with the integers modulo p, which are ℤ_p’s residue field; ℤ_p is the limit of ℤ/pᵏℤ over all k. A non-real object.

Cites: [25.11](ch25-refs.md#r-25.11), [25.12](ch25-refs.md#r-25.12), [25.6](ch25-refs.md#r-25.6)

See also: [the field of p-adic numbers](ch25-cards.md#g-Qp), [the p-adic absolute value](ch25-cards.md#g-absp), [the all-twos 3-adic integer](ch25-cards.md#g-threeadic)

### the p-adic expansion — `Σ a_{k} p^{k}` {#g-padicexp}

*Function or operator*

Digits running left forever instead of right: every p-adic number is a base-p numeral with infinitely many digits before the point.

The p-adic expansion, r = Σ aₖ pᵏ with 0 ≤ aₖ < p, the sum starting at k = v_p(r) and running to +∞. In the ordinary reals a base-p numeral runs off to the right, toward small powers; in ℚ_p it runs off to the left, toward large powers, which converge because |pᵏ|_p = p^(−k) → 0. The expansion is unique: unlike decimals, where 0.999… and 1 are two spellings of one number, no p-adic number has two expansions. Hensel’s 1897 construction was precisely this analogy with power series in a variable. A notation of the framework.

Cites: [25.1](ch25-refs.md#r-25.1), [25.2](ch25-refs.md#r-25.2)

See also: [the field of p-adic numbers](ch25-cards.md#g-Qp), [zero point nine repeating](ch04-cards.md#g-nines), [the real numbers](ch01-cards.md#g-reals)

### Hensel’s lemma — `Hensel’s lemma` {#g-hensel}

*Function or operator*

A simple root modulo p lifts to an exact root in ℤ_p, which is why √2 exists in ℤ₇ and √−1 in ℤ₅.

Hensel’s lemma: if a polynomial with p-adic integer coefficients has a root modulo p at which its derivative is not zero modulo p, that root lifts to a unique root in ℤ_p, digit by digit, by the same Newton iteration used for real numbers. Kurt Hensel stated a special case of it, for monic polynomials, in Neue Grundlagen der Arithmetik in 1904, and the monicity restriction came off in his Theorie der algebraischen Zahlen of 1908. It is the tool that puts √2 in ℤ₇ (from 3² ≡ 2 mod 7) and √−1 in ℤ₅ (from 2² ≡ −1 mod 5). A tool of the framework.

Cites: [25.8](ch25-refs.md#r-25.8), [25.9](ch25-refs.md#r-25.9)

See also: [the ring of p-adic integers](ch25-cards.md#g-Zp), [the real numbers](ch01-cards.md#g-reals), [square root of two](ch04-cards.md#g-sqrt2)

### Ostrowski’s theorem — `Ostrowski’s theorem` {#g-ostrowski}

*Function or operator*

Every nontrivial absolute value on the rationals is equivalent to the ordinary one or to one |·|_p: the reals and the p-adics are the only completions there are.

Ostrowski’s theorem, proved by Alexander Ostrowski in 1916: every nontrivial absolute value on ℚ is equivalent either to the usual absolute value or to a p-adic absolute value for exactly one prime p. So the rationals have exactly one completion per prime, plus ℝ, and no others; the primes together with the point at infinity are the places of ℚ. A theorem of the framework.

Cites: [25.10](ch25-refs.md#r-25.10), [25.12](ch25-refs.md#r-25.12)

See also: [the p-adic absolute value](ch25-cards.md#g-absp), [the real numbers](ch01-cards.md#g-reals)

### the local–global principle — `Hasse–Minkowski` {#g-hasse}

*Function or operator*

A quadratic form with rational coefficients has a zero other than the origin exactly when it has one in ℝ and in every ℚ_p: check every prime and infinity, and the rationals follow.

The Hasse–Minkowski theorem, the local-to-global principle for quadratic forms: a quadratic form with rational coefficients represents zero over ℚ if and only if it does so over ℝ and over every ℚ_p. Helmut Hasse proved it for ℚ in his 1921 thesis, extending Hermann Minkowski’s work, and published the extension to all number fields in 1924. It was among the first results to show the wider mathematical world what the p-adic numbers were for. A theorem of the framework.

Cites: [25.13](ch25-refs.md#r-25.13), [25.20](ch25-refs.md#r-25.20)

See also: [the field of p-adic numbers](ch25-cards.md#g-Qp), [the real numbers](ch01-cards.md#g-reals)

### Monsky’s theorem — `Monsky’s theorem` {#g-monsky}

*Function or operator*

A square cannot be cut into an odd number of triangles of equal area, and the only known proof uses the 2-adic valuation.

Monsky’s theorem, proved by Paul Monsky in 1970 after Fred Richman posed the problem in 1967: a square cannot be dissected into an odd number of triangles of equal area. The proof colors the plane by the 2-adic valuations of the coordinates, applies Sperner’s lemma, and finds a triangle whose area has negative 2-adic valuation, which 1/n cannot have when n is odd. No proof avoiding the p-adics is known, which makes it the cleanest example of the 2-adic ruler settling a question about ordinary geometry. A theorem of the framework.

Cites: [25.14](ch25-refs.md#r-25.14), [25.21](ch25-refs.md#r-25.21)

### the 10-adic all-nines, plus one — `…999_{10} + 1` {#g-nines10}

*Integer or rational*

The same carry as the numerator, run in base ten: …999 is −1 in ℤ₁₀, so this is 0.

The 10-adic number with every digit 9, plus 1: the carry runs left forever and leaves all zeros, so …999 = −1 in ℤ₁₀, the mirror image of 0.999… = 1 in the reals. It is the base-ten twin of the chapter’s numerator and is here as a key entry so a reader can see the trick in the base they already know. Its value is the integer 0.

Cites: [25.3](ch25-refs.md#r-25.3)

See also: [zero point nine repeating](ch04-cards.md#g-nines), [the real numbers](ch01-cards.md#g-reals)