# References — Chapter ᚙ(X): The bridge *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-26, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch26.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 26.1 {#r-26.1} Conway developed the surreal-number construction while studying Go endgames, and Knuth’s 1974 book introduced the name surreal numbers; Conway’s On Numbers and Games appeared in 1976. *Status: confirmed* - [Donald Knuth, Surreal Numbers (Stanford-hosted bibliographic page)](https://cs.stanford.edu/~knuth/sn.html) — gives the 1974 Addison-Wesley publication and quotes Knuth’s postscript — “as the two characters in this book gradually explore and build up Conway’s number system” — together with Gardner’s description of Conway waving “two simple rules in the air”; it records the 1972 lunch and the 1973 week in Oslo. It does not describe the setting of the story: the page was grepped in full for beach, rock, stone, island and tablet, with zero hits. - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — records Conway’s Go-endgame work, Knuth’s 1974 introduction of the name, and Conway’s 1976 On Numbers and Games. ### 26.2 {#r-26.2} A surreal number is constructed as a cut {L | R} of earlier numbers, with every left option less than every right option; the first birthdays give 0, ±1, and the dyadic rationals such as 1/2. The first infinite generation, Sω, whose new numbers are born on day ω, adds ω and −ω, the infinitesimals ±ε and every remaining real. *Status: confirmed* - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — defines the recursive left/right construction and gives the initial numbers and their birthdays; re-read 29 September 2026 (cache of 24 September): its list of the first generations has 3/4 arriving in S3, and of Sω, the first infinite generation, it says “The maximal subset of Sω that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors y ± ε of each nonzero dyadic fraction y”; the dyadic fractions are “the surreal numbers with birthday less than ω”; “A unique infinitely large positive number occurs in Sω”; “Not only do all the rest of the rational numbers appear in Sω; the remaining finite real numbers do too”; “The only infinities in Sω are ω and −ω”; and the infinitesimals “in Sω are ε and its additive inverse −ε” (spacing as normalized here). ### 26.3 {#r-26.3} The surreal numbers form a totally ordered real-closed field containing the real numbers and the ordinals; hence every positive surreal has a square root and nonzero surreals are invertible. *Status: confirmed* - [Costin, Ehrlich and Friedman, Integration on the Surreals (arXiv:1505.02478)](https://arxiv.org/abs/1505.02478) — abstract states that Conway introduced a real-closed field No containing the reals and ordinals. - [Wikipedia, Real closed field](https://en.wikipedia.org/wiki/Real_closed_field) — states that a real-closed ordered field contains square roots of all positive elements. ### 26.4 {#r-26.4} Ordinal multiplication uses copies of the left factor indexed by the right factor, so ω·0 is zero copies; ordinal ω has no immediate predecessor and cannot be treated as an ordinary divisible field element. *Status: confirmed* - [Wikipedia, Ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) — defines ordinal multiplication by replacing each element of the right factor with a copy of the left factor and gives ω·2 and 2·ω examples. ### 26.5 {#r-26.5} In the surreal field, ω = {0,1,2,…|} is the first infinite surreal and ε = {0|1,1/2,1/4,…} is a positive infinitesimal equal to 1/ω, so ε·ω−1=0. Both first appear in the generation Sω, so both are born on day ω. *Status: confirmed* - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — describes the first infinite surreal, infinitesimals, and the field operations; the displayed equality is also verified by the field identity ε=1/ω. Re-read 29 September 2026 (cache of 24 September): “A unique infinitely large positive number occurs in Sω,” and the infinitesimals “in Sω are ε and its additive inverse −ε” (spacing as normalized here). - **Own computation** (the author's own work) — Using ε=1/ω in a field: ε·ω=(1/ω)·ω=1, hence ε·ω−1=0. ### 26.6 {#r-26.6} The surreal field contains ω−1, ω/2, √ω and ω^ω as surreal values, while the corresponding ordinal notation does not support all of those field operations; √ω is infinite but smaller than every positive real multiple ω/n. In the surreals ω − 1 = {0, 1, 2, … | ω} has birthday ω + 1 and is not an ordinal; ω/2 is infinite but smaller than ω − n for every n, with birthday ω·2; ω^y is greater than rω^x for every real r when x < y. *Status: confirmed* - [Integration on the Surreals (arXiv:1505.02478)](https://arxiv.org/abs/1505.02478) — identifies No as a field containing ordinals and states that usual square-root operations extend naturally to the surreal field. - [Wikipedia, Ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) — documents the noncommutative ordinal operations and the absence of an ordinal immediately before ω; the comparisons in the chapter follow the surreal order and field arithmetic. - [Surreal number, Wikipedia (ω − 1, ω/2, powers of ω)](https://en.wikipedia.org/wiki/Surreal_number) — read as raw wikitext: “There is another positive infinite number in generation ω + 1: ω − 1 = {0, 1, 2, 3, 4, … | ω}”; “The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal”; “there is a surreal number ω/2 that is infinite but smaller than ω − n for any natural number n”; “If x < y then ω^y is “infinitely greater” than ω^x, in that it is greater than r ω^x for all real numbers r”; and, in the rendered page cached 24 September 2026, “The birthday of ω/2 is the limit ordinal ω2” (spacing as normalized). ### 26.7 {#r-26.7} Robinson’s nonstandard analysis made infinitesimals rigorous; the hyperreals form an ordered field extension of the reals containing infinite and infinitesimal elements. Robinson announced nonstandard analysis in 1961, in the eight-page paper “Non-standard analysis”, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings ser. A 64 (= Indagationes Mathematicae 23), 432–440, and gave the full development in his book Non-standard Analysis (North-Holland, 1966); the 1961 paper is an announcement, not the finished theory. *Status: confirmed* - [Wikipedia, Hyperreal number](https://en.wikipedia.org/wiki/Hyperreal_number) — describes hyperreals as an ordered field extension containing infinite and infinitesimal numbers and dates Robinson’s work to the 1960s; “The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz’s heuristic law of continuity..” - [Joseph W. Dauben, Abraham Robinson and Nonstandard Analysis (Minnesota Studies in the Philosophy of Science XI)](https://conservancy.umn.edu/server/api/core/bitstreams/1b052307-901a-4018-9c72-6d00757dd16c/content) — bibliography gives “Robinson, Abraham. 1961. Non-Standard Analysis. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, ser. A, 64: 432-40”, and records that the idea was first given wide publicity in a 1961 conference paper. - [MacTutor, Abraham Robinson](https://mathshistory.st-andrews.ac.uk/Biographies/Robinson/) — “non-standard analysis which he introduced in 1961” and “In 1966 Robinson published his famous text Non-standard analysis”. ### 26.8 {#r-26.8} Hyperreals can be constructed as an ultrapower of the reals, and Łoś’s theorem supplies the transfer principle for first-order statements. The principle is an equivalence, but a qualified one: a formula holds of the reals if and only if its starred transform holds of the hyperreals, with quantification bounded and ranging over internal sets. Unqualified, it is false — “every positive number exceeds 1/n for some positive integer n” is true of the reals and false of the hyperreals unless n is allowed to be a hyperinteger, which is why *ℝ is non-Archimedean while ℝ is Archimedean. The ultrapower needs a free ultrafilter, whose existence rests on the ultrafilter lemma, weaker than the axiom of choice, and which cannot be explicitly constructed; the resulting field has the cardinality of the continuum, so it is a set. *Status: confirmed* - [Wikipedia, Transfer principle](https://en.wikipedia.org/wiki/Transfer_principle) — states the hyperreal transfer principle as preservation of first-order truth and identifies its model-theoretic basis. - [Wikipedia, Nonstandard analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis) — gives the ultrapower presentation of *R. - [Wikipedia, Hyperreal number](https://en.wikipedia.org/wiki/Hyperreal_number) — (lead): the page that names the theorem — “Łoś’s theorem of 1955” — and states the transfer principle in terms of it. Neither of the two pages above contains the name; this one is already cited at 26.7, 26.9 and 26.10. On the construction: “This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed”; “It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice.”; and of the ultrapower field, “Since this field contains R it has cardinality at least that of the continuum” and it “has the same cardinality as R.” ### 26.9 {#r-26.9} For a finite hyperreal x there is a unique real infinitely close to it, its standard part st(x); for an infinite positive H, 1/H is infinitesimal and st(1/H)=0. *Status: confirmed* - [Wikipedia, Standard part function](https://en.wikipedia.org/wiki/Standard_part_function) — defines the standard-part map from finite hyperreals to their infinitely close real values. - [Wikipedia, Hyperreal number](https://en.wikipedia.org/wiki/Hyperreal_number) — describes infinite hyperreals, infinitesimals, and the standard-part operation in nonstandard calculus. ### 26.10 {#r-26.10} An infinitesimal is positive but smaller than every positive real; the real field has none by the Archimedean property, while hyperreal and surreal fields contain them. *Status: confirmed* - [Wikipedia, Infinitesimal](https://en.wikipedia.org/wiki/Infinitesimal) — defines infinitesimals and contrasts them with the Archimedean real numbers; “The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange.” - [Wikipedia, Hyperreal number](https://en.wikipedia.org/wiki/Hyperreal_number) — confirms genuine infinitesimal elements in hyperreal extensions. ### 26.11 {#r-26.11} No is a proper class rather than a set, contains the ordinals and reals, and is a real-closed ordered field in the standard Conway theory. *Status: confirmed* - [Costin, Ehrlich and Friedman, Integration on the Surreals (arXiv:1505.02478)](https://arxiv.org/abs/1505.02478) — abstract describes No as Conway’s real-closed field containing the reals and ordinals. - [Surreal substructures](https://www.texmacs.org/joris/sss/sss.html) — describes the surreal numbers as a proper-class ordered real-closed field. ### 26.12 {#r-26.12} Surreal sign expansions encode numbers as transfinite strings of plus and minus signs; the examples 1/2=+−, ω as ω pluses, and ε as a plus followed by ω minuses are standard descriptions. *Status: confirmed* - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — describes sign expansions and the recursive representation of surreal numbers; “According to Conway, intermediate values must be governed by his rule of simplicity. That is, numbers born on subsequent birthdays must be the simplest between the new and the prior day..” - [István Mező, Convergence on surreals, arXiv:1210.5675](https://arxiv.org/abs/1210.5675) — read from the PDF: “X(i) = + (i = 0, 1, …) denotes “the” infinite number ω = +^ω, while X(0) = +, X(i) = − (i = 1, 2, …) denotes the infinitesimal number ε = +−^ω.” ### 26.13 {#r-26.13} The hyperreal standard-part expression st((f(x+dx)−f(x))/dx) is the nonstandard-analysis form of a derivative, and the standard part discards an infinitesimal error. *Status: confirmed* - [Wikipedia, Standard part function](https://en.wikipedia.org/wiki/Standard_part_function) — describes standard part as the map used to extract ordinary real values from finite hyperreal calculations. - [Wikipedia, Nonstandard analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis) — describes the use of infinitesimals and standard parts in nonstandard calculus. ### 26.14 {#r-26.14} The chapter’s distinction between the surreal ω and the extended-real symbol ∞ is mathematical rather than typographical: ω is a field element, whereas extended-real infinity is not an ordinary field element; Conway’s ∞ is a gap/cut rather than a surreal number. *Status: judgement* - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — discusses surreal infinities and the distinction between surreal numbers and gaps in the ordered class. - [Wikipedia, Extended real number line](https://en.wikipedia.org/wiki/Extended_real_number_line) — treats ±∞ as adjoined symbols with non-field arithmetic. Editorial note: We use ω for a surreal field element and ∞ for the extended-real symbol, following the chapter’s explicit convention; Conway’s gap terminology is kept as a framework-specific clarification. ### 26.15 {#r-26.15} The epsilon-delta definition of a limit is Weierstrass’s, written down in his Berlin lectures of 1861, not in the 1870s; Bolzano (1817) and Cauchy (in the 1820s) are its forerunners, Cauchy having used the symbols without fixing delta as a function of epsilon. *Status: confirmed* - [G. I. Sinkevich, On History of Epsilontics, Antiquitates Mathematicae 10 (2016)](https://www.spbgasu.ru/upload/medialibrary/fe7/2o94a2geo3s81h760av6m43l1elmvbyt/Sinkiewicz,%20G.%20I.%20On%20History%20of%20Epsilontics.%20Antiquitates%20Mathematicae.pdf) — “It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass’ definition of a limit”; the earliest known text is Schwarz’s notes from Weierstrass’s summer-term 1861 lectures at the Berlin Gewerbeinstitut, and Cauchy introduced the symbols in 1823 without a functional relationship for delta. - [Wikipedia, Karl Weierstrass](https://en.wikipedia.org/wiki/Karl_Weierstrass) — records Bolzano’s 1817 definition of a limit and that the basic idea of delta-epsilon proofs is first found in Cauchy in the 1820s. ### 26.16 {#r-26.16} Formulated in von Neumann–Bernays–Gödel set theory with global choice, No is universal among ordered fields: every ordered field whose universe is a set or a proper class of NBG embeds in it, and No is the unique such homogeneous universal ordered field up to isomorphism. Because No is a proper class the word field is not strictly correct, and Conway writes Field. *Status: confirmed* - [Philip Ehrlich, Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners (arXiv:1808.03345)](https://arxiv.org/abs/1808.03345) — an ordered field is homogeneous universal when “every ordered field … whose universe is a set or a proper class of NBG can be embedded in A”, and “No is (up to isomorphism) the unique homogeneous universal ordered field”; the underlying set theory is stated to be NBG with global choice. - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — “If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field”, with the footnote that No is a proper class “so the term field is not precisely correct” and some authors write Field. ### 26.17 {#r-26.17} The omnific integers Oz are Conway’s integers inside the surreals, those x with x = {x − 1 | x + 1}; every ordinal is one, Oz is an integer part of No, and every surreal number is a quotient of two omnific integers, so No is the field of fractions of Oz. The omnific integers were first defined by Simon P. Norton; Conway wrote about them in On Numbers and Games. Because Oz contains the ordinals and is closed under subtraction, ω − 1 is an omnific integer though not an ordinal, while ½ and ε are not. *Status: confirmed* - [L’Innocente & Mantova, A factorisation theory for generalised power series and omnific integers (arXiv:1710.07304; Advances in Mathematics)](https://arxiv.org/abs/1710.07304) — “Inside No, Conway isolated the subring Oz of the omnific integers”, and “Oz is an integer part of No, namely for every b in No there is a unique c in Oz such that c ≤ b < c+1”. - [Awate, Chavan and Manva, An Overview of Surreal Numbers (Euler Circle, 2021)](https://simonrs.com/eulercircle/cgt2021/om-sarth-arkan-surreal.pdf) — “The class Oz includes On, the class of ordinal numbers. In addition, the omnific integers are closed under addition, subtraction, and multiplication; Oz is therefore a subRing of NO, with NO being the field of fractions of Oz”, and Theorem 3.3 (Conway): “Every surreal number x can be represented as a quotient of two omnific integers”. - **Own computation** (the author's own work) — ω and 1 are ordinals, so ω − 1 is in Oz by closure under subtraction; it exceeds every natural n (as n + 1 < ω) and is less than ω, and the only ordinals below ω are the naturals, so it is no ordinal. If ½ were in Oz, both c = 0 and c = ½ would satisfy c ≤ ½ < c + 1, against the unique integer part quoted above; 0 and ε rule out ε the same way. ### 26.18 {#r-26.18} Conway’s discovery of the surreal numbers dates from around 1970 and came out of his analysis of Go endgames; the name is not his but Knuth’s, from the 1974 novelette Surreal Numbers. Conway set the theory out in his book On Numbers and Games (1976). *Status: confirmed* - [MacTutor, John Horton Conway](https://mathshistory.st-andrews.ac.uk/Biographies/Conway/) — “Another topic for which Conway is famous is discovery of surreal numbers, which again dates from around 1970”; “Conway was not trying to develop number systems, but rather was analysing the game of Go … Conway discovered that certain games behaved like numbers and surreal numbers were born”; and “The name surreal numbers was not invented by Conway, however, but by Donald Knuth who … wrote Surreal Numbers (1974) in the form of a novelette”. This is the page Wikipedia’s Surreal number article itself cites for the Go-endgame origin, and it carries the chapter’s own hedge, “around 1970”, word for word. Among Conway’s books it lists “On numbers and games (1976),” with a reviewer’s “a new, exciting, and highly original theory is expounded by its creator.” Editorial note: Added in the V14 prose pass to close the V10 “could not confirm” item on the bare year 1970. There is no primary 1970 publication — On Numbers and Games is 1976 — so the chapter’s hedge is sourced rather than sharpened. ### 26.19 {#r-26.19} In Conway’s theory a gap is a cut {L | R} of the surreals whose two sides together exhaust the class No; at least one side is then a proper class, so a gap is not a surreal number. The gap separating the surreals that lie below some whole number from those above every whole number is greater than every real and less than every positive infinite surreal, and is the least upper bound of the reals in the completion of No; ∞ and On are named as gaps. *Status: confirmed* - [Wikipedia, Surreal number (§ Gaps and continuity)](https://en.wikipedia.org/wiki/Surreal_number) — “Conway defines a gap as {L | R} such that every element of L is less than every element of R, and L ∪ R = No; this is not a number because at least one of the sides is a proper class”; “there is no least positive infinite surreal, but the gap … is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals”; and “there are other types of gap that are not limits, such as ∞ and On”. Editorial note: The page names ∞ as a gap and separately describes the finite/infinite cut as a gap; it does not say in one sentence that Conway’s ∞ is that particular cut. The chapter’s identification of the two rests on r-26.14, which is a judgement call. ### 26.20 {#r-26.20} Leibniz was using infinitesimal differentials by 1675: his manuscript of 21 November 1675 is the first in which he wrote the notation ∫ f(x)dx. *Status: confirmed* - [MacTutor, Gottfried Wilhelm von Leibniz](https://mathshistory.st-andrews.ac.uk/Biographies/Leibniz/) — “On 21 November 1675 he wrote a manuscript using the ∫ f(x)dx notation for the first time”. The page also dates the d notation’s first publication to Nova Methodus pro Maximis et Minimis (1684) and the printed ∫ to Acta Eruditorum (1686). It gives no general date for Leibniz’s first use of infinitesimals as such, so this entry claims only what the manuscript date supports. Editorial note: Added in the V14 prose pass. Card `infinitesimal` asserted “Leibniz used them from 1675” footnoted to r-26.15, an epsilontics entry that carries no such year, while r-26.10 in the same chapter says instead that the modern concept was introduced “around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz”. The card now says “was using them by 1675, when he first wrote ∫ f(x)dx”, which is what a source carries; the ~1670 introduction and its open Mercator/Leibniz attribution is a different claim and is left to r-26.10. ### 26.21 {#r-26.21} The setting of Knuth’s Surreal Numbers — two people on a beach who find a stone carved with two rules — is the author’s own reading of the novelette, not a claim carried by either page cited beside it. *Status: judgement* - **Own reading** (the author's own work) — The two characters and the two rules are sourced: r-26.1 quotes Knuth’s postscript on “the two characters in this book” and Gardner on Conway waving “two simple rules in the air,” and r-26.18 gives the 1974 novelette. Neither page describes where the story is set. The Stanford-hosted bibliographic page cited by r-26.1 was grepped in full for beach, rock, stone, island and tablet, with zero hits. The detail is true of the book itself, so the sentence stands and this entry records that it rests on the author having read it. Editorial note: Added for Q26.2, answered by the default on 11 September 2026: mark the beach and the stone as the author’s own reading rather than trim them. One page number from a copy of Surreal Numbers would close this permanently and upgrade the entry to confirmed. ### 26.22 {#r-26.22} Every surreal number has a Conway normal form: a well-based series in powers of omega under Conway’s omega-map, which is an order-preserving bijection extending Cantor’s ordinal exponentiation, with real coefficients over a decreasing support. Through that map and the normal form the surreal field is naturally isomorphic to a field of well-based series. *Status: confirmed* - [Vincent Bagayoko and Joris van der Hoeven, Surreal substructures](https://www.texmacs.org/joris/sss/sss.html) — read as raw HTML, §2: “Conway also constructed an order preserving bijection that extends Cantor’s ordinal exponentiation. Through this [omega]-map and the so-called Conway normal form [11, ch. 5], it turns out that the field is naturally isomorphic to a field of well-based series, for which [the omega-powers] becomes the monomial group. For this series representation, any number has a set-sized support. The Conway normal form of [a number] coincides with its expression as a series.” The bracketed reference 11 is Conway, On Numbers and Games; the chapter number is the paper’s, and the V42 pass of 18 September 2026 could not check it against the book, whose contents list suggests the normal form sits in that book’s third chapter. Editorial note: Added 17 September 2026 for the B6 pass. The #cnf card’s whole substance is the normal form, and its quick line — which is what every popup repeats — stood on r-26.11, whose claim line is that No is a proper class and a real-closed ordered field and which never mentions the normal form, omega-powers or Cantor. The chapter prose has the same gap at the omega^omega sentence. Both now cite this entry. Note what it does and does not carry: it is the omega-map and the series representation, and the phrase “Cantor’s normal form for ordinals” that the card uses as the analogy is carried by r-27.20, which the card already cites for Cantor 1897. ### 26.23 {#r-26.23} The chapter epigraph is the opening line of Donald Knuth’s Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness, of 1974, the book that gave the numbers their name. *Status: confirmed* - [Surreal number, Wikipedia](https://en.wikipedia.org/wiki/Surreal_number) — read as raw HTML: “Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway’s construction was introduced in Donald Knuth’s 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness,” and the page records that the name surreal is Knuth’s. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book, its full title, its year and the naming are sourced here; that the epigraph is its opening line is the author’s own reading of the printed novelette and is offered as such. r-26.21 already rules on calling it a novelette rather than a novella. ### 26.24 {#r-26.24} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: The Conway normal form is set out in Conway’s On Numbers and Games, 1976 (‘Conway 1976’); The ultrapower construction of a hyperreal field goes back to Edwin Hewitt (1948); Conway’s surreals of 1976 (On Numbers and Games) contain infinitesimals; The {L | R} notation was set out in On Numbers and Games (1976), the construction having first appeared in print in Knuth’s Surreal Numbers (1974); Conway introduced the class No of surreal numbers in On Numbers and Games, 1976 (‘Conway 1976’); The omnific integers Oz, first defined by Simon Norton, were set out by Conway in On Numbers and Games, 1976 (‘Conway 1976’); Conway’s On Numbers and Games, which uses the word birthday, is of 1976; The sign expansion goes back to Conway, On Numbers and Games, 1976; Gonshor developed the sign-expansion treatment of the surreals in 1986 (An Introduction to the Theory of Surreal Numbers). Conway’s construction was introduced in Knuth’s 1974 book Surreal Numbers. *Status: confirmed* - [Bagayoko and van der Hoeven, Surreal substructures, arXiv:2305.02001](https://arxiv.org/pdf/2305.02001) — read raw: “[14] J. H. Conway. On numbers and games. Academic Press, 1976..” - [Philip Ehrlich, Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners, arXiv:1808.03345](https://arxiv.org/pdf/1808.03345) — read raw: “Hewitt (1948) had already employed the ultrapower construction to obtain a non-Archimedean real-closed extension of the reals”; “In his monograph On numbers and Games (1976), J. H. Conway introduced a real-closed ordered field of surreal numbers containing”; “Conway, J. H. (1976): On Numbers and Games, Academic Press, London. Second Edition, A K Peters, Ltd., Natick, Massachusetts (2001).”; “Conway’s sign-expansion construction (Conway 1976, 2001, p. 65), which was made popular by Gonshor (1986).” - [Surreal number, Wikipedia](https://en.wikipedia.org/wiki/Surreal_number) — read raw: “Conway later adopted Knuth’s term, and used surreals for analyzing games in his 1976 book On Numbers and Games.”; “A treatment of surreals based on the sign-expansion realization: Harry Gonshor, An Introduction to the Theory of Surreal Numbers, 1986, ..” - [On Numbers and Games, Wikipedia](https://en.wikipedia.org/wiki/On_Numbers_and_Games) — read raw: “The book was first published by Academic Press in 1976, , and a second edition was released by A K Peters in 2001.” - [L’Innocente and Mantova, A factorisation theory for generalised power series and omnific integers, arXiv:1710.07304](https://arxiv.org/pdf/1710.07304) — read raw: “John Horton Conway. On Number and Games. London Mathematical Society Monographs. London: Academic Press, 1976, pp. x+238..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.