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# Chapter ᚙ(X): The bridge

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T26, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [Its glossary cards](ch26-cards.md) · [Its references](ch26-refs.md) · [Previous chapter](ch25.md) · [Next chapter](ch27.md)

*A footnote such as [r-26.3] is reference entry 26.3 and links to it. Section 26 of 28; chapters are named by their Ogham letter, never by number.*

ᚙ – [Eamhancholl](ch26-cards.md#g-og-emancholl): associated with the witch hazel tree, Section 26

## Chapter ᚙ(X): The bridge

> “In the beginning, everything was void, and J. H. W. H. Conway began to create numbers.”

> — Donald Knuth · *Surreal Numbers*, 1974, the novelette that named them [r-26.23](ch26-refs.md#r-26.23)

Formula: ([{ | }](ch26-cards.md#g-surzero) + [ε·ω − 1](ch26-cards.md#g-epsomega))/([ω − 1](ch26-cards.md#g-omm1) + [√ω](ch26-cards.md#g-sqrtomega) + [ω<sup>ω</sup>](ch26-cards.md#g-omom) + [ω/2](ch26-cards.md#g-omhalf) + [{0 | 1}](ch26-cards.md#g-halfS)) {brace: [surreal numbers](ch26.md#ch-T26-surreal-numbers)} · ([st(1/H)](ch26-cards.md#g-hyper))/([H](ch26-cards.md#g-hyperH)) {brace: [nonstandard analysis](ch26.md#ch-T26-nonstandard-analysis)}

#### Type of number

Non-real object ·

Key concepts: Surreal numbers and the hyperreals of nonstandard analysis: two fields where you can honestly divide by an infinity. It includes symbols you may not have seen before, such as: [{ | }](ch26-cards.md#g-surzero), [ω](ch26-cards.md#g-omegaS), and [st](ch26-cards.md#g-hyper). Zero here is born first: the surreal 0 is made from nothing on day zero, and one over an infinite hyperreal rounds to 0.

Here it is: the bridge, and the view through its great window. We are in orbit, and the whole route lies below us at once: the garden and the park, the city streets and the rail line, the museum, the road with its mileposts and crossroads, the cutting, and the launch tower. Places we visited one at a time now sit in a single picture. Everything down there has an ordinary finite size, but from up here it is natural to wonder about sizes the [real line](ch01-cards.md#g-reals) cannot offer: a positive number smaller than every positive real, or one larger than every real. [The surreal numbers](ch26-cards.md#g-No) answer with one construction that holds every real number and those new ones too.

We can sit by the window with a story that takes place on a beach. Conway found [the surreal numbers](ch26-cards.md#g-No) through his work on games, the kind we watched from the stadium seats in Chapter ᚅ(N). Knuth introduced them to a wider audience in a short novel: two people on a beach discover rules on a stone and begin rebuilding arithmetic. The scene is well chosen. The construction starts with so little that a reader can make its first numbers by hand. [r-26.1](ch26-refs.md#r-26.1) [r-26.18](ch26-refs.md#r-26.18) [r-26.21](ch26-refs.md#r-26.21)

Field notes

### [surreal numbers](ch26.md#ch-T26-surreal-numbers) {#ch-T26-surreal-numbers}

Write [{L|R}](ch26-cards.md#g-lrnotation), with previously constructed numbers on the left and right, every left option less than every right option. The notation picks out the simplest surreal lying above all the left options and below all the right ones, where “simplest” means earliest in the construction. On day zero there is nothing to use; on day one, {0|} gives 1 and {|0} gives −1; on day two come 1/2, −1/2, and ±2. Every fraction with a power of two underneath (3/4, 5/8, 7/16, …) appears after finitely many days. The days do not stop at the finite ones: on the first infinite day every other real number arrives, together with the first infinite and the first infinitesimal numbers, and later days bring many more. [r-26.2](ch26-refs.md#r-26.2)

#### Numerator

[{ | }](ch26-cards.md#g-surzero) = 0; the very first surreal number, the one everything else is built from.

On day zero there are no earlier numbers, so the cut {|} has no constraints on either side. The simplest number satisfying no constraints is 0. [r-26.2](ch26-refs.md#r-26.2)

[ε·ω − 1](ch26-cards.md#g-epsomega) = 0; [an infinitesimal](ch26-cards.md#g-infinitesimal) and an infinity, chosen to cancel exactly.

On the first infinite day, [{0,1,2,3,…|}](ch26-cards.md#g-omegaS) gives ω, larger than every natural number and hence every real. The cut {0|1,1/2,1/4,1/8,…} gives a positive [infinitesimal](ch26-cards.md#g-infinitesimal) ε, and under surreal multiplication it is exactly 1/ω, so [εω − 1](ch26-cards.md#g-epsomega) = 0. This is an exact algebraic fact, not a limit in which a finite reciprocal keeps shrinking. [r-26.5](ch26-refs.md#r-26.5)

#### Denominator

[The surreal numbers](ch26-cards.md#g-No) form an ordered field: any two of them can be added, subtracted, multiplied, and compared, and every nonzero surreal has an inverse, so we may divide by it, infinite or not. [r-26.3](ch26-refs.md#r-26.3) There are too many surreals to gather into a set (they form a proper class), but the arithmetic works all the same. [r-26.11](ch26-refs.md#r-26.11)

Four different infinite [surreals](ch26-cards.md#g-No) share this denominator with the familiar fraction [½](ch26-cards.md#g-halfS).

[ω − 1](ch26-cards.md#g-omm1) > 0, infinite; one less than infinity.

In [the surreals](ch26-cards.md#g-No), subtracting 1 from ω is an ordinary field operation, and the result is still larger than every real. The [ordinals](ch27-cards.md#g-ordarith) of Chapter ᚚ(P), which count past infinity, have no such step: nothing comes just before their ω. [r-26.6](ch26-refs.md#r-26.6)

[√ω](ch26-cards.md#g-sqrtomega) > 0, infinite; a square root of infinity.

It is the positive surreal whose square is ω, and it is still infinite.

[ω<sup>ω</sup>](ch26-cards.md#g-omom) > 0, infinite; omega to the omega, a tower that stays a number.

Read as a surreal power, it is a definite positive infinite surreal, vastly larger than ω, and still nonzero. Powers of ω are the surreals’ building blocks: in [Conway’s normal form](ch26-cards.md#g-cnf), every surreal is a sum of them, each multiplied by a real number. [r-26.22](ch26-refs.md#r-26.22) [r-26.6](ch26-refs.md#r-26.6)

[ω/2](ch26-cards.md#g-omhalf) > 0, infinite; half of infinity, which is still infinite.

Dividing ω by 2 makes sense in a field, and the result still exceeds every real. [r-26.4](ch26-refs.md#r-26.4) [r-26.6](ch26-refs.md#r-26.6)

[{0 | 1}](ch26-cards.md#g-halfS) = 1/2; one of the numbers we can build by hand, [born on day](ch26-cards.md#g-sbirthday) two.

It is the simplest number between 0 and 1: a positive finite surreal.

The first fraction is 0/(a positive infinite surreal): the sum is positive, hence nonzero, even though it is infinite, and dividing zero by that field element gives exactly 0.

**Fine print.** This ε differs from [the dual unit](ch21-cards.md#g-dual) in Chapter ᚔ(I): a positive [surreal infinitesimal](ch26-cards.md#g-epsS) has a positive nonzero square, while [the dual unit](ch21-cards.md#g-dual)’s square is zero. They share a useful letter, not the same arithmetic. This division is also not the extended-real convention used on the book’s left side: ω is an element of a richer field, whereas the extended-real symbol +∞ is not an invertible field element. The setting determines the operation.

### [nonstandard analysis](ch26.md#ch-T26-nonstandard-analysis) {#ch-T26-nonstandard-analysis}

Robinson came to [infinitesimals](ch26-cards.md#g-infinitesimal) from a different direction. Conway was analyzing games; Robinson’s [hyperreals](ch26-cards.md#g-hyperreals) make rigorous the infinitely small quantities that Leibniz wrote as dx. [r-26.18](ch26-refs.md#r-26.18) [r-26.20](ch26-refs.md#r-26.20) They extend [the reals](ch01-cards.md#g-reals) with infinitesimals and infinite numbers, and a [transfer principle](ch26-cards.md#g-transfer), stated with care, carries the rules of the reals over to them, so that a proof with infinitesimals counts as a proof about ordinary numbers. [r-26.7](ch26-refs.md#r-26.7) [r-26.8](ch26-refs.md#r-26.8)

#### Numerator

[st(1/H)](ch26-cards.md#g-hyper) = 0; [the standard part](ch26-cards.md#g-stpart), the tool that turns [infinitesimal](ch26-cards.md#g-infinitesimal) reasoning back into ordinary calculus.

For a positive infinite [hyperreal](ch26-cards.md#g-hyperreals) H, 1/H is positive and smaller than every positive real number. The *[standard part](ch26-cards.md#g-stpart)* of a finite [hyperreal](ch26-cards.md#g-hyperreals) is the unique real number infinitesimally close to it, so [st(1/H)](ch26-cards.md#g-hyper) = 0. The operation extracts the real value next to the [hyperreal](ch26-cards.md#g-hyperreals); it does not claim that the nonzero [infinitesimal](ch26-cards.md#g-infinitesimal) itself equals zero. [r-26.9](ch26-refs.md#r-26.9)

Here is why this matters for calculus. For a real x and a nonzero [hyperreal](ch26-cards.md#g-hyperreals) [infinitesimal](ch26-cards.md#g-infinitesimal) h,

((x + h)² − x²)/(h) = 2x + h

The top of the fraction is (x + h)² − x² = 2xh + h², and dividing it by h is exact because h ≠ 0. Taking [the standard part](ch26-cards.md#g-stpart) gives 2x, the derivative of x². We keep the [infinitesimal](ch26-cards.md#g-infinitesimal) through the arithmetic and remove it only when extracting the ordinary real answer. [r-26.7](ch26-refs.md#r-26.7)

#### Denominator

[H](ch26-cards.md#g-hyperH) > 0, infinite; the same infinite [hyperreal](ch26-cards.md#g-hyperreals), placed where the [real line](ch01-cards.md#g-reals) would forbid it.

H is infinite relative to the embedded reals, but it is still a nonzero element of the [hyperreal](ch26-cards.md#g-hyperreals) field, with inverse 1/H. [r-26.9](ch26-refs.md#r-26.9)

The second and final fraction is 0/H: the numerator’s real 0 is embedded in the [hyperreal](ch26-cards.md#g-hyperreals) field, and 0/H is 0, an ordinary real number again.

**Fine print.** This H is a [hyperreal](ch26-cards.md#g-hyperreals), not the letter of a chapter, and not the extended-real symbol ∞.

What this chapter adds to the formula: 0 / (a positive infinite surreal) · 0 / H = 0 · 0 = 0

Smallness and largeness have acquired meanings our original [real line](ch01-cards.md#g-reals) could not supply. [Standard part](ch26-cards.md#g-stpart) has also given us a route back to ordinary real values.

Before we leave orbit, we should probably decide where we want to go. The navigator’s console beside the window holds the star charts, with more destinations than anyone could ever visit. Pull up a chair and let’s consider where we might go, because choosing a journey turns out to be a mathematical question.

Watch and read

- [Surreal Numbers (writing the first book) - Numberphile](https://www.youtube.com/watch?v=mPn2AdMH7UQ) Numberphile with Don Knuth: how Conway explained the numbers to him over lunch in 1972, and how Knuth wrote the first book about them, and named them, in a week in an Oslo hotel room in 1973

- [Surreal Numbers (Don Knuth Extra Footage) - Numberphile](https://www.youtube.com/watch?v=gMuS8ckbdZE) The extra footage, where Knuth actually builds the first few numbers from the two rules

- [Surreal number](https://en.wikipedia.org/wiki/Surreal_number) Wikipedia

- [Hyperreal number](https://en.wikipedia.org/wiki/Hyperreal_number) Wikipedia