Surreal Numbers: The Number System Born in a Game of Go

Q: What are surreal numbers?

Surreal numbers are a number system invented by John Conway that contains all real numbers, all ordinal numbers (which describe infinite sizes), and all infinitesimals (numbers smaller than any positive real but greater than zero) within a single coherent framework. Every surreal number is defined by two sets: a set of numbers it is 'greater than' (left set) and a set of numbers it is 'less than' (right set).

Q: How were surreal numbers discovered?

John Conway discovered them in 1970 while analyzing endgame positions in the board game Go. Certain Go positions decomposed naturally into sums of simpler positions, and Conway realized the positions themselves formed a number-like system. The full theory was published in his 1976 book 'On Numbers and Games.' Donald Knuth popularized the discovery by writing a mathematical novella called 'Surreal Numbers' (1974), the first published account.

Q: What is an infinitesimal in the surreal numbers?

An infinitesimal is a surreal number greater than zero but smaller than every positive real number. The simplest infinitesimal is ε = {0 | 1, 1/2, 1/4, 1/8, ...} — it is greater than 0 but less than every positive real. Its reciprocal 1/ε is an infinite surreal number larger than every real. Unlike the informal infinitesimals of early calculus, surreal infinitesimals are rigorously defined and obey all the usual arithmetic rules.

Q: Are surreal numbers used in practice?

Surreal numbers are primarily a theoretical construction with deep connections to combinatorial game theory and foundations of mathematics. They provide a rigorous framework for reasoning about infinite and infinitesimal quantities. In combinatorial game theory, they give exact values to abstract games. In model theory, they are related to non-standard analysis. Practical applications are limited, but their theoretical unification of disparate number concepts is considered a significant achievement.

A Number System Found in a Board Game

In 1970, the mathematician John Conway was thinking about Go — the ancient Chinese board game. In the endgame, a Go board fragments into independent regions that can be analyzed separately. Conway noticed that these regions combined like numbers: some were worth half a point, others a quarter, others strange fractional values he couldn’t quite name.

Following this observation led him somewhere unexpected. The positions in Go endgames were not just game-theoretically equivalent to numbers — they were numbers, in a precise mathematical sense. And the number system they formed was like nothing that had existed before.

Conway called his discovery the surreal numbers. The name was coined by Donald Knuth, who in 1974 wrote a mathematical novella — Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness — introducing Conway’s system to the world, a year before Conway’s own book appeared. The surreals contain all the real numbers, all infinite quantities, all infinitesimals, and much more — built from a single, elegant recursive definition that starts with nothing.

Everything From Nothing: The Construction

Every surreal number is written as $\{L \mid R\}$ , where $L$ (the left set) and $R$ (the right set) are sets of surreal numbers already constructed, with every element of $L$ less than every element of $R$ .

On Day 0, no surreal numbers have been constructed yet. The only available option is both sets empty:

$0 = \{\mid\}$

This is the surreal number zero. We declare it to be neither positive nor negative — it is the simplest number, born from emptiness.

On Day 1, we can use 0 in our sets:

$1 = \{0 \mid\} \qquad -1 = \{\mid 0\}$

The number $\{0 \mid\}$ is defined to be the simplest number greater than 0 — which is 1. The number $\{\mid 0\}$ is the simplest number less than 0 — which is −1.

On Day 2:

$2 = \{1 \mid\} \qquad -2 = \{\mid -1\} \qquad \frac{1}{2} = \{0 \mid 1\}$

The number $\{0 \mid 1\}$ is the simplest number greater than 0 and less than 1 — which is 1/2.

On Day 3: integers 3 and −3, and fractions 1/4, 3/4, 3/2, …

By Day $n$ : all integers up to $n$ and $−n$ , and all fractions with denominators that are powers of 2 (dyadic rationals) with numerators up to $2^n$ .

After infinitely many days (Day ω): all real numbers. Then, on Day ω+1, something new appears.

The First Infinite Number

On Day ω+1, we can form:

$\omega = \{0, 1, 2, 3, 4, \ldots \mid\}$

The left set contains all natural numbers; the right set is empty. The simplest surreal greater than all natural numbers is ω — the first infinite surreal number.

ω is not the same as ∞ used informally in calculus. It is a genuine number, obeying the arithmetic rules of the surreals:

$\omega + 1 = \{1, 2, 3, \ldots, \omega \mid\}$ $\omega - 1 = \{0, 1, 2, \ldots \mid \omega\}$ $2\omega = \{\omega, \omega+1, \omega+2, \ldots \mid\}$ $\frac{\omega}{2} = \{0, 1, 2, \ldots \mid \omega, \omega-1, \omega-2, \ldots\}$

These are all distinct surreal numbers. Infinite arithmetic works consistently. You can add, subtract, multiply, and divide infinities just as you do real numbers — the algebra is coherent.

The First Infinitesimal

On the same day ω+1, something equally remarkable appears on the other side:

$\varepsilon = \{0 \mid 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\}$

This number is greater than 0 but less than every positive real number (less than 1/2, less than 1/4, less than 1/8, …). It is the simplest infinitesimal — a number that is positive but smaller than any real positive number.

Notice that $\varepsilon = 1/\omega$ : the reciprocal of ω. This can be verified directly from the surreal arithmetic. Infinite quantities and infinitesimals are not separate concepts bolted onto the reals — they arise automatically from the same construction, as reciprocals of each other.

The infinitesimals of the surreal numbers are perfectly rigorous — unlike the “infinitely small quantities” that troubled the foundations of calculus in the 17th and 18th centuries. Every surreal number obeys the same arithmetic rules as every other.

The Full Picture: Everything, Together

The surreal number line is a vast extension of the real line. Moving outward from 0:

All integers: …, −3, −2, −1, 0, 1, 2, 3, …
All rationals between them: 1/2, 1/3, 1/4, …
All irrationals: √2, π, e, …
All real numbers, forming a continuum
Then infinitesimals: ε, ε/2, ε², …
And beyond: ω, ω+1, ω·2, ω², ω^ω, …
And their negatives on the left

Every ordinal number — the mathematical objects that describe transfinite quantities in set theory — embeds in the surreals. So does every non-standard real in the sense of Abraham Robinson’s non-standard analysis. The surreals contain all of these as a single unified system.

In fact, Conway proved a striking theorem: the surreal numbers form a proper class (too large to be a set in standard set theory) that contains every ordered field as a subfield. They are, in a precise sense, the largest possible ordered field.

Arithmetic of the Surreals

Surreal addition and multiplication are defined recursively, following the construction. The definitions are more complex than the construction of the numbers themselves — but they satisfy all the expected properties: commutativity, associativity, distributivity, and existence of additive and multiplicative inverses (except for 0).

Some striking computations:

ω is not a real number, but arithmetic with it works: $(\omega - 1) \cdot 2 = 2\omega - 2$ $\sqrt{\omega} = \{\sqrt{1}, \sqrt{2}, \sqrt{3}, \ldots \mid \omega, \omega/2, \omega/4, \ldots\}$

Yes, square roots of infinite surreals exist and are themselves surreal numbers.

ε satisfies: $\varepsilon^2 < \varepsilon < 1$

The square of an infinitesimal is an even smaller infinitesimal — exactly as one would hope.

The surreal number $\{1, 2, 3, \ldots \mid \omega\}$ is a surreal number strictly between all finite integers and ω — an “almost infinite” quantity with no counterpart in the real numbers.

Connection to Games

Conway developed surreal numbers in the context of combinatorial game theory — the mathematical analysis of perfect-information, two-player games like Go, Nim, and Hackenbush. The connection is not superficial.

Every combinatorial game is defined by a set of positions reachable by the Left player and a set reachable by the Right player — exactly the structure $\{L \mid R\}$ of a surreal number. Some games are equal to surreal numbers: they have a definite numerical value. Others are genuinely “fuzzy” — neither player has an advantage, but the game is not equal to 0.

The surreal numbers are exactly those games where $L < R$ for all elements of the sets — where the left positions are genuinely better for Left and the right positions better for Right. Fuzzy games (where $L \not< R$ in some cases) are not surreals, but they are numbers in an even broader sense: combinatorial games form a structure called a surreal number field with games, which properly contains the surreals.

This connection means that surreal number arithmetic translates directly into game strategy. A game worth $1/2$ means: in the endgame, playing in this position is worth half a move to Left. Infinite and infinitesimal game values correspond to positions where one player has an overwhelming or negligible advantage.

Calculus With Infinitesimals

Newton and Leibniz invented calculus using infinitesimals — quantities dx “infinitely small.” For over a century, no one could make this rigorous, and eventually the infinitesimals were replaced by limits. Abraham Robinson showed in 1960 that infinitesimals could be made rigorous using non-standard analysis — an approach via mathematical logic.

Surreal numbers offer a different home for infinitesimals. Since every surreal infinitesimal like ε satisfies 0 < ε < r for every positive real r, calculus can be reformulated using surreal infinitesimals directly.

For example, the derivative of $f(x) = x^2$ :

$\frac{f(x + \varepsilon) - f(x)}{\varepsilon} = \frac{(x+\varepsilon)^2 - x^2}{\varepsilon} = \frac{2x\varepsilon + \varepsilon^2}{\varepsilon} = 2x + \varepsilon$

The answer is $2x + \varepsilon$ . Since ε is infinitesimal, we “take the standard part” — drop the infinitesimal — and get $2x$ . This is exactly the derivative, recovered through infinitesimal arithmetic without limits.

The surreals thus rehabilitate, in rigorous form, the intuitive approach to calculus that Newton and Leibniz had in mind.

Conway: The Mind Behind the Surreals

John Horton Conway (1937–2020) was one of the most playful and prolific mathematicians of the 20th century. He invented the Game of Life (the cellular automaton), made fundamental contributions to group theory (including the Monster group — the largest sporadic simple group), knot theory, sphere packing, and combinatorial game theory.

Conway said he discovered the surreal numbers in a single day — though developing the theory fully took years. He was reportedly dismissive of the name “surreal numbers” (preferring “numbers” or “Conway numbers”) but the name stuck after Knuth’s novella.

His book On Numbers and Games (1976) remains the definitive reference, written in a conversational style unusual for advanced mathematics. Conway was known for doing serious research while playing games — and for blurring the line between the two so completely that the distinction ceased to matter.

Conclusion: The Largest Number System

The surreal numbers are an answer to a question nobody quite asked: is there a single number system that contains all real numbers, all transfinite ordinals, all infinitesimals, and all the exotic quantities in between?

The answer is yes, and it was found in a game of Go.

What makes the surreals remarkable is not just their size — it is their simplicity. The definition $\{L \mid R\}$ is a single recursive rule. From this rule, following nothing but logic, the entire structure emerges: every real number, every infinite cardinal, every infinitesimal, every ordinal. All of arithmetic, recovered from emptiness.

Conway once described mathematics as the study of what is “interesting.” By that measure, the surreals are extremely interesting indeed.

Related topics: Combinatorial game theory, non-standard analysis, ordinal numbers, foundations of mathematics, Conway’s Game of Life

Frequently asked

What are surreal numbers?

Surreal numbers are a number system invented by John Conway that contains all real numbers, all ordinal numbers (which describe infinite sizes), and all infinitesimals (numbers smaller than any positive real but greater than zero) within a single coherent framework. Every surreal number is defined by two sets: a set of numbers it is 'greater than' (left set) and a set of numbers it is 'less than' (right set).

How were surreal numbers discovered?

John Conway discovered them in 1970 while analyzing endgame positions in the board game Go. Certain Go positions decomposed naturally into sums of simpler positions, and Conway realized the positions themselves formed a number-like system. The full theory was published in his 1976 book 'On Numbers and Games.' Donald Knuth popularized the discovery by writing a mathematical novella called 'Surreal Numbers' (1974), the first published account.

What is an infinitesimal in the surreal numbers?

An infinitesimal is a surreal number greater than zero but smaller than every positive real number. The simplest infinitesimal is ε = {0 | 1, 1/2, 1/4, 1/8, ...} — it is greater than 0 but less than every positive real. Its reciprocal 1/ε is an infinite surreal number larger than every real. Unlike the informal infinitesimals of early calculus, surreal infinitesimals are rigorously defined and obey all the usual arithmetic rules.

Are surreal numbers used in practice?

Surreal numbers are primarily a theoretical construction with deep connections to combinatorial game theory and foundations of mathematics. They provide a rigorous framework for reasoning about infinite and infinitesimal quantities. In combinatorial game theory, they give exact values to abstract games. In model theory, they are related to non-standard analysis. Practical applications are limited, but their theoretical unification of disparate number concepts is considered a significant achievement.