Surreal numbers
E29943
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
Aliases (4)
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Conway number
→
number system → ordered field → proper class → |
| alternativeName |
Conway numbers
→
|
| birthProcess |
numbers are created in stages indexed by ordinals
→
|
| constructedFrom |
left set and right set of earlier numbers
→
|
| containsAsSubfield |
ordinal numbers
→
real numbers → |
| containsElement |
-1
→
0 → 1 → 1/ω (a positive infinitesimal) → all dyadic rationals → all ordinal numbers → all real numbers → ω (first infinite ordinal as a number) → |
| definedBy |
transfinite recursion
→
|
| extends |
ordered fields
→
real numbers → |
| generalizes |
Dedekind-complete ordered fields
→
|
| hasCanonicalForm |
Conway normal form
→
|
| hasOrderType |
proper class length
→
|
| hasProperty |
every set of surreals has a greatest lower bound
→
every set of surreals has a least upper bound → no maximal element → no minimal element other than zero → real-closed field → totally ordered → |
| hasSubset |
day-n numbers (numbers born on day n in the construction)
→
|
| includes |
infinite numbers
→
infinitesimal numbers → |
| introducedBy |
John Horton Conway
→
|
| introducedInPublication |
On Numbers and Games
→
|
| introducedInYear |
1974
→
|
| isClassOf |
all numbers generated from the empty set by Conway’s rules
→
|
| isModelOf |
ordered field axioms
→
real-closed field axioms → |
| relatedConcept |
Hahn series
→
hyperreal numbers → nonstandard analysis → |
| satisfiesCondition |
every left element is less than every right element
→
|
| supportsOperation |
addition
→
division → exponentiation (partial) → multiplication → subtraction → |
| usedIn |
combinatorial game theory
→
|
Referenced by (7)
| Subject (surface form when different) | Predicate |
|---|---|
|
Donald E. Knuth
("Surreal Numbers")
→
John H. Conway → John H. Conway → John Horton Conway → |
notableWork |
|
Surreal numbers
("Conway numbers")
→
|
alternativeName |
|
Surreal numbers
("Conway normal form")
→
|
hasCanonicalForm |
|
On Numbers and Games
("surreal numbers")
→
|
topic |