Editing Surreal number (section)

==Arithmetic==

The addition, negation (additive inverse), and multiplication of surreal number ''forms'' {{math|1=''x'' = {{mset| ''X''{{sub|''L''}} {{!}} ''X''{{sub|''R''}} }}}} and {{math|1=''y'' = {{mset| ''Y''{{sub|''L''}} {{!}} ''Y''{{sub|''R''}} }}}} are defined by three recursive formulas.

===Negation===

Negation of a given number {{math|1=''x'' = {{mset| ''X''{{sub|''L''}} {{!}} ''X''{{sub|''R''}} }}}} is defined by
<math display=block>-x = - \{ X_L \mid X_R \} = \{ -X_R \mid -X_L \},</math>
where the negation of a set {{mvar|S}} of numbers is given by the set of the negated elements of {{mvar|S}}:
<math display=block>-S = \{ -s: s \in S \}.</math>

This formula involves the negation of the surreal ''numbers'' appearing in the left and right sets of {{mvar|x}}, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form.  This makes sense only if the result is the same, irrespective of the choice of form of the operand.  This can be proved inductively using the fact that the numbers occurring in {{math|''X''{{sub|''L''}}}} and {{math|''X''{{sub|''R''}}}} are drawn from generations earlier than that in which the form {{mvar|x}} first occurs, and observing the special case:
<math display=block>-0 = - \{ {}\mid{} \} = \{ {}\mid{} \} = 0.</math>

===Addition===
The definition of addition is also a recursive formula:
<math display=block>x + y = \{ X_L \mid X_R \} + \{ Y_L \mid Y_R \} = \{ X_L + y, x + Y_L \mid X_R + y, x + Y_R \},</math>
where

<math display=block>X + y = \{ x' + y: x' \in X \} , \quad x + Y = \{ x + y': y' \in Y \}</math>

This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:
<math display=block>0 + 0 = \{ {}\mid{} \} + \{ {}\mid{} \} = \{ {}\mid{} \} = 0</math>
<math display=block>x + 0 = x + \{ {}\mid{} \} = \{ X_L + 0 \mid X_R + 0 \} = \{ X_L \mid X_R \} = x</math>
<math display=block>0 + y = \{ {}\mid{} \} + y = \{ 0 + Y_L \mid 0 + Y_R \} = \{ Y_L \mid Y_R \} = y</math>
For example:
:{{math|1={{sfrac|1|2}} + {{sfrac|1|2}} = {{mset| 0 {{!}} 1 }} + {{mset| 0 {{!}} 1 }} = {{mset| {{sfrac|1|2}} {{!}} {{sfrac|3|2}} }}}},
which by the birthday property is a form of 1. This justifies the label used in the previous section.

====Subtraction====
Subtraction is defined with addition and negation:
<math display=block>x - y = \{ X_L \mid X_R \} + \{ -Y_R \mid -Y_L \} = \{ X_L - y, x - Y_R \mid X_R - y, x - Y_L \}\,.</math>

===Multiplication===
Multiplication can be defined recursively as well, beginning from the special cases involving 0, the [[multiplicative identity]] 1, and its additive inverse −1:
<math display=block>\begin{align}
xy & = \{ X_L \mid X_R \} \{ Y_L \mid Y_R \} \\
   & = \left\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R \mid X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \right\} \\
\end{align}</math>
The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression <math display=inline>X_R y + x Y_R - X_R Y_R</math> that appears in the left set of the product of {{mvar|x}} and {{mvar|y}}. This is understood as <math display=inline>\left\{ x' y + x y' - x' y' : x' \in X_R,~ y' \in Y_R \right\}</math>, the set of numbers generated by picking all possible combinations of members of <math display=inline>X_R</math> and <math display=inline>Y_R</math>, and substituting them into the expression.

For example, to show that the square of {{sfrac|1|2}} is {{sfrac|1|4}}:
:{{math|1={{sfrac|1|2}} ⋅ {{sfrac|1|2}} = {{mset| 0 {{!}} 1 }} ⋅ {{mset| 0 {{!}} 1 }} = {{mset| 0 {{!}} {{sfrac|1|2}} }} = {{sfrac|1|4}}}}.

=== Division ===
The definition of division is done in terms of the reciprocal and multiplication:

<math display=block>\frac xy = x \cdot \frac 1y</math>

where<ref name="Con01" />{{rp|21}}

<math display=block>\frac 1y = \left\{\left.0, \frac{1+(y_R-y)\left(\frac1y\right)_L}{y_R}, \frac{1+\left(y_L-y\right)\left(\frac1y\right)_R}{y_L} \,\,\right|\,\, \frac{1+(y_L-y)\left(\frac1y\right)_L}{y_L}, \frac{1+(y_R-y)\left(\frac1y\right)_R}{y_R} \right\}</math>

for positive {{mvar|y}}. Only positive {{math|''y''{{sub|''L''}}}} are permitted in the formula, with any nonpositive terms being ignored (and {{math|''y''{{sub|''R''}}}} are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of {{mvar|y}}, but also recursion in that the members of the left and right sets of {{math|{{sfrac|1|''y''}}}} itself. 0 is always a member of the left set of {{math|{{sfrac|1|''y''}}}}, and that can be used to find more terms in a recursive fashion. For example, if {{math|1=''y'' = 3 = { 2 {{!}} }}}, then we know a left term of {{sfrac|1|3}} will be 0. This in turn means {{math|1={{sfrac|1 + (2 − 3)0|2}} = {{sfrac|1|2}}}} is a right term. This means
<math display=block>\frac{1+(2-3)\left(\frac12\right)}2=\frac14</math>
is a left term. This means
<math display=block>\frac{1+(2-3)\left(\frac14\right)}2 = \frac 38</math>
will be a right term. Continuing, this gives
<math display=block>\frac13 = \left\{\left. 0, \frac14, \frac5{16}, \ldots \,\right|\, \frac12, \frac38, \ldots\right\}</math>

For negative {{mvar|y}}, {{math|{{sfrac|1|''y''}}}} is given by
<math display=block>\frac1y=-\left(\frac1{-y}\right)</math>

If {{math|1=''y'' = 0}}, then {{math|{{sfrac|1|''y''}}}} is undefined.

===Consistency===
It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
* Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday {{mvar|n}} will eventually be expressed entirely in terms of operations on numbers with birthdays less than {{mvar|n}};
* Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday {{mvar|n}} will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than {{mvar|n}};
* As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
* The operations can be extended to ''numbers'' (equivalence classes of forms): the result of negating {{mvar|x}} or adding or multiplying {{mvar|x}} and {{mvar|y}} will represent the same number regardless of the choice of form of {{mvar|x}} and {{mvar|y}}; and
* These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a [[field (mathematics)|field]], with additive identity {{math|1=0 = {{mset| {{!}} }}}} and multiplicative identity {{math|1=1 = {{mset| 0 {{!}} }}}}.

With these rules one can now verify that the numbers found in the first few generations were properly labeled.  The construction rule is repeated to obtain more generations of surreals:

: {{math|1=''S''{{sub|0}} = {{mset| 0 }}}}
: {{math|1=''S''{{sub|1}} = {{mset| −1 < 0 < 1 }}}}
: {{math|1=''S''{{sub|2}} = {{mset| −2 < −1 < −{{sfrac|1|2}} < 0 < {{sfrac|1|2}} < 1 < 2 }}}}
: {{math|1=''S''{{sub|3}} = {{mset| −3 < −2 < −{{sfrac|3|2}} < −1 < −{{sfrac|3|4}} < −{{sfrac|1|2}} < −{{sfrac|1|4}} < 0 < {{sfrac|1|4}} < {{sfrac|1|2}} < {{sfrac|3|4}} < 1 < {{sfrac|3|2}} < 2 < 3 }}}}
: {{math|1=''S''{{sub|4}} = {{mset| −4 < −3 < ... < −{{sfrac|1|8}} < 0 < {{sfrac|1|8}} < {{sfrac|1|4}} < {{sfrac|3|8}} < {{sfrac|1|2}} < {{sfrac|5|8}} < {{sfrac|3|4}} < {{sfrac|7|8}} < 1 < {{sfrac|5|4}} < {{sfrac|3|2}} < {{sfrac|7|4}} < 2 < {{sfrac|5|2}} < 3 < 4 }}}}

===Arithmetic closure===

For each [[natural number]] (finite ordinal) {{mvar|n}}, all numbers generated in {{math|''S''{{sub|''n''}}}} are [[dyadic fraction]]s, i.e., can be written as an [[irreducible fraction]] {{math|{{sfrac|''a''|2{{sup|''b''}}}}}}, where {{mvar|a}} and {{mvar|b}} are [[integer]]s and {{math|0 ≤ ''b'' < ''n''}}.

The set of all surreal numbers that are generated in some {{math|''S''{{sub|''n''}}}} for finite {{mvar|n}} may be denoted as <math display=inline>S_* = \bigcup_{n \in N}  S_n</math>.  One may form the three classes
<math display=block>\begin{align}
S_{0} &= \{ 0 \} \\
S_{+} &= \{ x \in S_*: x > 0 \} \\
S_{-} &= \{ x \in S_*: x < 0 \}
\end{align}</math>
of which {{math|''S''{{sub|''∗''}}}} is the union.  No individual {{math|''S''{{sub|''n''}}}} is closed under addition and multiplication (except {{math|''S''{{sub|0}}}}), but {{math|''S''{{sub|∗}}}} is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbers {{mvar|β}} for which the set of surreal numbers with birthday less than {{mvar|β}} is closed under the different arithmetic operations.<ref name=vdDE2001>{{cite journal | last1 = van den Dries | first1 = Lou | last2 = Ehrlich | first2 = Philip | author2-link = Philip Ehrlich | title = Fields of surreal numbers and exponentiation | journal = Fundamenta Mathematicae | volume = 167 | issue = 2 | pages = 173–188 | publisher = Institute of Mathematics of the Polish Academy of Sciences | location = Warszawa | date = January 2001 | issn = 0016-2736 | doi = 10.4064/fm167-2-3 | doi-access = free }}</ref> For any ordinal {{mvar|α}}, the set of surreal numbers with birthday less than {{math|1=''β'' = ''ω''{{sup|''α''}}}} (using [[#Powers of ω|powers of {{mvar|ω}}]]) is closed under addition and forms a group; for birthday less than {{mvar|ω{{sup|ω{{sup|α}}}}}} it is closed under multiplication and forms a ring;{{efn|1=The set of dyadic fractions constitutes the simplest non-trivial group and ring of this kind; it consists of the surreal numbers with birthday less than {{math|1=''ω'' = ''ω''{{sup|1}} = ''ω''{{sup|''ω''{{sup|0}}}}.}}}} and for birthday less than an (ordinal) [[Epsilon number (mathematics)|epsilon number]] {{mvar|ε{{sub|α}}}} it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.<ref name=vdDE2001 /><ref name=G1986>{{cite book | last=Gonshor | first=Harry | title=An Introduction to the Theory of Surreal Numbers | year=1986 | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | volume=110 | isbn= 9780521312059 | doi=10.1017/CBO9780511629143 }}</ref>{{rp|at=ch. 10}}<ref name=vdDE2001 />

However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a [[proper class]]. With their ordering and algebraic operations they constitute an [[ordered field]], with the caveat that they do not form a [[Set (mathematics)|set]]. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers.<ref name=bajnok/> The class of all surreal numbers is denoted by the symbol <math display=inline>\mathbb{No}</math>.