Right, I saw these mentioned on the internet, and read up a bit about them, they're an alternate algebraic closure of the rational field.

So, after my maths teacher, an obsessive pure-maths sort I'll call T, mentions he'll be doing infinities in the next class (I'm in the top set of double maths, so he can afford to waste lessons on fun stuff like that). He also runs maths club, which I don't normally go to, because it'd mean walking back to college after I've gone home, and I'm lazy.

But, I thought it might be worth mentioning to him the concept of surreal numbers, inside of which can be constructed not only the real numbers, but infinite numbers, and infinitesimal numbers. He patiently listens to me go over the concept, and then says "Cool." Then I suggest that it would be a good concept to cover in maths club. He then says that's a good idea, and would I do the talk on surreal numbers, as I am the only one in college who understands them.

Now, my normal reaction when faced with the prospect of talking to a large group of people is to bolt for the door, but I decided that it might be worth it this time, what with the whole gaining the respect of my fellow nerds thing. So I agreed, and I just need to get my notes in order for the talk, and then give them a few weeks notice to publicise (!) it. Not only that, but my mum, a psychology teacher, and her friend, a sociology teacher, are going to show up. That ought to be fun.

So, that's my nervous involvement with them, and as well as making this post not completely useless for non-maths types, explains why I'm discussing such an esoteric subject all of a sudden.

Now for the maths:

This subject makes heavy use of set notation, and I have tried to make this as painless as possible, but I'm warning you not to continue if you have a headache.

There are two important definitions in the field of surreal numbers, the two you absolutely must have to get started.

**Definition of a surreal number**(Note: S is the set of surreal numbers, no blackboard bold, sorry. and the ∋ sign is the wrong way round, but it would be even more confusing to write these out backwards, so sorry, just keep on your mental toes)

{L|R} ∋ S ⇔ ∀l∋L:l∋S ^ ∀r∋R:r∋S ^ ∀l∋L∀r∋R:¬(r≤l)

**Definiton of less than or equal to**(note: Xl means the left set of the surreal number x, similarly with Xr)

x≤y ⇔ ¬∃m ∋ Xl:y≤m ^ ¬∃n ∋ Yr:n ≤ x

Now, these definitions may seem circular, but that's because they are. Mathematicians prefer "iterative", however.

Now, the simplest set possible is the empty set, {}, which is also represented by the symbol ∅.

So, let us consider the set {∅|∅}. Is this a surreal number? Well, first let us consider the statement

∀x∋∅:x∋S

Is it true? Yes, trivially so. Any condition is true for all members of the empty set. That covers the first and second part of the surreal number definition. Now let us look at the third:

∀x∋∅∀y∋∅:¬(y≤x)

Again, trivially true, as there are no members in the empty set.

So, {∅|∅} is a surreal number, and to save time, I shall assign it the label 0. At the moment these labels are arbitrary, but when it comes to mapping the reals as a subset of the surreals, they will be meaningful (and it can be done, the reals are a proper subset of the surreals).

Now, an important question: is 0≤0? Well, considering that 0={∅|∅}, and applying the definition of ≤, we get:

0≤0 ⇔ ¬∃n ∋ ∅:0≤xL ^ ¬∃m ∋ ∅:m ≤ 0

Both halves are trivially true. There does not exist an element in the empty set. So, 0≤0, and if we now establish a few conventions:

x≤y ⇔ y≥x

x≤y ^ ¬(y≤x) ⇔ x<y

x<y ⇔ y>x

x≤y ^ y≤x ⇔ x=y

These are definitions, and so need no justification. Using these conventions, we can say than 0=0. Interesting, huh?

Well, now let's get to creating more numbers. Using 0 and ∅, we can create:

{{0}|∅}

{{0}|{0}}

{∅|{0}}

They are all sets of surreal numbers, but the middle one is not well formed, as 0≤0. The others are well formed, as there is nothing in the null set to be less then or equal to.

Let's have a look at {{0}|∅}. Specifically, is {∅|∅}≤{{0}|∅}? For this to be so, these two conditions must be met:

¬∃m ∋ ∅:{∅|{0}}≤m

¬∃n ∋ {0}:n ≤ ∅

First is true, as there does not exist an element in the empty set. Second is true, because any comparisons to the empty set are false.

So, {∅|∅}≤{{0}|∅}. But, is {{0}|∅}≤{∅|∅}? Well, lets check:

¬∃m ∋ {0}:0≤m

¬∃n ∋ ∅:n ≤ {{0}|∅}

The first is false, as 0≤0. Therefore, the statement {{0}|∅}≤{∅|∅} is false, and by our conventions, we can say 0<{{0}|∅}. So, let us label this number 1. Lo! We hath achieved much. From nothing, and structure, we have created unity.

Anyway, glossing over the details, {∅|{0}}=-1, {{1}|∅}=2.

In general, to map a real integer to a surreal integer,

if x=0, f(x)={∅|∅}

if x>0, f(x)={f(x-1)|∅}

if x<0, f(x)={∅|f(x+1)}

Anyway, integers are boring, lets look at a couple of more interesting combinations: first, {{0}|{1}}. Skipping the proof, 'cause I'm lazy,

{∅|∅}<{{0}|{1}}<{{0}|∅}

So it shall be called

^{1}/

_{2}. This is a good label, because when surreal addition is defined, {{0}|{1}}+{{0}|{1}}=1.

Anyway, now {{-1}|{1}}. {{-1}|{1}}=0; that is to say, {{-1}|{1}}≤0 ^ 0≤{{-1}|{1}}.

That's a significantly surprising result that it justifies proving. So, in academic tradition, I'll leave it as an exercise to the reader.

Anyway, you probably get the idea that we can represent all integers in surreal number form, and

^{1}/

_{2}. But we can do

^{1}/

_{8}, and

^{1}/

_{16} too. Now, these statements would take a definition of multiplicative inverse over the surreals to justify, but let me just state the mapping for all fractions

^{m}/

_{2n}:

f(x)={f(

^{m-1}/

_{2n})|f(

^{m+1}/

_{2n})}

Now, one more definition: addition. Addition is the basic operation of arithmetic, and all others can be derived from it. Over the surreals it is defined as:

x+y={Xl+y, x+Yl|Xr+y, x+Yr}

Again, this is circular. But, as any operation done to the null set results in the null set, it is usable. Take 1+2,

{{0}|∅}+{{1}|∅}={{0}+{{1}|∅}, {{0}|∅}+{1}|∅+{{1}|∅}, {{0}|∅}+∅}

Which simplifies to:

{{0}+{{1}|∅}, {{0}|∅}+{1}|∅}

Which leaves two sums,

0+{{1}|∅}={∅+{{1}|∅}, {0}+{1}|∅+{{1}|∅}, 0+∅}

={{0}+{1}|∅}

{{0}|∅}+1={{0}+{1}, {1}+∅|∅+{1},{1}+∅}

={{0}+{1}|∅}

Which leaves us with one sum needed to do, 0+1.

{∅|∅}+{{0}|∅}={∅+{1}, {0}+{0}|∅+{1},{0}+∅}

={{0}+{0}|∅}

now, 0+0:

{∅|∅}+{∅|∅}={∅+{0}, {0}+∅|∅+{0}, {0}+∅}

={∅|∅}

=0

now, substituting back in:

0+1={{0}|∅}=1

0+2={{1}|∅}=2

1+1={{1}|∅}=2

1+2={{0}+{2}, {1}+{1}|∅}={{2},{2}|∅}

Which, dues to a property of surreal numbers, is equal to {{2},∅}. A sensible name for which, from our mapping, is 3.

Yes, I know it's a lot simpler to calculate in real arithmetic, but that's not the point. Anyway, S is abelian over +. And the number that was called '-1' really is the additive inverse of 1. Subtraction is merely shorthand for adding negative numbers.

And multiplication is shorthand for repeated addition. So, essentially, they are all defined. Yes, this is a sloppy job, but it's good enough for an introductory text.

Now, a few rules for the surreals (all have been proved, but I wouldn't want to spoil your fun by depriving you of mathematical exercise):

- The simplification theorem: you can remove all but the biggest of L and the smallest of R without changing the value of a surreal number.
- A surreal number is greater than all members of its left set and lesser than all members of its right set.
- A surreal number is equal to the oldest number between the largest member of its left set, and the smallest member of its right set. (oldest means the one that you would have to go through the least number of generative iterations to get to)

Anyway, this could all be done in the reals to far, so let me show you something interesting:

Consider the set Z

_{s}, the set of surreal integers. It is defined thusly:

0∋Z

_{s}n∋Z

_{s}⇒{n|∅}∋Z

_{s}n∋Z

_{s}⇒{∅|n}∋Z

_{s}As you can probably tell, this is identical to the set Z of integers. So, I'll call it Z for simplicity's sake. Anyway, consider the number: {Z|∅}. And, yes, it is a number. It meets all the requirements. But what is its value?

Well, the set Z contains all numbers that can be created in the form 1+1+1+... Therefore, by the second bulletpoint, it is greater than them. So therefore, it must be infinity. Oh, I'm sure you've had your maths teacher tell you infinity is not a number. But that's in the reals. But, we don't call it infinity. Infinity is too vague. The proper name for this number is ω. It is an ordinal. All ordinal infinities can be constructed in the surreals. You can calculate ω-1, ω+ω, ω

^{2}, ω

^{ω}, etc. And the smallest ordinal not constructible using ω, by using the set of all ordinals constructible from ω in the left set. It is called ω

_{1}, and the smallest ordinal not constructible using ω

_{1} is called ω

_{2}. Neat, eh? You could make up a complete set of ordinal arithmetic.

But also, consider the set D of unit dyadic fractions:

1∋D

n∋D ⇒

^{1}/

_{2n} ∋D

i.e., {1,

^{1}/

_{2},

^{1}/

_{4},

^{1}/

_{8}...}

Now, consider the number {∅|D}. What value does it have? Well, it is smaller than all the unit dyadic fractions. But whatever positive fraction you pick, there will be a member of D that is smaller. So, this number is smaller than all possible fractions of a unit. But, it is bigger than zero. This is an infinitesimal number, which I shall call ε. Even though ε is smaller than all possible fractions, you can still have

^{ε}/

_{2}. You can have 2ε, 1+ε, and ωε (which is equal to one). This creates another whole section of arithmetic to play with.

And this also leads to a real, genuinely interesting exercise for the reader: How do you represent a non-dyadic fraction as a surreal number? Of course, like all my 'exercises for the reader', you could look this up on Google. That's why this has so many holes in, it's just an introduction. But, if you think you've followed what I've explained so far, working this one out might actually be worth your time.

Neat, eh?