# Forum for Introduction to infinitesimals

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# Please give the calculation rules [#1337]

ThorAvaTahr: Please give the calculation rules (2008-03-21 16:41) [#4536]

I assume the infinitesimal gives a new sort algebra similar to for instance binary algebra.

For me to understand this matter better I would like to know how to work with infinitesimal theory.

Therefore it would like to have the algebraic rules:

^+^=^^ ^+DOWN=0 ^+*=^*

^=-DOWN

Also, at some point it is mentioned that infinitesimal algebra transforms to numbers. But also for the numbers the algebra is missing.

Am I approaching this from the wrong perspective, or can you give this information.

X
Bill: Re: Please give the calculation rules (2008-03-21 19:46) [#4537]

Combinatorial games, including infinitesimals, form a group under addition. As for infinitesimals, * + * = 0, and otherwise, an infinitesimal plus its negative equals 0. Otherwise, infinitesimal expressions do not reduce. Irreducible combinations of infinitesimals are often written without the plus sign. Thus ^* is the same as ^+*.

If you want to delve deeper into CGT, see On Numbers and Games or Winning Ways.

ThorAvaTahr: thx (2008-03-26 11:56) [#4554]

Thx Bill,

One remark about this, it seems to me that you reduced the expression ^+* to ^*, since i can imagine that you have a group that is ^* and that inherently this group equals two different groups, a * and a ^.

So to conclude we can have

^*+^+*+^^-^-^=^

or is this calculation wrong?

and then, when we have a surplus of ^^ we always have the final play, I read that somewhere, is it true?

X
Bill: Re: thx (2008-03-26 16:56) [#4556]

One remark about this, it seems to me that you reduced the expression ^+* to ^*, since i can imagine that you have a group that is ^* and that inherently this group equals two different groups, a * and a ^.

Yes, you can have a group that is ^* in chilled go. And it is equal to ^ + *.

So to conclude we can have

^*+^+*+^^-^-^=^

or is this calculation wrong?

As I count the ^s, it reduces to ^^.

and then, when we have a surplus of ^^ we always have the final play, I read that somewhere, is it true?

I think that has to do with atomic weight. Atomic weight is approximately equal to a number of ^s. An atomic weight of 2 is enough to get the final play.

^ itself is positive, which means that Left can get the last play. ^* also has an atomic weight of 1, but Right can win by playing first. However, ^^* has an atomic weight of 2, and Left wins even if Right plays first.