Surreal numbers are defined as an
arithmetic continuum that do not just accommodate real numbers but infinite and
infinitesimal numbers as well. What this indicates is that between two numbers,
there is a gap that can be filled. This gap is then filled with the simplest
number. Before elaborating on this, some examples of the format for surreal
numbers should be given. Consider the following: the subsets consist of the
members L and R and are represented by the form {L|R} where L is always less
than R. Again, the purpose of this is to find the simplest number between L and
R. Some examples would be:
{0| } = 1 { |0} = -1 {0|1}
= ½
{1| } = 2 { |-1} = -2 {0|
½ } = ¼
{2| } = 3 { |-2} = -3 {
½ |1} = ¾
As the
examples above show, the answer cannot just be any number between L and R but
the middle number since the middle number is the simplest number. In the cases
for the first and second columns of examples, the answers are essentially the
simplest number that would fill the blank space.
In this chapter, there are
multiple types of two player games that use surreal numbers as guidelines for
the games. These games are very simple in the fact that the only way to lose is
by not being able to make a move. For example, one of the games is played with
dominoes and one of the scenarios can be represented by the following crude
diagram:
[ ]
[ ] (one domino takes up two boxes)
[ ][ ]
If the game
is played with L placing the dominoes vertically and R placing the dominoes
horizontally, L can block R by placing a domino in the lowermost vertical
position which would leave a position value of zero and since R only has one
possible move in the scenario, it has a position value of one. The score is
then represented by the notation {0|1} and is defined as ½ ; therefore, left
wins since the position counts as half a move in favor of left (left wins when
the notation is defined as positive and right wins when the notation is defined
as negative).
I
found this chapter to be quite confusing due to the fact that you needed to
have previous knowledge of a game called Nim in order to understand the rest of
the games. Despite that, I also found the chapter very fascinating because it
was a concept that my friend and I have previously discussed over the summer
without any knowledge of surreal numbers. The concept revolved around infinity
and most people think of infinity as being something along the lines of
1,2,3,4,5,6,7,8,9, etc. Not many people think about the idea that there are an
infinite amount of numbers between 0 and 1 due to decimal places. While we
discussed this, we essentially thought that since there are an infinite amount
of decimals between each whole number and an infinite amount of whole numbers
that our perception of infinity was really infinity times infinity (just think
about that for a second). While infinity times infinity isn’t possible, it’s
still kind of a cool concept. Food for thought I guess…
I found this chapter confusing but your summary helped make it more understanding. I liked your explanation and examples of the {L|R} form and the crude diagram. It was very interesting.
ReplyDeleteThe topic is interesting and the concept is easy to understand but its applications still remain confusing to me. Good summary though!
ReplyDeleteGreat job summarizing the chapter, it helped me understand the reading a lot better. The examples you included were very cool and interesting.
ReplyDelete