Surreal number

It is nearly impossible to introduce surreal numbers without either trivializing difficult concepts, or describing these same concepts in a way no novice could possibly understand. What I can do is locate points on the real number line that correspond to some of the surreal numbers. And, I can briefly introduce a few concepts that while peripheral to the topic of surreal numbers, appear to be similar to related topics that are essential to far more difficult concepts that are essential to the topic of surreal numbers. I emphasize "appear to be", because I myself to not fully grasp the more difficult topics. This begs the question of why a non-expert would be attempting to explain surreal numbers: First, we are allowed to do things that the other WMF wikis forbid. Also, in applied mathematics (especially physics) one is permitted to judge a mathematical on the problems it solves and the insight it yields, without worrying too much about rigor.

Finally, a conversation with chatbot Gemini (14 May 2024) informed me that I should study: Set Theory (especially with well-ordered sets like ordinals and proper classes). Ordinal Arithmetic (defines addition and multiplication for infinite numbers). Axiomatic Set Theory (Zermelo-Fraenkel set theory with Choice, and Alternative Set Theory (von Neumann-Bernays-Gödel.)

• Surreal number:
• Set Theory:
• This page introduce fundamentals: Dyadic rational numbers are easy to grasp, and will be introduced sequentially in a way that superficially seem to "fill the number line".

This discussion mixes prerequisite knowledge required by the Wikipedia article with insights that might make surreal numbers more interesting. Our focus is on concepts involving to countability and infinity that are easy to explain. To this end, we employ the language of naive set theory in a way that cannot fully explain or describe surreal numbers. The current author of this resource is not an expert on this subject, and for that reason, corrections and elaborations are welcome. While this might seem like a chaotic way to start a resource, it will generate a number of projects for students that take the form, "Is this really true?".

${\displaystyle {\begin{matrix}&&&&&&&&&&&&&&&1&&&&&&&&&&&&&&&\\&&&&&&&{\tfrac {1}{2}}&&&&&&&&&&&&&&&2&&&&&&&\\&&&{\tfrac {1}{4}}&&&&&&&&{\tfrac {3}{4}}&&&&&&&&1{\tfrac {1}{2}}&&&&&&&&3&&&\\&{\tfrac {1}{8}}&&&&{\tfrac {3}{8}}&&&&{\tfrac {5}{8}}&&&&{\tfrac {7}{8}}&&&&1{\tfrac {1}{4}}&&&&1{\tfrac {3}{4}}&&&&2{\tfrac {1}{2}}&&&&4&\\{\tfrac {1}{16}}&&{\tfrac {3}{16}}&&{\tfrac {5}{16}}&&{\tfrac {7}{16}}&&{\tfrac {9}{16}}&&{\tfrac {11}{16}}&&{\tfrac {13}{16}}&&{\tfrac {15}{16}}&&1{\tfrac {1}{8}}&&1{\tfrac {3}{8}}&&1{\tfrac {5}{8}}&&1{\tfrac {7}{8}}&&2{\tfrac {1}{4}}&&2{\tfrac {3}{4}}&&3{\tfrac {1}{2}}&&5\end{matrix}}}$

The first 31 positive surreal numbers are created on days 1-5, and are all rational dyadic fractions. They range from 1 through 5, and almost half of them are less than 1.

1. The story begins with 0 and dyadic rational fractions, which are ratios are of the form p/q where p is an integer and q=2n, where n is a non-negative integer. These dyadic rationals are shown in figure 1 as as "0", "±1", "±½","±2",…. The quotation marks around the "numbers" will be explained later.
2. In a strange sort of way, these dyadic ratios can define the set of all rational and irrational numbers.
3. The surreal rational and irrational numbers are defined in the language of set theory in a way that has little to do with numbers as we know them.
4. An axiomatic version[1]?of set theory can be used to create definitions that reminds one of entities such as 0/0 and ∞/∞, that virtually all textbooks dismiss as "indeterminant". This variation of set theory can also give meaning to an algebraic expression like, ${\displaystyle \infty ^{2}-3\infty +1}$ (except that it is customary to use ${\displaystyle \omega }$ instead of ${\displaystyle \infty }$ to represent one of the many (infinite) versions of infinity associated with surreal numbers.)

Dyadic rationals are fractions where the denominators are powers of 2, i.e., fractions form p/qn, where p is an integer and n is a positive integer, as shown in figure 2.

For reasons to be explained later, the counting is done in groups known as "birthays" (or simply days when each dyadic rational is "born".)

${\displaystyle 0}$ is born on day 0. On day 1, two numbers are born: ${\displaystyle -1}$, and ${\displaystyle 1}$) are born on day 1. Note that ${\displaystyle 2^{0}=1}$ and ${\displaystyle 2^{1}=2}$, so that on day 2 we have ${\displaystyle 2^{2}=}$ four new numbers: ${\displaystyle -2}$, ${\displaystyle -{\tfrac {1}{2}}}$, ${\displaystyle {\tfrac {1}{2}}}$, and ${\displaystyle 2.}$) Each day the the count doubles, so that on day 3, we obtain eight numbers: ${\displaystyle -3}$, ${\displaystyle -1{\tfrac {1}{2}}}$, ${\displaystyle -2{\tfrac {1}{4}}}$, ${\displaystyle -2{\tfrac {1}{2}}}$, ${\displaystyle 2{\tfrac {1}{4}}}$, ${\displaystyle 2{\tfrac {1}{2}}}$, ${\displaystyle 2{\tfrac {3}{4}}}$, ${\displaystyle 3}$.

The rules for creating new positive numbers are as follows:

1. Each day a new positive integer is created by adding 1 to the previous day's new positive integer.
2. All other new numbers are dyadic fractions created as the midpoint between the fractions created on the previous day.
3. The negative numbers are created in a symmetric fashion. On the same day that p/q is created, −p/q is also created.

It is impossible to create all of the ${\displaystyle 2^{n}}$ surreal numbers created on day ${\displaystyle n}$ until all ${\displaystyle 2^{n-1}}$ surreal numbers had been created on on day ${\displaystyle n-1.}$ But after the previous day's numbers were created, the next day's numbers can be created in any order.

What does it mean to count numbers?

By definition, "counting" doesn't mean you finish, but that you can create a list that includes every item if you count long enough. Stating a formal definition of "countability" is not a trivial task. But figure 3 illustrate a well-known countable infinity.

The rational numbers are countable
 0 . 9 3 3 2 4 7 0 2 0 ... 0 . 6 3 7 4 9 3 3 6 1 ... 0 . 2 7 7 5 5 2 4 6 6 ... 0 . 4 7 5 5 6 8 2 9 7 ... 0 . 8 3 8 1 4 2 2 4 9 ... 0 . 2 5 2 3 5 2 5 7 6 ... 0 . 8 4 4 2 2 1 1 4 1 ... 0 . 6 8 6 2 1 5 1 8 3 ... 0 . 1 7 8 6 6 2 8 1 5 ... ⋮ . ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ Fig. 4: Failed attempt to count real numbers

Figure 3 shows you can count all fractions of the form ${\displaystyle p/q}$ where ${\displaystyle p}$ and ${\displaystyle q}$ are integers. Simply follow the arrows. You should be able to convince yourself that if ${\displaystyle X}$ is larger than both ${\displaystyle p}$ and ${\displaystyle q}$, then you will reach ${\displaystyle p/q}$ in no more than ${\displaystyle X^{2}}$ steps. This fact is essential to the formal proof that the rational numbers are countable.

Real numbers are not countable

Figure 4 establishes that the real numbers are uncountable. To prove this, imagine that a list of all numbers between 0 and 1 has been created, as shown in the table to the right. It is possible to find an irrational number that is not on this list as follows:

• The first number on the list is 0.9332... . Change the first digit after the decimal point from 9 to 0.
• On the second number (0.6374...), change the second digit from 3 to 4
• On the third number, change the third digit (7) to 8.

You have to do this to infinity, but when you are "done", you will have a number not on your original list. Therefore it is impossible to make an infinite list that contains all the irrational numbers in the interval (0,1). This argument can be found on a website found on Carnegie Mellon University,[2] and it seems to contradict two ideas essential to the construction of surreal numbers:

1. Fractions are countable, while the real numbers are not.
2. The real numbers can be represented by a sequence of (dyadic) fractions.

The resolution of this faux-paradox is to note that irrational numbers and fractions like 1/3 can only be defined as surreal numbers using a sequence of didactic numbers:

${\displaystyle 1/3=\lim _{n\to \infty }a_{n},}$

where ${\displaystyle a_{n}}$ is the n-th element a sequence such as,

${\displaystyle 1/4=0.25,\quad 1/2=0.50,\quad 3/8=0.378,\quad 11/32=0.34375,\quad \ldots }$

Each of these dycactic fractions can be labeled with a unique integer on the day of its birth. But the sequence can only be defined as an infinite collection of integers. For that reason, counting the real numbers is impossible for the same reason that real numbers in decimal form cannot be counted.

Figure 1 equates what look like sets with what look like numbers. This is an attempt to translate the set-theory language into conversational english.

"0"={Φ|Φ} ...(Day zero)
"0" is created. "Nothing" is to the left and "nothing" to the right.
Here, "nothing" means the null set Φ, and to the left (right) means "smaller" ("larger")

"−1"={Φ|0}  &  "1"={0|Φ} ...(Day one)
"−1" is created with "nothing" to the left and "0" to the right.
"1" is created with "0" to the left and "nothing" to the right.

"−2"={Φ|−1}  ,  "−½"={−1|−1}  ,  "½"={1|2}  &  "2"={1|Φ} ...(Day two)

"−2" is created with "nothing" to the left and "−1" to the right.
"−½" is created with "−1" to the left and "0" to the right.
"½" is created with "0" to the left and "1" to the right.
"2" is created with "−1" to the left and "nothing" to the right.

The instructions might have been more clear if the (positive) integer N was declared one unit to the right of N−1. Similarly, it could have been mentioned that "½" was midway between "0" and "1". Perhaps the author wanted to keep the reader in suspense. Or, perhaps this discussion fails to capture the beauty of surreal numbers. The next section suggests surreal numbers are indeed quite dazzling.

The rest of this resource is under construction

This section is under construction. It will summarize some (but not all) of the following Wikipedia articles:

Under construction

${\displaystyle {\begin{matrix}&&&&&&&&&&&&&&&1&&&&&&&&&&&&&&&\\&&&&&&&{\tfrac {1}{2}}&&&&&&&&&&&&&&&2&&&&&&&\\&&&{\tfrac {1}{4}}&&&&&&&&{\tfrac {3}{4}}&&&&&&&&1{\tfrac {1}{2}}&&&&&&&&3&&&\\&{\tfrac {1}{8}}&&&&{\tfrac {3}{8}}&&&&{\tfrac {5}{8}}&&&&{\tfrac {7}{8}}&&&&1{\tfrac {1}{4}}&&&&1{\tfrac {3}{4}}&&&&2{\tfrac {1}{2}}&&&&4&\\{\tfrac {1}{16}}&&{\tfrac {3}{16}}&&{\tfrac {5}{16}}&&{\tfrac {7}{16}}&&{\tfrac {9}{16}}&&{\tfrac {11}{16}}&&{\tfrac {13}{16}}&&{\tfrac {15}{16}}&&1{\tfrac {1}{8}}&&1{\tfrac {3}{8}}&&1{\tfrac {5}{8}}&&1{\tfrac {7}{8}}&&2{\tfrac {1}{4}}&&2{\tfrac {3}{4}}&&3{\tfrac {1}{2}}&&5\end{matrix}}}$

proof of conways simplicity rule

${\displaystyle 0=\{|\}}$

${\displaystyle n+1=\{n|\}}$

${\displaystyle -n-1=\{|-n\}}$

${\displaystyle {\tfrac {2p+1}{2^{q+1}}}=\left\{{\tfrac {p}{2q}}|{\tfrac {p+1}{2q}}\right\}}$

Limits and analysis

Infinity is easy to imagine, but difficult to incorporate

Infinity is easy to imagine, but difficult to incorporate into rigorous mathematics. The following calculation certainly violates the rules of mathematics:

${\displaystyle \lim _{\epsilon \to 0}{\frac {1}{\epsilon }}=\infty \;}$ and ${\displaystyle \;\lim _{\epsilon \to 0}{\frac {2}{\epsilon }}=\infty \;}$ implies ${\displaystyle \;{\frac {\infty }{\infty }}=2.}$

This is why expressions like ${\displaystyle \infty /\infty }$ and ${\displaystyle 0/0}$ of often called indeterminant. Most of the time, mistakes like this can be avoided by utilizing concepts taught in a course on mathematical analysis. Both the Wikipedia article and a query sent to an online chatbot suggest that no widely known problems in this field have been solved using surreal numbers. For that reason, any forrey into surreal numbers should probably be viewed as recreational.

Categories

1. The question mark ? is needed here because currently the only reference is the ChatBot Bard, which is known to be unreliable. (See Chatbot math for examples of recent false statements made by online chatbots.) If you know of a reliable reference, please insert it here!
2. https://www.math.cmu.edu/~wgunther/127m12/notes/CSB.pdf