Surreal number/The dyadics

Dyadic fractions subdivide an inch

■ Surreal number/The dyadics is still under construction. It will be a very simple introduction that might make surreal numbers seem easy to understand (if not a bit silly.) The figure to the right illustrates how rulers often subdivide the inch into dyadic fractions. The construction of surreal numbers begins with these dyadics (ranging from −∞ to +∞.) Each is assigned "birthday" and is associated with a pair of sets. Curiously, each set consists only of the null set $\{\,\},$ as well as sets that contain the null set in some way. Making matters even more complicated is each surreal can be associated with an endless variety of these pairs of sets.

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Important facts about surreal numbers:

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^?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, $\infty ^{2}-3\infty +1$ (except that it is customary to use $\omega$ instead of $\infty$ to represent one of the many (infinite) versions of infinity associated with surreal numbers.)

About the quotation marks: "0", "1", "1½",...

Most authors emphasize that these numbers are just labels for a pair of sets. My sense is that students should view these labels as actual numbers. In other words, "1" and "1½" might be mere "labels", but I found it impossible to understand any introduction to surreal numbers without knowing that "1" plus "1½" equals "2½". Wikipedia's Surreal number defines addition in a way that seems to require a priori knowledge that these labels act like numbers when added. Conway's title emphasizes that this is "pure math", while my background is in physics. I guess I am like a cabinet maker who like tools more than cabinets ..., but have no interest in knowing how tools are manufactured.

Henceforth, we shall refer to surreal numbers without the quotation marks.

The "birth" of surreal dyadics

Ordered pairs of sets

Fig. 1: Surreal numbers in the interval [−1,4] created up to and including day 4 are shown: On day zero, "0" is created. On day one, "−1" & "+1" are "born". Four more numbers are "born" on day two (not shown is −2.)

Surreal numbers that are also dyadic numbers are defined using ordered pairs of sets (called the left and right set). They cannot be fully understood without a deep understanding of set theory (something this author lacks.) But we might be able to get the gist of these numbers using naive set theory.

Suppose ${\mathcal {L}}$ and ${\mathcal {R}}$ represent an ordered pair of sets, and let ${\mathcal {N}}$ represent a surreal number. For now, let our "universe" consist of the capital letters A-Z. Just for fun, we rank them, with A being somehow the "smallest": A<B<C<...<Z.

In set theory, setting ${\mathcal {L}}$ equal to $\{A,B,C\}$ is the same as setting it equal to $\{C,A,B\}.$ The order in which elements appear has no significance in set theory. On the other hand, we cannot interchange the sets ${\mathcal {L}}$ and ${\mathcal {R}}$ because these sets are part of an ordered pair. It is not uncommon to call such an ordered pair a two-dimensional vector and express it using square brackets. For example,

A

$${\mathcal {N}}={\big [}{\mathcal {L}},{\mathcal {R}}{\big ]}={\big [}\{A,B,C\},\{F,W\}{\big ]}$$

It is convenient to write this as,

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$${\mathcal {N}}=\{{\mathcal {L}}{\big |}{\mathcal {R}}\}=\{A,B,C{\big |}F,W\}$$

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$$0=\left[\{\,\},\{\,\}\right]$$

$$\left\lbrack {\frac {a}{b}}\right\rbrack$$

B

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

"−1"={ |0}  &  "1"={0| } ...(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"={ |−1}  ,  "−½"={−1|−1}  ,  "½"={1|2}  &  "2"={1| } ...(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.