Surreal number/Simple hackenbush

Surreal number (subpages/draftspace)

The construction of surreal numbers with birthdays yields an obvious way to count the real dyadic numbers. Here we look at a plausibility argument for a pair of sequences of dyadic numbers that converge to ${\sqrt {2}}$ , one from above, and one from below. I like plausibility arguments because sometimes you have to read an entire book to understand a proof.

If a mathematical statement is both plausible and useful, I honestly don't care if it is true.

The parent page Surreal number described the subset of all surreal numbers associated with ordinary real numbers. Oddly, all such "real" surreal numbers includes the irrationals, despite the fact that this subset can be written as didactic rationals:

${\frac {m}{2^{n}}}$

Two sequences converge to root 2

Hackenbush

https://hackenbush.xyz/ w:Hackenbush

See also this MUST READ: [1] saved as Bartlett.pdf

Stalk rule

In fact, assuming that the base segment of the stalk is blue (for red, just take the negatives of these numbers), there is a simple algorithm to calculate the value of a stalk:

Count the number of blue segments that are connected to the in one continuous path. If there are n of them, start with the number n.
For each new segment going up, assign the value of that segment to be half of the one below it, and add it to the sum if it is blue, and subtract if it is red. :#When you reach the top of the stalk, that’s your final value.

For example, consider the stalk whose segments are, starting from the ground: BBBRRBRRBR. We begin with 3 because of the three blue segments. The next red adds −1/2, the next red adds −1/4, the following blue adds +1/8, and so on. Thus the value for this stalk will be:

3 − 1/ 2 − 1/ 4 + 1/ 8 − 1 /16 − 1/ 32 + 1 /64 − 1/ 128

Conversion to base 2 expression of a positive surreal number

It is customary to associate blue with a + sign and red with a − sign. If the string is all + signs, the surreal number is length of the string. For example +++ equals three; converting from base 10 to base 2, we have: +++=3₁₀ = 11₂.

The first minus sign converts the +− into a decimal point. This causes, +++−, to smaller than three because the third minus sign now serves to produce the decimal point. For that reason +++−=2.??₁₀ = 10.??₂. Here the ?? represents unknown digits after what is called the "decimal point" (even in base 2.)

To find these digit(s) (??) we have two rules:

Each sign change produces a 1, and each repetition from the previous sign produces a 0 (in the base 2 form of the number.)
After the sequence terminates, append a final 1 to the number.

The very first minus sign does not count as a sign change, since it was used to create the decimal point. For example, +−=0.1₂ (one−half), where the "1" after the decimal point is the appended digit as per the second rule above. On the other hand, there is one (and only one) sign change in, +−+=0.11₂.

Exercise: Show that +++−−+−−= 10.01101₂=2.40625₁₀

See also stackexchange question 1237974

References

mit surrealSlides & mit surrealDoc

ω+1>ω is true

ω+1>ω is false

Is there such an ordinal as ω-1? There is an extension of ordinal numbers, the surreal numbers, that include ω−1, but let me give you two warnings about that. First, the addition of the surreal numbers is not exactly the same as ordinal addition. Second, I don't recommend reading about the surreal numbers until you get a better grasp of w:ordinal numbers.
is-omega-1-finite

I need to learn about

Excellent site that mentions Dedekind cuts: https://thatsmaths.com/2012/11/22/the-root-of-infinity-its-surreal/
Excellent simple answer: https://www.quora.com/What-are-surreal-numbers-in-laymans-terms
https://math.stackexchange.com/questions/3652852/how-does-conways-proposed-compromise-for-constructing-the-real-numbers-actually
Excellent explanation of addition https://issuu.com/oz1cz/docs/surreal