Surreal number/Simple hackenbush

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/ Hackenbush

See also this MUST READ: 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

See also https://math.stackexchange.com/questions/556014/what-is-worth-of-a-stalk-in-red-blue-hackenbush