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A Hackenbush stalk is a Hackenbush position whose graph is a single path attached to the ground at one endpoint. Read its blue and red graph edges upward from the ground; Left may cut blue edges and Right red edges, with every edge above the cut falling away. Finite red-blue stalks are number games with dyadic rational values.

Here is the explicit value rule. Give blue sign $+1$ and red sign $-1$. If the initial monochromatic run has length $m$ and sign $\epsilon$, it contributes $\epsilon m$. If more edges remain, edge $j>m$, of sign ${\epsilon }_j$, contributes ${\epsilon }_j/2^{j-m}$. ThusFor example, a blue-red stalk has value $1-1/2=1/2$, blue-red-red has value $1/4$, and blue-blue-red has value $3/2$. The recursive number game rule verifies the first example directly: its Left option is $0$, its Right option is $1$, and the simplest surreal number between them is $1/2$. Repeating this comparison after successive cuts gives the dyadic rule.

A number game is a recursively defined normal-play partizan game whose options are number games and every left option is smaller than every right option. Its value is the simplest surreal number between those options. Finite number games have dyadic rational values.

The surreal numbers form an ordered field built recursively from cuts $\{L\mid R\}$, where every already constructed left option is smaller than every right option. The earliest examples are $0=\{\mid \}$, $1=\{0\mid \}$, $-1=\{\mid 0\}$ and $1/2=\{0\mid 1\}$. A cut names the simplest number between its options; different cuts can name the same number.

Allowing construction through arbitrary ordinals introduces both infinite numbers, such as $\omega =\{0,1,2,\dots \mid \}$, and positive infinitesimals, such as $1/\omega$. All real numbers embed in the resulting ordered field, but its entire collection is a proper class. Unlike ordinal arithmetic, surreal multiplication and addition obey the usual commutative field laws.