Dyadic numbers #

Dyadic numbers are obtained by localizing ℤ away from 2. They are the initial object in the category of rings with no 2-torsion.

Dyadic surreal numbers #

We construct dyadic surreal numbers using the canonical map from ℤ[2 ^ {-1}] to surreals. As we currently do not have a ring structure on Surreal we construct this map explicitly. Once we have the ring structure, this map can be constructed directly by sending 2 ^ {-1} to half.

Embeddings #

The above construction gives us an abelian group embedding of ℤ into Surreal. The goal is to extend this to an embedding of dyadic rationals into Surreal and use Cauchy sequences of dyadic rational numbers to construct an ordered field embedding of ℝ into Surreal.

For a natural number n, the pre-game powHalf (n + 1) is recursively defined as {0 | powHalf n}. These are the explicit expressions of powers of 1 / 2. By definition, we have powHalf 0 = 1 and powHalf 1 ≈ 1 / 2 and we prove later on that powHalf (n + 1) + powHalf (n + 1) ≈ powHalf n.

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    For all natural numbers n, the pre-games powHalf n are numeric.

    Powers of the surreal number half.

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      theorem Surreal.zsmul_pow_two_powHalf (m : ) (n : ) (k : ) :
      theorem Surreal.dyadic_aux {m₁ : } {m₂ : } {y₁ : } {y₂ : } (h₂ : m₁ * 2 ^ y₁ = m₂ * 2 ^ y₂) :

      The additive monoid morphism dyadicMap sends ⟦⟨m, 2^n⟩⟧ to m • half ^ n.

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        We define dyadic surreals as the range of the map dyadicMap.

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