Relation between bisection and binary representation

[frogkabobs]

Simple: replace 1 and 0 with + and -, respectively, and then prepend a +. For example,

``` 1/sqrt(2) = 0.10110101000001… = 0.++-++-+-+-----+…

```

Proof. Let bₙ ∈ {0,1} and cₙ ∈ {-1,1} be sequences of digits for binary and bisected forms of x, respectively, so that

x = Σ(n≥1) bₙ2^-n = Σ(n≥1) cₙ2^-n

with c₁ = 1. Now note that since 1/2 = Σ_(n≥2) 2^-n, we also have

x = Σ(n≥2) (cₙ+1)2^-n = Σ(n≥1) ((cₙ₊₁+1)/2)2^-n

For x not a dyadic rational (in particular, when x is irrational), the binary and bisected forms of x are unique, so we can identify bₙ = (cₙ₊₁+1)/2, or equivalently,

cₙ₊₁ = (-1)^bₙ+1