nLab dyadic rational number

Idea

A dyadic rational number is a rational number $r \in \mathbb{Q}$ such that the binary expansion of $r$ has finitely many digits.

Definition

As a subset of the rational numbers

A dyadic rational number is a rational number $r \in \mathbb{Q}$ such that there exists $n \in \mathbb{N}$ and $a \in \mathbb{Z}$ such that $r = \frac{a}{2^n}$.

As the localisation of the integers at 2

The commutative ring of dyadic rational numbers $\mathbb{Z}[1/2]$ is the localization of the integers $\mathbb{Z}$ away from $2$.

As an initial object of a category

For lack of a better name, let us define a set with dyadic rational numbers to be a set $A$ with a function $\iota \in \mathbb{Z} \times \mathbb{N} \to A$, such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. \forall c \in \mathbb{Z}. \forall d \in \mathbb{N}. (a \cdot 2^d = c \cdot 2^b) \implies (\iota(a, b) = \iota(c, d))$$

The integer $a:\mathbb{Z}$ represents the integer if one ignores the separator in the binary numeral representation of the dyadic rational number, and the natural number $b:\mathbb{N}$ represents the number of digits to the left of the final digit where the separator ought to be placed after in the binary numeral representation of the dyadic rational number. The axiom above is used to state that equivalent numeral representations are equal: i.e. 1.00 = 1.000 with binary numeral representations.

A homomorphism of sets with dyadic rational numbers between two sets with dyadic rational numbers $A$ and $B$ is a function $f:A \to B$ such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. f(\iota_A(a, b)) = \iota_B(a, b)$$

The category of sets with dyadic rational numbers is the category $SwDRN$ whose objects $Ob(SwDF)$ are sets with dyadic rational numbers and whose morphisms $Mor(A,B)$ for sets with dyadic rational numbers $A \in Ob(SwDRN)$, $B \in Ob(SwDRN)$ are homomorphisms of sets with dyadic rational numbers. The set of dyadic rational number, denoted $\mathbb{Z}[1/2]$, is defined as the initial object in the category of sets with dyadic rational number.

Properties

Algebraic closure

The algebraic closure $\overline{\mathbb{Z}[1/2]}$ of the dyadic rational numbers is called the field of algebraic numbers, and is thus isomorphic to $\overline{\mathbb{Q}}$, the algebraic closure of the rational numbers.

Topologies

There are several interesting topologies on $\mathbb{Z}[1/2]$ that make $\mathbb{Z}[1/2]$ into a topological group under addition, allowing us to define interesting fields by taking the completion with respect to this topology:

1. The discrete topology is the most obvious, which is already complete.

2. The absolute-value topology is defined by the metric $d(x,y) \coloneqq {|x - y|}$; the completion is the field of real numbers.

   (This topology is totally disconnected.)

3. The $2$-adic topology is defined by the ultrametric $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $2$ in the prime factorization of ${|x - y|}$; the completions of each metric are the fields of $2$-adic numbers.

Related entries

• decimal rational number

• Urysohn's lemma

• halving group

• symmetric midpoint algebra

• dyadic rational module

• dyadic rational algebra

• dyadic interval coalgebra

References

• Wikipedia, Dyadic rational