nLab infinite decimal representation of a unit interval

Definition

Let $R$ be an Archimedean integral domain and let $[0, 1]_R$ be the unit interval in $R$.

The infinite decimal representation of $[0, 1]_R$ is a function $\mathcal{I}:[0, 1]_R \to (\mathbb{N} \to [0, 9]_\mathbb{N})$ from the unit interval in $R$ to the type of sequences in the natural numbers that are bounded below by $0$ and bounded above by $9$, such that $r$ is equal to the limit of the following sequence

$$\prod_{r:[0, 1]_R} r = \lim_{n \to \infty} \sum_{i=0}^{n} \frac{\mathcal{I}(r)_i}{10^i}$$

Properties

• The infinite decimal representation of the unit interval in the rational numbers consist of all the eventually periodic sequences in the natural numbers that are bounded below by $0$ and bounded above by $9$. Every rational number in the unit interval could be represented by two natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$ such that $p:m \leq n$. Let us define the sequences $\delta:\mathbb{N} \to \mathbb{N}$ and $\rho:\mathbb{N} \to \mathbb{N}$ inductively as

  $$\delta_0 \coloneqq m \div n$$
  $$\rho_0 \coloneqq m\ \%\ n$$
  $$\delta_{s(i)} \coloneqq (\rho_i \cdot 10) \div n$$
  $$\rho_{s(i)} \coloneqq (\rho_i \cdot 10)\ \%\ n$$

  The sequence $\delta$ is the infinite decimal representation of the rational number $m/n$.

• The infinite decimal representation of the unit interval in the decimal numbers consist of all the sequences such that there is a natural number $N$ such that for all decimal number $i$ and natural numbers $n \geq N$, $\mathcal{I}(i)(n) = 0$ or $\mathcal{I}(i)(n) = 9$.

See also

• real numbers

• pre-algebra real numbers

• sequence

• limit of a sequence

• unit interval