nLab decimal number

Idea

Talk about the decimal number system for integers and decimal fractions, and then infinite sequences of decimals as a terminal coalgebra for an endofunctor.

Natural numbers

As a free monoid

Define a set of digits $D$, and the free monoid $D^*$ on $D$ with unit $\epsilon$, quotiented by an equivalence relation. Then define a function $s$ on $D^*$ such that $D^*,\epsilon,s$ is a natural numbers object.

Long division algorithm

Let $[0, 9]^*$ be the free monoid on the closed interval $[0, 9]$, which represents the list of digits in the decimal numeral representation of the natural numbers, with length function $\mathrm{len}:[0, 9]^* \to \mathbb{N}$. There is a surjection $f:[0, 9]^* \to \mathbb{N}$ defined as

$$f(a) \coloneqq \sum_{i = 0}^{\mathrm{len}(a) - 1} a_i 10^{\mathrm{len}(a) - i - 1}$$

representing the numerical value of the list of digits.

The algorithm is as follows: Let $a:[0, 9]^*$ and $b:[0, 9]^*$ be lists of digits, with $f(a)$ the dividend and $f(b)$ the divisor. If $\mathrm{len}(a) \lt \mathrm{len}(b)$, then $f(a) \div f(b) = 0$ and $f(a)\ \%\ f(b) = f(a)$. Otherwise, we iterate for $\mathrm{len}(a)$ times before stopping:

For each iteration $i:[0, \mathrm{len}(a)]$, let $q_i$ be the quotient extracted from the algorithm at iteration $i$, let $d_i$ be the dividend at iteration $i$, let $r_i$ be the remainder at iteration $i$, and let $c_i$ be the next digit of the quotient, with the restriction that $0 \leq r_i$ and $r_i \lt f(b)$

We set initial conditions to be

$$q_{-1} \coloneqq 0$$

$$r_{-1} \coloneqq \sum_{i = 0}^{\mathrm{len}(b) - 2} a_i 10^{\mathrm{len}(b) - i - 2}$$

The successive conditions are defined inductively as

$$d_{i + 1} \coloneqq 10 r_i + a_{i + \len(b)}$$

$$r_{i + 1} \coloneqq d_{i + 1} - m c_{i + 1} = 10 r_i + a_{i + \len(b)} - m c_{i + 1}$$

$$q_{i + 1} \coloneqq 10 q_{i} - c_{i + 1}$$

Order theoretic definitions

Integers

work in progress…

Finite decimals

See below, but define in terms of an initial algebra instead of a terminal coalgebra.

Infinite decimals

Consider the category of intervals $Int$, i.e., linearly ordered sets with identified elements $1$ and $0$, and let

$$F: Int \to Int$$

be the endofunctor which takes an interval $X$ to $\bigvee_{0 \leq i \lt 10} X$, the the interval obtained by taking ten copies of $X$ and identifying the $1$ of the $(i-1)$-th copy with the $0$ of the $i$-th copy, for $0 \lt i \lt 10$. The real interval $[0, 1]$ becomes a coalgebra if we identify $\bigvee_{0 \leq i \lt 10} X$ with $[0, 10]$ and consider the multiplication-by-10 map $[0, 1] \to [0, 10]$ as giving a coalgebra structure.

Theorem

The interval $[0, 1]$ is terminal in the category of such coalgebras.

Proof (sketch)

Given any coalgebra structure $f: X \to \bigvee_{0 \leq i \lt 10} X$, any value $f(x)$ lands either in the $n$-th tenth (the $n$-th $X$ in $\bigvee_{0 \leq i \lt 10} X$) for $0\lt n \leq 10$, or at the precise spot between them, where the $1$ in the $n$-th copy is glued to the $0$ in the $(n+1)$-th for $0\lt n \lt 10$. Intuitively, one could think of a coalgebra structure $\theta: X \to \bigvee_{0 \leq i \lt 10}$ as giving an automaton where on input $x_0$ there is output of the form $(x_1, h_1)$, where $h_1$ is one of 19 states, “0”, “1”, “2”, “3”, “4”, “5”, “6”, “7”, “8”, “9”, “either 0 or 1”, “either 1 or 2”, “either 2 or 3”, “either 3 or 4”, “either 4 or 5”, “either 5 or 6”, “either 6 or 7”, “either 7 or 8”, and “either 8 or 9”. By iteration, this generates a behavior stream $(x_n, h_n)$. Let “0”, “1”, “2”, “3”, “4”, “5”, “6”, “7”, “8”, and “9” be decimal digits $D$, the $h_n$ form a decimal expansion to give a number between 0 and 1, and therefore we have an interval map $X \to [0, 1]$ which sends $x_0$ to that number. Of course, if $h_n$ is ever one of the “either $d_l$ or $d_u$” states, for $d_l,d_u:D$ we have a choice to resolve it as either $(1_X, d_l)$ or $(0_X, d_u)$ and continue the stream, but these streams are identified, and this corresponds to the identifications of decimal expansions

$$h_1... h_{n-1} 100000... = .h_1... h_{n-1}099999...$$

$$h_1... h_{n-1} 200000... = .h_1... h_{n-1}199999...$$

$$h_1... h_{n-1} 300000... = .h_1... h_{n-1}299999...$$

$$h_1... h_{n-1} 400000... = .h_1... h_{n-1}399999...$$

$$h_1... h_{n-1} 500000... = .h_1... h_{n-1}499999...$$

$$h_1... h_{n-1} 600000... = .h_1... h_{n-1}599999...$$

$$h_1... h_{n-1} 700000... = .h_1... h_{n-1}699999...$$

$$h_1... h_{n-1} 800000... = .h_1... h_{n-1}799999...$$

$$h_1... h_{n-1} 900000... = .h_1... h_{n-1}899999...$$

as real numbers. In this way, we produce a unique well-defined interval map $X \to [0, 1]$, so that $[0, 1]$ is the terminal coalgebra.

Let $(\mathbb{Z},0,s,n,\lt)$ be the set of integers, the initial set with an element $0$, a linear order $\lt$, a monotone $s$ such that $a \lt s(a)$ for all $a \in \mathbb{Z}$, and an antitone $n$ such that $n(0)=0$ and $n = s \circ n \circ s$. Let $\mathbb{R}$ be a set with a structure preserving function $g:\mathbb{Z}\to\mathbb{R}$ and a function into the monotone poset $f:\mathbb{Z} \to [0,1]\to \mathbb{R}$ such that $f(a)(0) = g(a)$ and $f(a)(1) = s(g(a))$ for all $a \in \mathbb{Z}$. The set $\mathbb{R}$ of real numbers is the initial such system. By uncurrying the function $f$, one gets a function $i:\mathbb{Z} \times [0,1]\to \mathbb{R}$. An element $(a,b):\mathbb{Z} \times [0,1]$ is called an infinite decimal representation, where the comma used to represent pairs in set theory or type theory is literally the decimal separator commonly seen in non-English speaking countries.

The arithmetic operations and topological properties on $\mathbb{R}$ can be defined by the properties of the function algebra of $\mathbb{R}$ and currying.

Rig theoretic definitions

Integers

$$\mathbb{N} \cong \mathbb{N}[10] \coloneq \mathbb{N}[x]/(x=10)$$

Finite decimals

Localisation of the rig of natural numbers at 10 $\mathbb{N}[1/10]$, finite decimals as canonical representatives of $\mathbb{N}[1/10]$, and then group completion of the additive monoid to $\mathbb{Z}[1/10]$.

Infinite decimals

Sequence algebra$\mathbb{N}\to\mathbb{Z}[1/10]$ and Cauchy sequences

Doubly infinite decimals

Let $\mathbb{Z}/10\mathbb{Z}$ be the cyclic group consisting of 10 elements, and let $C_\bullet$ be a chain complex of abelian groups consisting of a sequence of $\mathbb{Z}/10\mathbb{Z}$ indexed by $\mathbb{Z}$. The indices $i:\mathbb{Z}$ are called place values, and $i$-cochains are called digits.

A 10-adic number is a cochain such that $c_i = 0$ for all $i \lt j$ or $c_i = 9$ for all $i \lt j$ for $i,j:\mathbb{Z}$. A 10-adic integer is a cochain such that $c_i = 0$ for all $i \lt 0$ or $c_ i = 9$ for all $i \lt 0$. A real number is a cochain such that $c_i = c_j$ for all $i \gt k$ and $j \gt k$ for $i,j,k:\mathbb{Z}$. A decimal rational is a real 10-adic number, and an integer is a real 10-adic integer.

10-adic solenoids

The cochain complex $C_\bullet$ defined in the previous section has a structure of an abelian group, making it into a 10-adic solenoid.

A cyclic group $G$ has a canonical cyclic order $[(-),(-),(-)]:(G \times G \times G) \to \Omega$. We define the cyclic order on $\mathbb{Z}/10\mathbb{Z}$ such that $[0,n,1]$ is false for all $n:\mathbb{Z}/10\mathbb{Z}$.

For each $i:\mathbb{Z}$, there exists a cocycle $f_i: (C_i \times C_i) \to C_{i + 1}$ called the digitwise carry function at place value $i$, defined such that for all $a,b:C_i$, $f_i(a,b) = 0$ if $[a,0,a+b]$ is false, and $f_i(a,b) = 1$ if $[a,0,a+b]$ is true.

We define the addition without carry on the cochain complex $(-)\oplus(-): (C_\bullet \times C_\bullet) \to C_\bullet$ as the addition of all digits using the abelian group operation, and we define the carry $carry: (C_\bullet \times C_\bullet) \to C_\bullet$ as the digitwise carry of all digits. Then, addition $(-)+(-): (C_\bullet \times C_\bullet) \to C_\bullet$ is defined recursively as $a+b = (a\oplus b)+carry(a,b)$.

The cochains consisting of all $n$s for all $n:\mathbb{Z}/10\mathbb{Z}$ are additive identity elements of the addition operation defined above. As such, they are algebraically equal to the same chain $0$ zero, the chain consisting of all zeroes. We define negation $-(-): C_\bullet \to C_\bullet$ such that for all chains $c:C_\bullet$, the digits in $-c$ are $(-c)_i = 9 + -c_i$. As a result, the chain complex itself is an abelian group.

Analytic completion of Laurent series

Let $1$ one denote the cochain with all digits equal to zero except at place value $0$, where the digit is equal to $1$. The cochain with all digits equal to zero except at place value $i$ is called the $i$-th power of ten and is denoted as $10^i$.

Let us define an $\mathbb{N}$-action $act: (\mathbb{N}^+\times C_\bullet) \to C_\bullet$ such that $act(0,0) = 0$ and $act(n + 1,c) = act(n,c) + c$ for all $n:\mathbb{N}^+$ and $c:C_\bullet$. This represents the $n$-fold sum of a cochain $c$.

One could also establish a ring structure on $\mathbb{Z}/10\mathbb{Z}$ and construct a multiplication operation on the chain complex such that the chain complex itself with the defined abelian group structure and multiplication should be equivalent to the quotient ring of the Laurent series $\mathbb{Z}[[x,x^{-1}]]/(x-10)\mathbb{Z}[[x,x^{-1}]]$.

Related concepts

• decimal rational number

• numeral

References

• Wikipedia, Decimal

• Wikipedia, Decimal representation