hcju/proofwikips huggingface dataset:
Definition:Prime Number/Definition 2
Let $p$ be a positive integer. Then $p$ is a prime number  $p$ has exactly four integral divisors: $\pm 1$ and $\pm p$.
Definition:Prime Number/Definition 3
Let $p$ be a positive integer. Then $p$ is a prime number : : $\tau \left({p}\right) = 2$ where $\tau \left({p}\right)$ denotes the tau function of $p$.
Definition:Pairwise Coprime
=== GCD Domain === {{:Definition:Pairwise Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Pairwise Coprime/Euclidean Domain}} === Integers === {{:Definition:Pairwise Coprime/Integers}}
Definition:Pairwise Coprime/Integers
A set of integers $S$ is '''pairwise coprime''' : :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.
Definition:Pairwise Coprime/GCD Domain
Let $\struct {D, +, \times}$ be a GCD domain. A subset $S \subseteq D$ is '''pairwise coprime (in $D$)''' : :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.
Definition:Pairwise Coprime/Euclidean Domain
Let $\struct {D, +, \times}$ be a Euclidean domain. A subset $S \subseteq D$ is '''pairwise coprime (in $D$)''' : :$\forall x, y \in S: x \ne y \implies x \perp y$ where $x \perp y$ denotes that $x$ and $y$ are coprime.
Definition:Polynomial Congruence/Number of Solutions
Let $S = \left\{{b_1, b_2, \ldots, b_n}\right\}$ be a complete set of residues modulo $n$. The '''number of solutions''' of $P \left({x}\right) \equiv 0 \pmod n$ is the number of integers $b \in S$ for which $P \left({b}\right) \equiv 0 \pmod n$.
Definition:Signature (Logic)
Let $\mathcal L$ be a formal language. A choice of vocabulary for $\mathcal L$ is called a '''signature''' for $\mathcal L$. === Signature for Predicate Logic === {{:Definition:Signature (Logic)/Predicate Logic}}
Definition:Non-Archimedean/Norm (Vector Space)
A norm $\norm {\,\cdot\,} $ on a vector space $X$ is '''non-Archimedean'''  $\norm {\, \cdot \,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = N4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}
Definition:Non-Archimedean/Norm (Vector Space)/Archimedean
A norm $\norm {\,\cdot\,} $ on a vector space $X$ is '''Archimedean'''  it is not non-Archimedean.
Definition:P-adic Norm/P-adic Metric
The '''$p$-adic metric''' on $\Q$ is the metric induced by $\norm{\cdot}_p$: :$\forall x, y \in \Q: \map d {x, y} = \norm{x - y}_p$
Definition:Non-Archimedean/Metric/Archimedean
A metric is '''Archimedean'''  it is not non-Archimedean.
Definition:Non-Archimedean/Metric
A metric $d$ on a metric space $X$ is '''non-Archimedean''' : :$\map d {x, y} \le \max \set {\map d {x, z}, \map d {y, z} }$ for all $x, y, z \in X$.
Definition:Max Operation/General Definition
Let $S^n$ be the cartesian $n$th power of $S$. The '''max operation''' is the $n$-ary operation on $\struct {S, \preceq}$ defined recursively as: :$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in S^n: \map \max x = \begin{cases} x_1 & : n = 1 \\ \map \max {x_1, x_2} & : n = 2 \\ \map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$ where $\map \max {x, y}$ is the binary max operation on $S^2$.
Definition:Language of Propositional Logic/Alphabet/Letter
The letters of $\LL_0$, called '''propositional symbols''', can be any infinite collection $\PP_0$ of arbitrary symbols. It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts. A typical set of '''propositional symbols''' would be, for example: :$\PP_0 = \set {p_1, p_2, p_3, \ldots, p_n, \ldots}$
Definition:Language of Propositional Logic/Alphabet/Sign/Bracket
{{begin-eqn}} {{eqn | ll= \bullet       | l = (       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = )       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}}
Definition:Language of Propositional Logic/Alphabet/Sign/Connective
{{begin-eqn}} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{eqn | ll= \bullet       | l = \top       | o = :       | r = \)the tautology sign\(       | c =  }} {{eqn | ll= \bullet       | l = \bot       | o = :       | r = \)the contradiction sign\(       | c =  }} {{end-eqn}} These comprise: * The nullary connectives $\top$ and $\bot$, representing the canonical tautology and contradiction, respectively * The unary connective $\neg$, representing negation * The binary connectives $\land, \lor, \implies$ and $\iff$, representing, respectively, conjunction, disjunction, implication and biconditional.
Definition:Language of Propositional Logic/Alphabet
The alphabet $\mathcal A$ of the language of propositional logic $\mathcal L_0$ is defined as follows: <section notitle="true" name="tc"> === Letters === {{:Definition:Language of Propositional Logic/Alphabet/Letter}} {{transclude:Definition:Language of Propositional Logic/Alphabet/Sign   |section = def   |title = Signs   |header = 3   |link = true   |increase = 1 }} </section>
Definition:Language of Propositional Logic/Formal Grammar/Backus-Naur Form
In Backus-Naur form, the formal grammar of the language of propositional logic takes the following form: {{begin-eqn}} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = p \ \mid \ \top \ \mid \ \bot       | c = where $p \in \PP_0$ is a letter }} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = \neg\) <tt><formula></tt>\( }} {{eqn | l = \)<tt><formula></tt>\(       | o = \)<tt>::=</tt>\(       | r = (\) <tt><formula></tt> <tt><op></tt> <tt><formula></tt> \() }} {{eqn | l = \)<tt><op></tt>\(       | o = \)<tt>::=</tt>\(       | r = \land \ \mid \ \lor \ \mid \implies \mid \iff }} {{end-eqn}} Note that this is a top-down grammar: :we start with a metasymbol <tt><formula></tt> :progressively replace it with constructs containing other metasymbols and/or primitive symbols until finally we are left with a well-formed formula of $\LL_0$ consisting of nothing but primitive symbols.
Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification
The following rules of formation constitute a bottom-up grammar for the formation of well-formed formulas (WFFs) of the language of propositional logic $\LL_0$. :Let $\PP_0$ be the vocabulary of $\LL_0$. :Let $\mathrm {Op} = \set {\land, \lor, \implies, \iff}$. The rules are: {| |- | $\mathbf W: \text {TF}$ || | $:$ | $\top$ is a WFF, and $\bot$ is a WFF. |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. |- | $\mathbf W: \text {Op}$ || | $:$ | If $\mathbf A$ is a WFF and $\mathbf B$ is a WFF and $\circ \in \mathrm {Op}$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF. |} Any string which can not be created by means of the above rules is not a WFF.
Definition:Language of Propositional Logic
In order to define $\mathcal L_0$, it is necessary to specify: * An alphabet $\mathcal A$ * A collation system with the unique readability property for $\mathcal A$ * A formal grammar (which determines the WFFs of $\mathcal L_0$)
Definition:Max Operation/General Definition/Real Numbers
The '''max operation''' on the real cartesian space $\R^n$ is the real-valued function $\max: \R^n \to \R$ defined recursively as: :$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in \R^n: \map \max x = \begin{cases} x_1 & : n = 1 \\ \map \max {x_1, x_2} & : n = 2 \\ \map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$ where $\map \max {x, y}$ is the binary max operation on $\R \times \R$.
Definition:Cartesian Product/Cartesian Space/Real Cartesian Space
Let $n \in \N_{>0}$. Then $\R^n$ is the cartesian product defined as follows: :$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$ Similarly, $\R^n$ can be defined as the set of all real $n$-tuples: :$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$
Definition:Language of Propositional Logic/Keisler-Robbin
There are many formal languages expressing propositional logic. The formal language used on  is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\PP_0$. See the  definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''square brackets''': {{begin-eqn}} {{eqn | ll= \bullet       | l = [       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = ]       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}} See the  definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{end-eqn}} See the  definition. === Collation System === The collation system used is that of words and concatenation. See the  definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. Let $Op = \set {\land, \lor, \implies, \iff}$. The rules are: {| |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. |- | $\mathbf W: Op$ || | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\sqbrk {\mathbf A \circ \mathbf B}$ is a WFF. |} See the  definition.
Definition:Order of Pole/Simple Pole
Let the order of the pole at $x$ be $1$. Then $x$ is a '''simple pole'''.
Definition:Order of Zero/Simple Zero
Let the order of the zero $x$ be $1$. Then $x$ is a '''simple zero'''.
Definition:Isolated Singularity/Pole
Let $z_0$ be an isolated singularity of $f$. Then $z_0$ is a '''pole''' : :$\displaystyle \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$
Definition:Characteristic of Ring/Definition 1
Let $n \cdot x$ be defined as in Definition:Power of Element. The '''characteristic''' $\Char R$ of $R$ is the smallest $n \in \Z, n > 0$ such that $n \cdot 1_R = 0_R$. If there is no such $n$, then $\Char R = 0$.
Definition:Characteristic of Ring/Definition 2
Let $g: \Z \to R$ be the initial homomorphism, with $\map g n = n \cdot 1_R$. Let $\ideal p$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $p$. The '''characteristic''' $\Char R$ of $R$ is the positive integer $p \in \Z_{\ge 0}$ such that $\ideal p$ is the kernel of $g$.
Definition:Characteristic of Ring/Definition 3
The '''characteristic of $R$''', denoted $\Char R$, is defined as follows. Let $p$ be the order of $1_R$ in the additive group $\struct {R, +}$ of $\struct {R, +, \circ}$. If $p \in \Z_{>0}$, then $\Char R := p$. If $1_R$ is of infinite order, then $\Char R := 0$.
Definition:Monoid Axioms
A monoid is an algebraic structure $\struct {S, \circ, e_S}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \text S 0         | lc= Closure         | q = \forall a, b \in S         | m = a \circ b \in S }} {{axiom | n = \text S 1         | lc= Associativity         | q = \forall a, b, c \in S         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text S 2         | lc= Identity         | q = \exists e_S \in S: \forall a \in S         | m = e_S \circ a = a = a \circ e_S }} {{end-axiom}} The element $e_S$ is called the identity element.
Definition:Unique Parsability
Let $\mathcal L$ be a formal language. Let every WFF of $\mathcal L$ result from a unique rule of formation. Then $\mathcal L$ has '''unique parsability'''. === Bottom-Up Grammar === For a bottom-up grammar, a WFF $\phi$ results from a unique rule of formation iff: :$\phi$ results from applying the rule of formation $\mathbf R$ to WFFs $\phi_1, \ldots \phi_n$ :$\phi$ results from applying the rule of formation $\mathbf R'$ WFFs $\psi_1, \ldots \psi_m$ imply that $\mathbf R = \mathbf R'$, $m = n$, and for $i = 1, \ldots, n$, $\phi_i = \psi_i$. === Top-Down Grammar === For a top-down grammar, a WFF $\phi$ results from a unique rule of formation iff: :$\phi$ results from applying the rule of formation $\mathbf R$, substituting the WFFs $\phi_1, \ldots, \phi_n$ for the metasymbols of $\mathbf R$ :$\phi$ results from applying the rule of formation $\mathbf R'$, substituting the WFFs $\psi_1, \ldots, \psi_n$ for the metasymbols of $\mathbf R'$ imply that $\mathbf R = \mathbf R'$, $m = n$, and for $i = 1, \ldots, n$, $\phi_i = \psi_i$.
Definition:Bottom-Up Form of Top-Down Grammar
Let $\mathcal L$ be a formal language. Let $\mathcal L$ be given by a top-down formal grammar $\mathcal T$. The '''bottom-up form of $\mathcal T$''' is the formal grammar $\mathcal B$ defined by declaring that: :Each letter of $\mathcal L$ is a $\mathcal B$-WFF and, for each rule of formation $\mathbf R$ of $\mathcal T$ of the form: :A metasymbol may be replaced by the collation $\phi$ with metasymbols $\phi_1, \ldots, \phi_n$ declaring that $\mathcal B$ has the rule of formation $\mathbf R_{\mathcal B}$: :If $\phi_1, \ldots, \phi_n$ are $\mathcal B$-WFFs, so is $\phi$.
Definition:Semigroup Axioms
A semigroup is an algebraic structure $\struct {S, \circ}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = \text S 0         | lc= Closure         | q = \forall a, b \in S         | m = a \circ b \in S }} {{axiom | n = \text S 1         | lc= Associativity         | q = \forall a, b, c \in S         | m = a \circ \paren {b \circ c} = \paren  {a \circ b} \circ c }} {{end-axiom}}
Definition:Latin Square Property
Let $\left({S, \circ}\right)$ be an algebraic structure. $\left({S, \circ}\right)$ has the '''Latin square property''' : :$\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: x \circ a = b$ :$\forall a, b \in S: \exists ! y: a \circ y = b$
Definition:Properties of Algebraic Structures of One Operation
The purpose of this page is to gather into one place the various types of algebraic structure of one binary operation and classify them according to the properties they hold. {| border="1" !  ! Closure ! Associativity ! Identity ! Inverses ! Commutativity ! Latin Square Property |- | Magma | align="center"| $\checkmark$ | | | | | |- | Quasigroup | align="center"| $\checkmark$ | | | | | align="center"| $\checkmark$ |- | Algebra Loop | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ |- | Semigroup | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | | |- | Commutative Semigroup | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ |  |- | Monoid | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | |- | Commutative Monoid | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | |- | Group | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $(\checkmark)$: see here |- | Abelian Group | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $(\checkmark)$: see here |- |} A '''checkmark''' in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.
Definition:Support of Mapping to Algebraic Structure/Real-Valued Function
Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. The '''support of $f$''' is the set of elements $x$ of $S$ whose values under $f$ are non-zero. That is: :$\operatorname{supp} \left({f}\right) := \left\{{x \in S: f \left({x}\right) \ne 0}\right\}$
Definition:Continuous Real-Valued Vector Function
Let $\R^n$ be the cartesian $n$-space. Let $f: \R^n \to \R$ be a real-valued function on $\R^n$. Then '''$f$ is continuous on $\R^n$''' iff: :$\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: d \left({x, a}\right) < \delta \implies \left|{f \left({x}\right) - f \left({a}\right)}\right| < \epsilon$ where $d \left({x, a}\right)$ is the distance function on $\R^n$: :$\displaystyle d: \R^n \to \R: d \left({x, y}\right) := \sqrt {\left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)}\right)}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right)$ are general elements of $\R^n$.
Definition:Support of Continuous Mapping/Real-Valued
Let $f: \R^n \to \R$ be a continuous real-valued function. The '''support of $f$''' is the closure of the set of elements $x$ of $\R^n$ whose values under $f$ are non-zero. That is: :$\operatorname{supp} \left({f}\right) = \overline{\left\{{x \in \R^n: f \left({x}\right) \ne 0}\right\}}$
Definition:Support of Distribution
Let $\Omega \subseteq \R^n$ be an open set. {{definition wanted|The link goes to a definition of an open set which has been made only for the real number line. Needs to be expanded to include the general real cartesian space. This has already been defined in the context of Metric Spaces; it's just a matter of creating a page which connects the concepts of an open set on a real number line to that on a general space.}} Let $\mathcal D \left({\Omega}\right)$ be the space of continuous functions compactly supported in $\Omega$. Let $T \in \mathcal D\,' \left({\Omega}\right)$ be a distribution. {{explain|$D\,'$: presumably a derivative?}} The '''support''' $\operatorname{supp} \left({T}\right) \subseteq \Omega$ of $T$ is defined by: : $\displaystyle x \notin \operatorname{supp} \left({T}\right)$ : :: there exists an open neighborhood $U$ of $x$ such that: ::: for all $\phi \in \mathcal D \left({\Omega}\right)$ such that $\operatorname{supp} \left({\phi}\right) \subseteq U$: :::: $T \left({\phi}\right) = 0$
Definition:Partition of Unity (Topology)/Subordinate
Let $X$ be a topological space. Let $\left\{{\phi_\alpha : \alpha \in A}\right\}$ be a partition of unity. Let $\mathcal B = \left\{{U_\beta: \beta \in B}\right\}$ be an open cover of $X$. Suppose the set $\left\{{\operatorname{supp} \left({\phi_\alpha}\right)^\circ : \alpha \in A}\right\}$ of interiors of supports is a refinement of $\mathcal B$ Then $\mathcal A$ is said to be '''subordinate''' to the cover $\mathcal B$.
Definition:Literal
A '''literal''' is either: :an atom $p$ of propositional logic, that is, a statement, or :the negation $\neg p$ of an atom $p$. In the language of propositional logic, these correspond to: :a letter $p$; :the WFF $\neg p$, where $p$ is a letter. === Positive Literal === {{:Definition:Literal/Positive}} === Negative Literal === {{:Definition:Literal/Negative}}
Definition:Conjunctive Normal Form
A propositional formula $P$ is in '''conjunctive normal form'''  it consists of a conjunction of: :$(1):\quad$ disjunctions of literals and/or: :$(2):\quad$ literals.
Definition:Set of Truth Values
The '''set of truth values''' of propositional logic is the $2$-element set: :$\Bbb B = \set {\T, \F}$ of (Aristotelian) truth values.
Definition:Continued Fraction/Finite
Let $n \geq 0$ be a natural number. Informally, a '''finite continued fraction of length $n$''' in $F$ is an expression of the form: :$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$ where $a_0, a_1, a_2, \ldots, a_n \in F$. Formally, a '''finite continued fraction of length $n$''' in $F$ is a finite sequence, called '''sequence of partial quotients''', whose domain is the integer interval $\left[0 \,.\,.\, n\right]$. A finite continued fraction should not be confused with its '''value''', when it exists.
Definition:Continued Fraction/Infinite
Informally, an '''infinite continued fraction''' in $F$ is an expression of the form: :$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n + \cfrac 1 {\ddots}}} }}}$ where $a_0, a_1, a_2, \ldots, a_n, \ldots \in F$. Formally, an '''infinite continued fraction''' in $F$ is a sequence, called '''sequence of partial quotients''', whose domain is $\N_{\geq 0}$. An infinite continued fraction should not be confused with its '''value''', when it exists.
Definition:Partial Quotient
Let $(a_n)_{n\geq0}$ be a continued fraction in $k$, either finite or infinite. Let $n \geq 0$ be a natural number. The '''$n$th partial quotient''' is the $n$th term $a_n$.
Definition:Continued Fraction/Simple/Finite
Let $n\geq 0$ be a natural number. A '''simple finite continued fraction of length $n$''' is a finite continued fraction in $\R$ of length $n$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a finite sequence $a : \left[0 \,.\,.\, n\right] \to \Z$ with $a_n > 0$ for $n >0$.
Definition:Continued Fraction/Simple/Infinite
A '''simple infinite continued fraction''' is a infinite continued fraction in $\R$ whose partial quotients are integers that are strictly positive, except perhaps the first. That is, it is a sequence $a : \N_{\geq 0} \to \Z$ with $a_n > 0$ for $n >0$.
Definition:Continued Fraction/Simple
Let $\R$ be the field of real numbers. === Simple Finite Continued Fraction === {{:Definition:Continued Fraction/Simple/Finite}} === Simple Infinite Continued Fraction === {{:Definition:Continued Fraction/Simple/Infinite}}
Definition:Value of Continued Fraction
Let $F$ be a field, such as the field of real numbers $\R$. === Finite Continued Fraction === {{Definition:Value of Continued Fraction/Finite}} === Infinite Continued Fraction === {{:Definition:Value of Continued Fraction/Infinite}}
Definition:Convergent of Continued Fraction
Let $F$ be a field, such as the field of real numbers. Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $C = \left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$. Let $k \leq n$ be a natural number. === Definition 1 === {{Definition:Convergent of Continued Fraction/Definition 1}} === Definition 2 === {{Definition:Convergent of Continued Fraction/Definition 2}}
Definition:Convergent of Continued Fraction/Even
The '''even convergents''' of $\left[{a_0, a_1, a_2, \ldots}\right]$ are the convergents $C_0, C_2, C_4, \ldots$, that is, those with an even subscript.
Definition:Convergent of Continued Fraction/Odd
The '''odd convergents''' of $\left[{a_0, a_1, a_2, \ldots}\right]$ are the convergents $C_1, C_3, C_5, \ldots$, that is, those with an odd subscript.