hcju/proofwikips huggingface dataset:
Definition:Ovals of Cassini/Shape
When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped. When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$. When $b = a$, $M$ degenerates into the lemniscate of Bernoulli.
Definition:Ovals of Cassini/Focus
Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the ovals.
Definition:Sigma-Algebra Generated by Collection of Subsets/Generator
One says that $\GG$ is a '''generator''' for $\map \sigma {\GG}$. Also, elements $G$ of $\GG$ may be called '''generators'''.
Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1
The '''$\sigma$-algebra generated by $\GG$''', $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$. That is, $\map \sigma \GG$ is subject to: :$(1): \quad \GG \subseteq \map \sigma \GG$ :$(2): \quad \GG \subseteq \Sigma \implies \map \sigma \GG \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$
Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2
The '''$\sigma$-algebra generated by $\GG$''', $\map \sigma \GG$, is the intersection of all $\sigma$-algebras on $X$ that contain $\GG$.
Definition:Limaçon of Pascal
Let $C$ be a circle of diameter $a$ with a distinguished point $O$ on the circumference. Let $OQ$ be a chord of $C$. The '''limaçons of Pascal''' are the loci of points $P$ in the plane such that: :$PQ = b$ where: :$OPQ$ is a straight line :$b$ is a real constant.
Definition:Limaçon of Pascal/Shape
Let $L$ denote a limaçon of Pascal. Depending on the value of $b$, the shape of $L$ is as follows: :For $b \ge 2 a$, $L$ is wholly convex. :For $a < b < 2 a$, $L$ has a concavity. :For $b = a$, $L$ degenerates to a cardioid. :For $0 < b < a$, $L$ has a loop inside its generating circle. :For $b = \dfrac a 2$, the internal loop of $L$ passes through the center of the generating circle. :For $b = 0$, $L$ degenerates to a circle. :For $b < 0$, $L$ is the same curve as for $-b$. :200px 160px 125px 108px 96px 90px 74px 70px
Definition:Differential Equation/Solution/Particular Solution
Let $S$ denote the solution set of $\Phi$. A '''particular solution''' of $\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\Phi$.
Definition:Convergent Sequence/Normed Vector Space
Let $\tuple {X, \norm {\,\cdot \,}}$ be a normed vector space. Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$. Let $L \in X$.  The sequence $\sequence {x_n}_{n \mathop \in \N}$ '''converges to the limit $L \in X$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$
Definition:Compact Space/Normed Vector Space
Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.  Let $K \subseteq X$. Then $K$ is '''compact'''  every sequence in $K$ has a convergent subsequence with limit $L \in K$. That is, if: :$\sequence {x_n}_{n \mathop \in \N} :\forall n \in \N : x_n \in K \implies \exists \sequence {x_{n_k} }_{k \mathop \in \N} : \exists L \in K: \displaystyle  \lim_{k \mathop \to \infty} x_{n_k} = L$
Definition:Producer of Dedekind Cut
Let $\struct {S, \preceq}$ be a totally ordered set. Let $S' \subseteq S$. Let $\tuple {L, R}$ be a Dedekind cut of $S'$. An $\alpha \in S$ is referred to as a '''producer''' of $\tuple {L, R}$ : :$l \preceq \alpha$ for all $l \in L$ :$\alpha \preceq r$ for all $r \in R$.
Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients
A '''linear first order ordinary differential equation with constant coefficients''' is a linear first order ordinary differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d y} {\d x} + a y = \map Q x$ where: :$\map Q x$ is a function of $x$ :$a$ is a constant.
Definition:Complementary Function of Linear First Order ODE With Constant Coefficients
Consider the linear first order ODE with constant coefficients: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ The general solution to the reduced equation: :$\dfrac {\d y} {\d x} + a y = 0$ is the '''complementary function''' of $(1)$. From First Order ODE: $\dfrac {\d y} {\d x} = k y$, the '''complementary function''' of $(1)$ is $C e^{-a x}$.
Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order
Consider the linear first order ODE with constant coefficients: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ The equation: :$\dfrac {\d y} {\d x} + a y = 0$ is the '''reduced equation''' of $(1)$.
Definition:Reduced Equation of Linear ODE with Constant Coefficients
Consider the linear $n$th order ODE with constant coefficients: :$(1): \quad \displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ The equation: :$\displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = 0$ is the '''reduced equation''' of $(1)$. === First Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order}} === Second Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order}}
Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order
Consider the linear second order ODE with constant coefficients: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$ The equation: :$\dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = 0$ is the '''reduced equation''' of $(1)$.
Definition:Linear Second Order ODE with Constant Coefficients
A '''linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = \map R x$ where: :$p$ and $q$ are real constants :$\map R x$ is a function of $x$. Thus it is a linear second order ODE: :$y'' + \map P x y' + \map Q x y = \map R x$ where $\map P x$ and $\map Q x$ are constant functions.
Definition:Open Ball/Normed Vector Space
Let $\struct{X, \norm {\,\cdot\,} }$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$''' is defined as: :$\map {B_\epsilon} x = \set {y \in X: \norm{x - y} < \epsilon}$
Definition:Open Set/Normed Vector Space
Let $V = \struct{X, \norm {\,\cdot\,} }$ be a normed vector space. Let $U \subseteq X$. Then $U$ is an '''open set in $V$''' : :$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$ where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.
Definition:Closed Set/Normed Vector Space
Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space. Let $F \subset X$. ==== Definition 1 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 1}} ==== Definition 2 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 2}}
Definition:P-adic Unit
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ denote the $p$-adic integers. The set of '''$p$-adic units''', denoted $\Z_p^\times$, is the set of invertible elements of $\Z_p$. From P-adic Unit has Norm Equal to One, the set of '''$p$-adic units''' is: :$\Z_p^\times = \set {x \in \Q_p: \norm x_p = 1}$
Definition:Linear nth Order ODE with Constant Coefficients
Let $n \in \Z_{>0}$ be a (strictly) positive integer. A '''linear $n$th order ODE with constant coefficient''' is an ordinary differential equation of order $n$ which can be manipulated into the form: :$\displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ where $a_k$ is a real constant for all $0 \le k \le n$. That is: :$a_n \dfrac {\d^n y} {d x^n} + a_{n - 1} \dfrac {\d^{n - 1} y} {d x^{n - 1} } + \dotsb + a_1 \dfrac {\d y} {d x} + a_0 y = \map R x$
Definition:Lie Theory
'''Lie theory''' is the study of Lie groups and Lie algebras.
Definition:Induced Norm on Quotient of Cauchy Sequences
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\mathcal C$ be the ring of Cauchy sequences over $R$ Let $\mathcal N$ be the set of null sequences. For all $\sequence {x_n} \in \mathcal C$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \mathcal N$ Let $\norm {\, \cdot \,}_1: \mathcal C \,\big / \mathcal N \to \R_{\ge 0}$ be defined by: :$\displaystyle \forall \eqclass {x_n} {} \in \mathcal C \,\big / \mathcal N: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$ $\norm {\, \cdot \,}_1$ is called the '''induced norm on the quotient ring of Cauchy sequences'''.
Definition:Linear Differential Operator
A '''linear differential operator''' is a differential operator $\mathscr L$ with the property that: :$\map {\mathscr L} {\alpha \phi_1 + \beta \phi_2} = \alpha \map {\mathscr L} {\phi_1} + \beta \map {\mathscr L} {\phi_2}$ Thus if $\phi_1$ and $\phi_2$ are solutions to the differential equation $\map {\mathscr L} {\phi_i} = 0$, then so is any linear combination of $\phi_1$ and $\phi_2$.
Definition:Linear Integral Operator
A '''linear integral operator''' is a integral operator $\mathscr L$ with the property that: :$\map {\mathscr L} {\alpha \phi_1 + \beta \phi_2} = \alpha \map {\mathscr L} {\phi_1} + \beta \map {\mathscr L} {\phi_2}$ Thus if $\phi_1$ and $\phi_2$ are solutions to the integral equation $\map {\mathscr L} {\phi_i} = 0$, then so is any linear combination of $\phi_1$ and $\phi_2$.
Definition:Set/Distinction between Element and Set
It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$. That is $a$ and $\set a$ are ''not'' the same thing. While it is true that: :$a \in \set a$ it is not true that: :$a = \set a$ neither is it true that: :$a \in a$
Definition:Experiment/Informal Definition
An '''experiment''' is defined as: : ''a course of action whose consequence is not predetermined.''
Definition:Experiment/Formal Definition
An '''experiment''', which can conveniently be denoted $\EE$, is a measure space $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr \Omega = 1$.
Definition:Sample Space/Discrete
If $\Omega$ is a countable set, whether finite or infinite, then it is known as a '''discrete sample space'''.
Definition:Event/Simple Event
A '''simple event''' in $\EE$ is an event in $\EE$ which consists of exactly $1$ elementary event. That is, it is a singleton subset of the sample space $\Omega$ of $\EE$.
Definition:Complementary Event
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A \in \Sigma$ be an event in $\EE$. The '''complementary event''' to $A$ is defined as $\relcomp \Omega A$. That is, it is the subset of the sample space of $\EE$ consisting of all the elementary events of $\EE$ that are not in $A$.
Definition:Trivial Event
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. The empty set $\O$ is a subset of $\Sigma$, by Empty Set is Subset of All Sets, and so is an event in $\EE$. It can be referred to as the '''the trivial event''' of $\EE$.
Definition:Probability Measure/Definition 1
Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$. Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.
Definition:Probability Measure/Definition 2
Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms: {{:Axiom:Kolmogorov Axioms}}
Definition:Probability Measure/Definition 3
Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = \text I         | q = \forall A \in \Sigma         | ml= \map \Pr A         | mo= \ge         | mr= 0 }} {{axiom | n = \text {II}         | ml= \map \Pr \Omega         | mo= =         | mr= 1 }} {{axiom | n = \text {III}         | q = \forall A \in \Sigma         | ml= \map \Pr A         | mo= =         | mr= \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e}         | rc= where $e$ denotes the elementary events of $\EE$ }} {{end-axiom}}
Definition:Probability Measure/Definition 4
Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = 1         | q = \forall A, B \in \Sigma: A \cap B = \O         | ml= \map \Pr {A \cup B}         | mo= =         | mr= \map \Pr A + \map \Pr B }} {{axiom | n = 2         | ml= \map \Pr \Omega         | mo= =         | mr= 1 }} {{end-axiom}}
Definition:Decimal
=== Decimal System === {{:Definition:Decimal System}} === Decimal Notation === {{:Definition:Decimal Notation}} === Decimal Expansion === {{:Definition:Decimal Expansion}} === Decimal Point === {{:Definition:Decimal Point|}} === Decimal Place === {{:Definition:Decimal Place}} === Decimal Part === Erroneously used to mean fractional part: {{:Definition:Decimal Part}}
Definition:Decimal Expansion/Decimal Place
Let the '''decimal expansion''' of $x$ be: :$x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$ Then $d_k$ is defined as being the digit in the $k$th '''decimal place'''.
Definition:Number Names
This page gathers together instances of names of numbers in various languages and contexts.
Definition:Number Names/Welsh
These are the names of the numbers in Welsh: {{begin-axiom}} {{axiom | n = 1         | lc = '''un''' }} {{axiom | n = 2         | lc = '''dau''', '''dwy''' }} {{axiom | n = 3         | lc = '''toi''', '''tair''' }} {{axiom | n = 4         | lc = '''pedwar''', '''pedair''' }} {{axiom | n = 5         | lc = '''pump''' (or '''pum''') }} {{axiom | n = 6         | lc = '''chwech''' (or '''chwe''') }} {{axiom | n = 7         | lc = '''saith''' }} {{axiom | n = 8         | lc = '''wyth''' }} {{axiom | n = 9         | lc = '''naw''' }} {{axiom | n = 10         | lc = '''deg''' }} {{axiom | n = 11         | lc = '''un ar ddeg''' }} {{axiom | n = 12         | lc = '''deuddeg''' }} {{axiom | n = 13         | lc = '''toi ar ddeg''', '''tair ar ddeg''' }} {{axiom | n = 14         | lc = '''pedwar ar ddeg''', '''pedair ar ddeg''' }} {{axiom | n = 15         | lc = '''pymtheg''' }} {{axiom | n = 16         | lc = '''un ar bymtheg''' }} {{axiom | n = 17         | lc = '''dau ar bymtheg''', '''dwy ar bymtheg''' }} {{axiom | n = 18         | lc = '''deunaw''' }} {{axiom | n = 19         | lc = '''pedwar ar bymtheg''', '''pedair ar bymtheg''' }} {{axiom | n = 20         | lc = '''ugain''' }} {{axiom | n = 30         | lc = '''deg ar hugain''' }} {{axiom | n = 40         | lc = '''dugain''' }} {{axiom | n = 50         | lc = '''deg ar deugain''' or '''hanner cant''' }} {{axiom | n = 60         | lc = '''trigain''' }} {{axiom | n = 70         | lc = '''trigain a deg''' or '''deg ar thrigain''' }} {{axiom | n = 80         | lc = '''pedwar ugain''' }} {{axiom | n = 90         | lc = '''pedwar ugain a deg''' or '''deg a phedwar ugain''' }} {{axiom | n = 100         | lc = '''cant''' }} {{end-axiom}} The different forms for $2$, $3$, $4$ and their derivatives are for masculine and feminine. The different forms for $5$ and $6$ are elisions for use in front of certain letters. The different forms for $50$, $70$ and $90$ are separate ways of saying the numbers. Note: :there are no distinct words for $11$ and $12$, indicating that there has been less influence of duodecimal systems on Welsh :there are clear indications of base $5$ and vigesimal influences as well as decimal :the interesting forms for $18$ (expressed as $2 \times 9$) and $50$ (expressed both as $10 + 40$ and $\dfrac 1 2 \times 100$).
Definition:Continuous Real Function/Point/Definition by Epsilon-Delta
Then '''$f$ is continuous at $x$'''  the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and: :$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$
Definition:Continuous Real Function/Point/Definition by Neighborhood
Then '''$f$ is continuous at $x$'''  the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and: :$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$ :for every $\epsilon$-neighborhood $N_\epsilon$ of $\map f x$ in $\R$, there exists a $\delta$-neighborhood $N_\delta$ of $x$ in $A$ such that $\map f x \in N_\epsilon$ whenever $x \in N_\delta$.
Definition:Approximation
An '''approximation''' is an estimate of a quantity. It is usually the case that there exists some knowledge about the accuracy of the estimate. The notation: :$a \approx b$ indicates that $b$ is an '''approximation''' to $a$.
Definition:Cut (Analysis)
Let $\alpha \subset \Q$ be a subset of the set of rational numbers $\Q$ which has the following properties: :$(1): \quad \alpha \ne \O$ and $\alpha \ne \Q$, that is: $\alpha$ contains at least one rational number but not all rational numbers :$(2): \quad$ If $p \in \Q$ and $q \in \Q$ such that $q < p$, then $q \in \Q$ :$(3): \quad \alpha$ does not contain a greatest element. Then $\alpha$ is called a '''cut'''.
Definition:Cut (Analysis)/Lower Number
Let $\alpha$ be a cut. Let $p \in \alpha$. Then $p$ is referred to as a '''lower number''' of $\alpha$.
Definition:Cut (Analysis)/Upper Number
Let $\alpha$ be a cut. Let $q \in \Q$ such that $q \notin \alpha$. Then $p$ is referred to as an '''upper number''' of $\alpha$.
Definition:Cut (Analysis)/Rational
Let $r \in \Q$ be a rational number. Let $\alpha$ be the cut consisting of all rational numbers $p$ such that $p < r$. Then $\alpha$ is referred to as a '''rational cut'''.
Definition:Equality of Cuts
Let $\alpha$ and $\beta$ be cuts such that: :$p \in \alpha \implies p \in \beta$ :$q \in \beta \implies q \in \alpha$ Then $\alpha$ and $\beta$ are '''equal'''.
Definition:Ordering of Cuts/Strict
:$\alpha$ '''is less than ''' $\beta$, denoted $\alpha < \beta$ : :there exists a rational number $p \in \Q$ such that $p \in \alpha$ but $p \notin \beta$.
Definition:Ordering of Cuts
Let $\alpha$ and $\beta$ be cuts. $\alpha$ and $\beta$ conventionally have the following ordering imposed on them, as follows: :$\alpha$ '''is less than or equal to''' $\beta$, denoted $\alpha \le \beta$ : :$\alpha = \beta$ or $\alpha < \beta$ where $<$ denotes the strict ordering relation between $\alpha$ and $\beta$. This can also be expressed as $\beta \ge \alpha$. === Strict Ordering === {{:Definition:Ordering of Cuts/Strict}}
Definition:Positive Cut/Strictly Positive
Let $\alpha > 0^*$. Then $\alpha$ can be referred to as a '''strictly positive cut'''.
Definition:Positive Cut
Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\alpha \ge 0^*$. Then $\alpha$ can be referred to as a '''positive cut'''. === Strictly Positive Cut === {{:Definition:Positive Cut/Strictly Positive}}
Definition:Negative Cut
Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\alpha \le 0^*$. Then $\alpha$ can be referred to as a '''negative cut'''. === Strictly Negative Cut === {{:Definition:Negative Cut/Strictly Negative}}
Definition:Negative Cut/Strictly Negative
Let $\alpha < 0^*$. Then $\alpha$ can be referred to as a '''strictly negative cut'''.
Definition:Addition of Cuts
Let $\alpha$ and $\beta$ be cuts. Let the operation of '''addition''' be defined on $\alpha$ and $\beta$ as: :$\gamma := \alpha + \beta$ where $\gamma$ is the set of all rational numbers $r$ such that: :$\exists p \in \alpha, q \in \beta: r = p + q$ In this context, $\gamma$ is known as the '''sum of $\alpha$ and $\beta$'''.
Definition:Negative of Cut
Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Let $\beta$ be the unique cut such that: :$\alpha + \beta = 0^*$ where $+$ denotes the operation of addition of cuts. Then $\beta$ is referred to as the '''negative''' of $\alpha$. It is usually denoted $-\alpha$.
Definition:Subtraction of Cuts
Let $\alpha$ and $\beta$ be cuts. Let $\beta - \alpha$ denote the operation on $\alpha$ and $\beta$ defined as $\beta + \paren {-\alpha}$. Then the operator $-$ is known as '''subtraction''', and $\beta - \alpha$ is itself called the '''difference''' between $\beta$ and $\alpha$.
Definition:Multiplication of Positive Cuts
Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be positive cuts, that is, cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts. Let the operation of '''multiplication''' be defined on $\alpha$ and $\beta$ as: :$\gamma := \alpha \beta$ where $\gamma$ is the set of all rational numbers $r$ such that either: :$r < 0$ or :$\exists p \in \alpha, q \in \beta: r = p q$ where $p \ge 0$ and $q \ge 0$. In this context, $\gamma$ is known as the '''product of $\alpha$ and $\beta$'''.
Definition:Multiplication of Cuts
Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be cuts. The operation of '''multiplication''' is defined on $\alpha$ and $\beta$ as: :$\alpha \beta := \begin {cases} \size \alpha \, \size \beta & : \alpha \ge 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha < 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha \ge 0^*, \beta < 0^* \\ \size \alpha \, \size \beta & : \alpha < 0^*, \beta < 0^* \end {cases}$ where: :$\size \alpha$ denotes the absolute value of $\alpha$ :$\size \alpha \, \size \beta$ is defined as in Multiplication of Positive Cuts :$\ge$ denotes the ordering on cuts. In this context, $\alpha \beta$ is known as the '''product of $\alpha$ and $\beta$'''.
Definition:Absolute Value of Cut
Let $\alpha$ be a cut. The '''absolute value of $\alpha$''' is denoted and defined as: :$\size \alpha = \begin {cases} \alpha & : \alpha \ge 0^* \\ -\alpha & : \alpha < 0^* \end {cases}$ where: :$0^*$ denotes the rational cut associated with the (rational) number $0$ :$\ge$ denotes the ordering on cuts.
Definition:Ordered Field Isomorphism
Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered fields. An '''ordered field isomorphism''' from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$ :$(2): \quad$ A group isomorphism from the group $\struct {S_{\ne 0}, \circ}$ to the semigroup $\struct {T_{\ne 0}, *}$ where $S_{\ne 0}$ and $T_{\ne 0}$ denote the sets $S$ and $T$ without the zeros of $S$ and $T$ respectively.
Definition:Real Number/Also known as
When the term '''number''' is used in general discourse, it is often tacitly understood as meaning '''real number'''. They are sometimes referred to in the pedagogical context as '''ordinary numbers''', so as to distinguish them from '''complex numbers''' However, depending on the context, the word '''number''' may also be taken to mean '''integer''' or '''natural number'''. Hence it is wise to be specific.
Definition:Real Number/Number Line Definition
A '''real number''' is defined as a '''number''' which is identified with a point on the '''real number line'''. ==== Real Number Line ==== {{:Definition:Real Number/Real Number Line}}
Definition:Real Number/Cauchy Sequences
Consider the set of rational numbers $\Q$. For any two Cauchy sequences of rational numbers $X = \sequence {x_n}, Y = \sequence {y_n}$, define an equivalence relation between the two as: :$X \equiv Y \iff \forall \epsilon \in \Q_{>0}: \exists n \in \N: \forall i, j > n: \size {x_i - y_j} < \epsilon$ A '''real number''' is an equivalence class $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Definition:Real Number/Digit Sequence
Let $b \in \N_{>1}$ be a given natural number which is greater than  $1$. The set of '''real numbers''' can be expressed as the set of all sequences of digits: :$z = \sqbrk {a_n a_{n - 1} \dotsm a_2 a_1 a_0 \cdotp d_1 d_2 \dotsm d_{m - 1} d_m d_{m + 1} \dotsm}$ such that: :$0 \le a_j < b$ and $0 \le d_k < b$ for all $j$ and $k$ :$\displaystyle z = \sum_{0 \mathop \le j \le n} a_j b^j + \sum_{k \mathop \ge 1} d_k b^{-k}$ It is usual for $b$ to be $10$.