hcju/proofwikips huggingface dataset:
Definition:Successive Values of Time Series
Let $T$ be a time series whose observations are $\map z {\tau_1}, \map z {\tau_2}, \dotsb, \map z {\tau_t}, \dotsb$ at an unbroken sequence of timestamps $\tau_1, \tau_2, \dotsb, \tau_t, \dotsb$. When we have $N$ such '''successive''' observations, without a missing value, we write: :$z_1, z_2, \dotsb, z_t, \dotsb, z_N$ and it is understood that the subscripts correspond directly to the timestamps. === Equispaced Time Series === {{:Definition:Successive Values of Time Series/Equispaced}}
Definition:Successive Values of Time Series/Equispaced
Let $T$ be equispaced with time interval $h$ between adjacent observations.  Then $N$ '''successive observations''', written as: :$z_1, z_2, \dotsb, z_t, \dotsb, z_N$ occur at timestamps: :$\tau_0 + h, \tau_0 + 2 h, \dotsb, \tau_0 + t h, \dotsb, \tau_0 + N h$ Hence we can refer to the observations at timestamp $\tau_0 + t h$ as $z_t$.
Definition:Discrete Time Series/Sampling
A '''discrete time series''' can be obtained by taking observations at predetermined instants from a comtinuous time series of measurements of the process in question.
Definition:Discrete Time Series/Accumulation
A '''discrete time series''' can be obtained by collecting a varying quantity over a period of time. The observations are the measurements of the quantities at the end of each period.
Definition:Space of Bounded Sequences
The '''space of bounded sequences''', denoted $\ell^\infty$ is defined as: :$\displaystyle \ell^\infty := \set{\sequence{z_n}_{n \mathop \in \N} \in \C^\N: \sup_{n \mathop \in \N} \size {z_n} < \infty}$ As such, $\ell^\infty$ is a subspace of $\C^\N$, the space of all complex sequences.
Definition:Field Negative
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $x \in F$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {F, +}$ of $F$ is referred to as the '''(field) negative''' of $x$ and is denoted $-x$. That is, the '''(field) negative''' of $x$ is the element $-x$ of $F$ such that: :$x + \paren {-x} = 0_F$
Definition:Realization of Stochastic Process
Let $S$ be a stochastic process. Let $T$ be a time series of observations of $S$ which has been acquired as $S$ evolves, according to the underlying probability distribution of $S$. Then $T$ is referred to as a '''realization''' of $S$. Thus we can regard the observation $z_t$ at some timestamp $t$, for example $t = 25$, as the '''realization''' of a random variable with probability mass function $\map p {z_t}$. Similarly the observations $z_{t_1}$ and $z_{t_2}$ at times $t_1$ and $t_2$ can be regarded as the '''realizations''' of two random variables with joint probability mass function $\map p {z_{t_1} }$ and $\map p {z_{t_2} }$. Similarly, the observations making an equispaced time series can be described by an $N$-dimensional random variable $\tuple {z_1, z_2, \dotsc, z_N}$ with associated probability mass function $\map p {z_1, z_2, \dotsc, z_N}$.
Definition:Strictly Stationary Stochastic Process
Let $S$ be a stochastic process giving rise to a time series $T$. $S$ is a '''strictly stationary stochastic process''' if its properties are unaffected by a change of origin of $T$.
Definition:Mean of Stochastic Process
Let $S$ be a stochastic process giving rise to a time series $T$. The '''mean''' of $S$ over an interval $Q$ is the arithmetic mean of the observations of $T$ over all the timestamps within $Q$.
Definition:Variance of Stochastic Process
Let $S$ be a stationary stochastic process giving rise to a time series $T$. The '''variance''' of $S$ is calculated as: :$\sigma_z^2 = \expect {\paren {z_t - \mu}^2} = \displaystyle \int_{-\infty}^\infty \paren {z - \mu}^2 \map p z \rd z$ where $\map p z$ is the (constant) probability mass function of $S$. It is a measure of the spread about the constant mean level $\mu$.
Definition:Histogram
A '''histogram''' is a form of bar graph used to represent the frequency distribution grouped by class intervals.
Definition:Sample Mean of Stochastic Process
Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample mean''' of $S$ over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$ is defined as: :$\overline z := \dfrac 1 N \displaystyle \sum_{t \mathop = 1}^N z_t$
Definition:Sample Variance of Stochastic Process
Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample variance''' of $S$ over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$ is defined as: :$\hat \sigma_z^2 := \dfrac 1 N \displaystyle \sum_{t \mathop = 1}^N \paren {z_t - \overline z}^2$ where $\overline z$ denotes the sample mean of $S$ over $\set {z_1, z_2, \dotsb, z_N}$.
Definition:Scatter Diagram
A '''scatter diagram''' is a technique of graphical representation of the distribution of ordered pairs of random variables. The ordered pairs are treated as points in a Cartesian plane: the first coordinate along the $x$-axis and the second coordinate along the $y$-axis.
Definition:Lag
Let $T$ be a time series. A '''lag''' is a constant time interval between two timestamps of $T$. Thus, for every observation $z_t$ of $T$, a pair of observations can be created with a given '''lag''' $k$ between them: :$\tuple {z_t, z_{t + k} }$
Definition:Field (Abstract Algebra)/Addition
The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as '''field addition''', or just '''addition'''.
Definition:Field (Abstract Algebra)/Product
The distributive operation $\times$ in $\struct {F, +, \times}$ is known as the '''(field) product'''.
Definition:Additive Group of Field
The group $\struct {F, +}$ is known as the '''additive group of $F$'''.
Definition:Additive Inverse
=== Additive Inverse in Ring === {{:Definition:Additive Inverse/Ring}} === Additive Inverse in Field === {{:Definition:Additive Inverse/Field}} === Additive Inverse of Number === The concept is often encountered in the context of numbers: {{:Definition:Additive Inverse/Number}}
Definition:Additive Inverse/Field
Let $\struct {F, +, \times}$ be a field whose addition operation is $+$. Let $a \in R$ be any arbitrary element of $F$. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \paren {-a} = 0_F$ where $0_F$ is the zero of $R$.
Definition:Unity (Abstract Algebra)
=== Unity of Ring === {{:Definition:Unity (Abstract Algebra)/Ring}} === Unity of Field === {{:Definition:Unity (Abstract Algebra)/Field}}
Definition:Unity (Abstract Algebra)/Field
Let $\struct {F, +, \times}$ be a field. The identity of the multiplicative group $\struct {F, \times}$ is referred to as '''the unity of the field $\struct {F, +, \times}$'''. It is (usually) denoted $1_F$, where the subscript denotes the particular field to which $1_F$ belongs (or often $1$ if there is no danger of ambiguity).
Definition:Space of Convergent Sequences
The '''space of convergent sequences''', denoted $c$ is defined as: :$\displaystyle c := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \exists L \in \C : \forall \epsilon \in \R_{>0} : \exists N \in \R: n > N \implies \cmod {z_n - L} < \epsilon}$ As such, $c$ is a subspace of $\C^\N$, the space of all complex sequences.
Definition:Scalar Addition
=== Scalar Addition on Module === {{:Definition:Scalar Addition/Module}} === Scalar Addition on Vector Space === {{:Definition:Scalar Addition/Vector Space}}
Definition:Scalar Addition/Vector Space
Let $\struct {K, +_K, \times_K}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_K$ be the corresponding '''vector space over $K$'''. The field addition operation $+_K$ on $V$ is known as '''scalar addition''' on $V$.
Definition:Scalar Addition/Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. Let $M := \struct {G, +_G, \circ}_R$ be the corresponding '''module over $R$''' (either a left module or a right module). The ring addition operation $+_R$ on $M$ is known as '''scalar addition''' on $M$.
Definition:Vector Addition/Vector Space
Let $\struct {F, +_F, \times_F}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_R$ be the corresponding '''vector space over $F$'''. The group operation $+_G$ on $V$ is known as '''vector addition''' on $V$.
Definition:Scalar Multiplication/Module
Let $\struct {G, +_G, \circ}_R$ be an module (either a left module or a right module or both), where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {G, +_G}$ is an abelian group. The operation $\circ: R \times G \to G$ is called '''scalar multiplication'''.
Definition:Scalar Multiplication/Vector Space
Let $\struct {G, +_G, \circ}_K$ be an vector space, where: :$\struct {K, +_K, \times_K}$ is a field :$\struct {G, +_G}$ is an abelian group. The operation $\circ: K \times G \to G$ is called '''scalar multiplication'''.
Definition:Negative of Element
=== Ring Negative === {{:Definition:Ring Negative}} === Field Negative === As a field is also a ring, the same definition can be used: {{:Definition:Field Negative}}
Definition:Space of Zero-Limit Sequences
The '''space of zero-limit sequences''', denoted $c$ is defined as: :$\displaystyle c_0 := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \forall \epsilon \in \R_{>0} : \exists N \in \R_{> 0}: n > N \implies \cmod {z_n} < \epsilon}$ As such, $c_0$ is a subspace of $\C^\N$, the space of all complex sequences.
Definition:Space of Almost-Zero Sequences
The '''space of almost-zero sequences''', denoted $c_{00}$ is defined as: :$\displaystyle c_{00} := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \exists N \in \R_{> 0}: n > N \implies z_n =0}$ As such, $c_{00}$ is a subspace of $\C^\N$, the space of all complex sequences.
Definition:C^k Norm
Let $f \in \map {\CC^k} {\R}$ be a function of differentiability class $k$. Then '''$C^k$ norm''' is defined as: :$\displaystyle \norm {f}_{\map {C^k} \R} := \sum_{i \mathop = 0}^k \sup_{x \in \R} \size {\map {f^{\paren i}} x} = \sum_{i \mathop = 0}^k \norm{f^{\paren i} }_\infty$ where $f^{\paren i}$ denotes the $i$-th derivative  $x$, and $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.
Definition:Autocovariance
Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocovariance''' of $S$ at lag $k$ is defined as: :$\gamma_k := \cov {z_t, z_{t + k} } = \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$ where: :$z_t$ is the observation at time $t$ :$\mu$ is the mean of $S$ :$\expect \cdot$ is the expectation.
Definition:Autocorrelation
Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocorrelation''' of $S$ at lag $k$ is defined as: :$\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$ where: :$z_t$ is the observation at time $t$ :$\mu$ is the mean of $S$ :$\expect \cdot$ is the expectation.
Definition:Autocovariance Matrix
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ The '''autocovariance matrix''' associated with $S$ for $\sequence {s_n}$ is: :$\boldsymbol \Gamma_n = \begin {pmatrix} \gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix}$ where $\gamma_k$ is the autocovariance of $S$ at lag $k$. That is, such that: :$\sqbrk {\Gamma_n}_{i j} = \gamma_{\size {i - j} }$
Definition:Autocorrelation Matrix
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ The '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$ is: :$\mathbf P_n = \begin {pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$ where $\rho_k$ is the autocorrelation of $S$ at lag $k$. That is, such that: :$\sqbrk {P_n}_{i j} = \rho_{\size {i - j} }$
Definition:Positive Definite Matrix
Let $\mathbf A$ be a square matrix of order $n$. $\mathbf A$ is '''positive definite''' : :$(1): \quad \mathbf A$ is symmetric :$(2): \quad$ for all column matrices $\mathbf x$ of order $n$, $\mathbf x^\intercal \mathbf A \mathbf x$ is strictly positive.
Definition:Matrix/Order/Column
Let $\mathbf A$ be an $n \times 1$ column matrix. Then the '''order''' of $\mathbf A$ is defined as being $n$.
Definition:Induced Norm
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $Y \subseteq X$ be a subspace. Then the '''induced norm (on $Y$)''' is defined as the restriction of $\norm {\, \cdot \,}_X$ to $Y$. :$\norm {\, \cdot \,}_Y := \norm {\, \cdot \,}_X {\restriction_Y}$
Definition:Wholly Real/Abbreviated Notation
Let $z = a + i b$ be a complex number such that $b = 0$. That is, let $z$ be wholly real: $z = a + 0 i$, or $\tuple {a, 0}$ Despite the fact that $z$ is still a complex number, it is commonplace to use the same notation as if it were a real number, and hence say $z = a$. While it is in theory important to distinguish between a real number and its corresponding wholly real complex number, in practice it makes little difference.
Definition:Pure Mathematics
'''Pure mathematics''' is the branch of mathematics which is not concerned with real-world applications.
Definition:Parallel (Geometry)/Line to Plane
Let $L$ be a straight line. Let $P$ be a plane. Then $L$ and $P$ are '''parallel''' , when produced indefinitely, they do not intersect at any point.
Definition:Method of Least Squares
=== Approximation Theory === {{:Definition:Method of Least Squares (Approximation Theory)}} === Statistics === {{:Definition:Method of Least Squares (Statistics)}}
Definition:Cardinality/Natural Numbers
When the natural numbers are defined as von Neumann construction of natural numbers, the cardinality function can be viewed as the identity mapping on $\N$. That is: :$\forall n \in N: \card n := n$
Definition:Spline
=== Draftsman's tool === {{:Definition:Spline (Tool)}} === Spline Function === {{:Definition:Spline Function}}
Definition:Spline (Tool)
:400px A '''spline''' is a drawing tool which consists of a long strip of uniformly flexible material fixed in position at a number of points whose tension creates a smooth curve passing through those points. Its purpose is to transferring that curve to another material.
Definition:Spline Function
Let $\closedint a b$ be a closed real interval. Let $T : = \set {a = t_0, t_1, t_2, \ldots, t_{n - 1}, t_n = b}$ form a subdivision of $\closedint a b$. Let $S: \closedint a b \to \R$ be a continuous function on $\closedint a b$ whose values on $t_0, t_1, \ldots, t_n$ are known. On each of the intervals $\closedint {t_k} {t_{k + 1} }$, let $P_k: \closedint {t_k} {t_{k + 1} }: \R$ be a polynomial function such that: :for $t$ on each of $t_k < t < t_{k + 1}$: $\map S t = \map {P_k} t$ The function $S: \closedint a b \to \R$ is known as a '''spline function''' on $T$.
Definition:Spline Function/Knot
The points $T := \set {t_0, t_1, t_2, \ldots, t_{n - 1}, t_n}$ of $S$ are known as the '''knots'''.
Definition:Spline Function/Knot/Knot Vector
The ordered $n + 1$-tuple $\mathbf t := \tuple {t_0, t_1, t_2, \ldots, t_{n - 1}, t_n}$ of $S$ is known as the '''knot vector'''.
Definition:Spline Function/Uniform
$S$ is a '''uniform spline'''  $T$ is a normal subdivision. That is,  the knots of $S$ are equally spaced.
Definition:Spline Function/Degree
The '''degree''' of $S$ is the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$.
Definition:Spline Function/Order
Some sources, instead of referring to the '''degree''' of a spline, use the '''order'''. Let the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$ be $n$. The '''order''' of $S$ is then $n + 1$.
Definition:Fundamental Circuit (Matroid)
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $x \in S \setminus B$. The '''fundamental circuit of $x$ in the base B''', denoted $\map C {x, B}$, is the unique circuit such that: :$x \in \map C {x, B} \subseteq B \cup \set x$
Definition:Set/Uniqueness of Elements/Multiple Specification
For a given '''set''', an object is either '''in''' the set or '''not in''' the set. So, if an element is in a set, then it is in the set '''only once''', however many times it may appear in the definition of the set. Thus, the set $\set {1, 2, 2, 3, 3, 4}$ is the same set as $\set {1, 2, 3, 4}$. $2$ and $3$ are in the set, and '''listing''' them twice makes no difference to the set's '''contents'''. Like the membership of a club, if you're in, you're in -- however many membership cards you have to prove it.
Definition:Set/Uniqueness of Elements/Order of Listing
It makes no difference in what order the elements of a set are specified. This means that the sets $S = \set {1, 2, 3, 4}$ and $T = \set {3, 4, 2, 1}$ are the ''same set''.
Definition:Set/Uniqueness of Elements/Equality of Sets
By definition of '''set equality''' {{:Definition:Set Equality/Definition 1}} So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as ''the same club'', although they may be given different '''names'''. This follows from the definition of '''equals''' given above. Note that there '''are''' mathematical constructs which '''do''' take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are ''not'' '''sets''' as such.
Definition:Parabola/Vertex
The '''vertex''' of $P$ is the point where the axis intersects $P$.
Definition:Parabola/Axis
The '''axis''' of $P$ is the straight line passing through the focus of $P$ perpendicular to the directrix $D$.
Definition:Age (Time)
The '''age''' of a physical objectis defined as the period of time over which it has been in existence.
Definition:Classical Algorithm
The '''classical algorithms''' are the conventional techniques for performing arithmetic on numbers with more than $1$ digit.
Definition:Multi-Digit Integer
Let $b \in \N_{>1}$ be a natural number greater than $1$. An '''$n$-digit integer base $b$''' is an integer which has no more than $n$ digits when expressed in base $b$. That is, it is strictly less than $b^n$.
Definition:Classical Algorithm/Primitive Operation
The '''primitive operations''' of the '''classical algorithms''' are the following operations which are performed on '''two''' $1$-digit integers $x$ and $y$: === Addition === {{:Definition:Classical Algorithm/Primitive Addition}} === Subtraction === {{:Definition:Classical Algorithm/Primitive Subtraction}} === Multiplication === {{:Definition:Classical Algorithm/Primitive Multiplication}} === Division === {{:Definition:Classical Algorithm/Primitive Division}}
Definition:Carry Digit
A '''carry digit''' is a digit that appears as a result of the operation of one of the primitive operations of the classical algorithms which is to be applied to a further primitive operation in some manner appropriate to the classical algorithm in question.
Definition:Classical Algorithm/Primitive Addition
Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = 1$. Addition of $x$, $y$ and $z$ gives a $1$-digit '''sum''' $s$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = s       | r = \paren {x + y + z} \pmod b }} {{eqn | l = c       | r = \floor {\dfrac {x + y + z} b} }} {{end-eqn}}
Definition:Classical Algorithm/Primitive Subtraction
Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = -1$. Subtraction of $y$ from $x$ with $z$ gives a $1$-digit '''difference''' $d$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = d       | r = \paren {x - y + z} \pmod b }} {{eqn | l = c       | r = \floor {\dfrac {x - y + z} b} }} {{end-eqn}}
Definition:Classical Algorithm/Primitive Multiplication
Multiplication of '''two''' $1$-digit integers $x$ and $y$ gives a $1$-digit '''product''' $p$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = p       | r = x \times y \pmod b }} {{eqn | o =  }} {{eqn | l = c       | r = \dfrac {x \times y - p} b }} {{end-eqn}}
Definition:Classical Algorithm/Primitive Division
Let $y$ be a $1$-digit integer. Let $x$ be a $2$-digit integer $x_1 b + x_2$, where $x_1$ and $x_2$ are $1$-digit integers. Suppose $x_1 \ge y$. Then division of $x_1$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x_1       | r = q \times y + r }} {{end-eqn}} Suppose $x_1 < y$. Then division of $x = x_1 b + x_2$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x       | r = q \times y + r }} {{end-eqn}}
Definition:Classical Algorithm/Primitive Addition/Base 10 Addition Table
The primitive addition operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} s & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4  \\ 6 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5  \\ 7 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6  \\ 8 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7  \\ 9 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 4 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 5 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 6 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 7 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 8 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 9 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}$
Definition:Classical Algorithm/Primitive Subtraction/Base 10 Subtraction Table
The primitive subtraction operation for conventional base $10$ arithmetic on two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} d & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ 1 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 \\ 2 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 \\ 3 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 \\ 4 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 \\ 5 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6  \\ 6 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7  \\ 7 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8  \\ 8 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9  \\ 9 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 4 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$