hcju/proofwikips huggingface dataset:
Definition:Trisection
To '''trisect''' a finite geometrical object is to cut it into $3$ equal parts.
Definition:Neusis Ruler
A '''neusis ruler''' is a straightedge on which two points have been marked with a specific fixed length between them.
Definition:Neusis Construction
A '''neusis construction''' is a technique of drawing geometric figures using only a neusis ruler and a compass.
Definition:Neusis Ruler/Diastema
On a neusis ruler, the '''diastema''' is the specific fixed length between the two marked points.
Definition:Line/Curve/Plane
A '''plane curve''' is a curve which can be embedded in the plane.
Definition:Conic Section/Focus-Directrix Property
A '''conic section''' is a plane curve which can be specified in terms of: :a given straight line $D$ known as the directrix :a given point $F$ known as a focus :a given constant $\epsilon$ known as the eccentricity. Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition: :$(1): \quad q = \epsilon \, p$ Then $K$ is a '''conic section'''. Equation $(1)$ is known as the '''focus-directrix property''' of $K$.
Definition:Conic Section/Intersection with Cone
:600px Let $C$ be a double napped right circular cone whose base is $B$. Let $\theta$ be half the opening angle of $C$. That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$. Let a plane $D$ intersect $C$. Let $\phi$ be the inclination of $D$ to the axis of $C$. Let $K$ be the set of points which forms the intersection of $C$ with $D$. Then $K$ is a '''conic section''', whose nature depends on $\phi$.
Definition:Parabola/Focus-Directrix
:300px Let $D$ be a straight line. Let $F$ be a point. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$: :$p = q$ Then $K$ is a '''parabola'''.
Definition:Hyperbola/Focus-Directrix
:300px Let $D$ be a straight line. Let $F_1$ be a point. Let $\epsilon \in \R: \epsilon > 1$. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F_1$ are related by the condition: :$\epsilon \, p = q$ Then $K$ is a '''hyperbola'''.
Definition:Conic Section/Directrix
The line $D$ is known as the '''directrix''' of the conic section.
Definition:Conic Section/Focus
The point $F$ is known as the '''focus''' of the conic section.
Definition:Conic Section/Eccentricity
The constant $\epsilon$ is known as the '''eccentricity''' of the conic section.
Definition:Ellipse/Focus
The point $F$ is known as the '''focus''' of the ellipse.
Definition:Ellipse/Directrix
The line $D$ is known as the '''directrix''' of the ellipse.
Definition:Ellipse/Eccentricity
The constant $\epsilon$ is known as the '''eccentricity''' of the ellipse.
Definition:Hyperbola/Focus
The point $F_1$ is known as a '''focus''' of the hyperbola. The symmetrically-positioned point $F_2$ is also a '''focus''' of the hyperbola.
Definition:Hyperbola/Directrix
The line $D$ is known as the '''directrix''' of the hyperbola.
Definition:Hyperbola/Eccentricity
The constant $\epsilon$ is known as the '''eccentricity''' of the hyperbola.
Definition:Parabola/Focus
The point $F$ is known as the '''focus''' of the parabola.
Definition:Parabola/Directrix
The line $D$ is known as the '''directrix''' of the parabola.
Definition:Greek Numerals
=== Attic System === {{:Definition:Greek Numerals/Attic System}} === Classical Period === {{:Definition:Greek Numerals/Classical Period}}
Definition:Greek Numerals/Attic System
The '''Greek numerals''' from the Attic period (c. 500 BCE to c. 300 BCE) are similar to the Roman system: {{begin-eqn}} {{eqn | l = \textsf{I}:       | o =        | r = \)one\( }} {{eqn | l = \Pi:       | o = \)'''penta'''\(:       | r = \)five\( }} {{eqn | l = \Delta:       | o = \)'''deka'''\(:       | r = \)ten\( }} {{eqn | l = \Eta:       | o = \)'''hekaton'''\(:       | r = \)one hundred\( }} {{eqn | l = \Xi:       | o = \)'''chilioi'''\(:       | r = \)one thousand\( }} {{eqn | l = \Mu:       | o = \)'''myriad'''\(:       | r = \)ten thousand\( }} {{end-eqn}} $\Pi$ was later written in a different form, with a shorter right leg, and from there it appears to have evolved into $\Gamma$. Numbers were combined in a similar way to Roman numbers, but with additive forms only. For example, $32718$ would have been written $\Mu \Mu \Mu \Xi \Xi \Eta \Eta \Eta \Eta \Eta \Eta \Eta \Delta \Pi \textsf {III}$. Later evolutions introduced symbols for $50$, $500$ and $5000$, consisting of a tiny version of the appropriate power of $10$ nestled under the right hand (short) branch of the short-leg version of $\Pi$. Such characters are difficult to render neatly, and as they are of limited importance, this has not been attempted on .
Definition:Greek Numerals/Classical Period
The '''Greek numerals''' from the Classical period (c. $600$ BCE to c. $300$ BCE) are as follows: :$\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \alpha & \beta & \gamma & \delta & \epsilon & stigma & \zeta & \eta & \theta \\ \hline \\ 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\ \iota & \kappa & \lambda & \mu & \nu & \xi & \omicron & \pi & koppa \\ \hline \\ 100 & 200 & 300 & 400 & 500 & 600 & 700 & 800 & 900 \\ \rho & \sigma & \tau & \upsilon & \phi & \chi & \psi & \omega & sampi \\ \end{array}$ where $stigma$, $koppa$ and $sampi$ are characters that are not supported by MathJax. These were the letters of the everyday Greek alphabet, with extra ones added from the Phoenician alphabet. As it could be easy to confuse numbers with the letters of Greek words, a line was written over the numbers to distinguish them. Numbers bigger than $999$ could be written by putting a stroke in front of the symbols.
Definition:Rational Number/Fraction/Denominator
The term $b$ is known as the '''denominator''' of $\dfrac a b$.
Definition:Rational Number/Fraction/Numerator
The term $a$ is known as the '''numerator''' of $\dfrac a b$.
Definition:Trigonometry
'''Trigonometry''' is the branch of mathematics which studies the relationships between the lengths of sides and angles of triangles.
Definition:Spherical Geometry
'''Spherical geometry''' is the branch of mathematics which concerns spatial relationships on the surface of a sphere.
Definition:Spherical Trigonometry
'''Spherical trigonometry''' is the branch of mathematics which concerns the measurement of spherical triangles and related figures on the surface of a sphere.
Definition:Napier's Bones
rightNapier's Rods '''Napier's bones''' is a device for performing the operation of multiplication of numbers. The device consists of: : a set of '''rods''', one for each digit $0$ to $9$, containing the multiples of that number, again from $0$ to $9$, in a column, written as $2$ digits, divided by a diagonal line. : a '''board''' upon which the '''rods''' can be placed, labeled $1$ to $9$ down a column on the left hand side, each digit corresponding to the numbers on the '''rods''' The multiplicand is assembled by placing the rods for each of its digit next to each other on the '''board'''. The multiplier corresponds to the number down the left hand side of the '''board''' For each digit in the multiplicand, the numbers adjacent to each other on the row corresponding to the multiplier in each of the compartments formed by the diagonal lines aligned together are added together. The product is thus formed.
Definition:Prosthaphaeresis
'''Prosthaphaeresis''' is the process whereby multiplication and division are performed by: :$(1): \quad$ converting the numbers into a form where they can be added or subtracted :$(2): \quad$ performing the addition or subtraction :$(3): \quad$ converting the result back again, using the inverse process to step $(1)$.
Definition:Prosthaphaeresis/Trigonometry
The trigonometric technique of prosthaphaeresis is based on the Prosthaphaeresis Formula for Sine plus Sine: :$\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$
Definition:Napierian Logarithm
The '''Napierian logarithms''' are a system of logarithms where: :$\log_{\text {Nap} } 10 \, 000 \, 000 = 0$ :$\log_{\text {Nap} } 9 \, 999 \, 999 = 1$ Thus: :$\log_{\text {Nap} } 10^7 x y = \log_{\text {Nap} } x + \log_{\text{Nap} } y$
Definition:Prosthaphaeresis/Logarithms
The logarithmic technique of prosthaphaeresis is based on the rule of the Sum of Logarithms: :$\log_b x y = \log_b x + \log_b y$ Any base $b$ can be used. The original tecnhique used Napierian logarithms where the base was $0.9999999$ In due course the common (base $10$) logarithm was used, as developed by {{AuthorRef|Henry Briggs}}, called Briggsian logarithms.
Definition:Epicycle
An '''epicycle''' is the path described by an object moving in a uniform circular motion around a point which is itself moving in a uniform circular motion around another point. :400px That point may itself also be moving in a uniform circular motion around yet another point.
Definition:Oblique Axes
'''Oblique axes''' are a pair of coordinate axes which are not at right angles to each other. :400px
Definition:Equation of Geometric Figure
Let there be a coordinate system. Let the variables $\alpha, \beta, \gamma, \ldots$ be used to identify points within that coordinate system. Let the variables be related to each other by means of an equation. Then the equations define the set of points which form a geometric figure. This equation is known as the '''equation of the geometric figure'''.
Definition:Tensor Product of Modules
=== Commutative ring === Let $R$ be a commutative ring with unity. Let $M$ and $N$ be $R$-modules. === Definition 1 === Their '''tensor product''' is a pair $(M \otimes_R N, \theta)$ where: :$M \otimes_R N$ is an $R$-module :$\theta : M \times N \to M \otimes_R N$ is an $R$-bilinear mapping satisfying the following '''universal property''': :For every pair $(P, \omega)$ of an $R$-module and an $R$-bilinear mapping $\omega : M \times N \to P$, there exists a unique $R$-module homomorphism $f : M \otimes_R N \to P$ with $\omega = f \circ \theta$. === Definition 2 === Their '''tensor product''' is the pair $\struct {M \otimes_R N, \theta}$, where: :$M \otimes_R N$ is the quotient of the free $R$-module $R^{\paren {M \times N} }$ on the direct product $M \times N$, by the submodule generated by the set of elements of the form: ::$\tuple {\lambda m_1 + m_2, n} - \lambda \tuple {m_1, n} - \tuple {m_2, n}$ ::$\tuple {m, \lambda n_1 + n_2} - \lambda \tuple {m, n_1} - \tuple {m, n_2}$ ::for $m, m_1, m_2 \in M$, $n, n_1, n_2 \in N$ and $\lambda \in R$, where we denote $tuple {m, n}$ for its image under the canonical mapping $M \times N \to R^{\paren {M \times N} }$. :$\theta : M \times N \to M \otimes_R N$ is the composition of the canonical mapping $M \times N \to R^{\paren {M \times N} }$ with the quotient module homomorphism $R^{\paren {M \times N} } \to M \otimes_R N$. === Noncommutative ring === Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. First construct a left module as a direct sum of all free left modules with a basis that is a single ordered pair in $M \times N$ which is denoted $\map R {m, n}$. :$T = \displaystyle \bigoplus_{s \mathop \in M \mathop \times N} R s$ That this is indeed a module is demonstrated in Tensor Product is Module. Next for all $m, m' \in M$, $n, n' \in N$ and $r \in R$ we construct the following free left modules. :$L_{m, m', n}$ with a basis of $\tuple {m + m', n}$, $\tuple {m, n}$ and $\tuple {m', n}$ :$R_{m, n, n'}$ with a basis of $\tuple {m, n + n'}$, $\tuple {m, n}$ and $\tuple {m, n'}$ :$A_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m r, n}$ :$B_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m, r n}$ Let: :$D = \displaystyle \map {\bigoplus_{r \in R, n, n' \in N, m, m' \in M} } {L_{m, m', n} \oplus R_{m, n, n'} \oplus A_{r, m, n} \oplus B_{r, m, n} }$ The '''tensor product''' $M \otimes_R N$ is then our quotient module $T / D$.
Definition:Folium of Descartes
=== Cartesian Form === {{:Definition:Folium of Descartes/Cartesian Form}} === Parametric Form === {{:Definition:Folium of Descartes/Parametric Form}} :500px
Definition:Archimedean Spiral
The '''Archimedean spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = a \theta$ :600px
Definition:Number Theory
'''Number theory''' is the branch of mathematics which studies the properties of the natural numbers.
Definition:Algebraic Number Theory
'''Algebraic number theory''' is the branch of abstract algebra which studies structures in which the usual number fields are embedded. As such it can also be considered to be a branch of number theory.
Definition:Topology (Mathematical Branch)
'''Topology''' is a geometry of transformations in which the only invariant is continuity. Some sources suggest that it can indeed be described simply as '''the study of continuity'''. As such, it is closely interwoven with the branch of '''analysis'''.
Definition:Plane Geometry
'''Plane geometry''' is the study of geometric figures in two dimensions.
Definition:Computer Science
'''Computer science''' is the branch of mathematics studying questions arising from the operation of [https://en.wikipedia.org/wiki/Computer digital computers]. As such it can be considered as an offshoot of both mathematical logic and discrete mathematics.
Definition:Computability Theory
'''Computability theory''' is a branch of mathematical logic which concerns itself with the algorithmic implementation of mathematical proofs.
Definition:Functional Analysis
'''Functional analysis''' is a branch of analysis, which studies vector spaces endowed a structure such as inner product, norm or topology.
Definition:Analysis
'''Analysis''' is a branch of mathematics that studies continuous change. It subsumes the fields of calculus, differential equations, the calculus of variations, power series, Fourier series, real analysis and complex analysis. It is one of the main branches of mathematics, alongside geometry, number theory and algebra.
Definition:Analysis/Real
'''Real analysis''' is a branch of mathematics that studies real functions.
Definition:Analysis/Complex
'''Complex analysis''' is a branch of mathematics that studies complex functions.
Definition:Ring Theory
'''Ring Theory''' is a branch of abstract algebra which studies rings and other related algebraic structures.
Definition:Analytic Number Theory
'''Analytic number theory''' is a branch of number theory which uses tools from analysis to solve problems about the integers.
Definition:Calculus/Differential
'''Differential calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities change.
Definition:Calculus/Integral
'''Integral calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities accumulate. Equivalently, given the rate of change of a quantity '''integral calculus''' provides techniques of providing the quantity itself. The equivalence of the two uses are demonstrated in the Fundamental Theorem of Calculus. The technique is also frequently used for the purpose of calculating areas and volumes of curved geometric figures.
Definition:Calculus
'''Calculus''' is the branch of mathematics which studies change.
Definition:Mathematical Physics
'''Mathematical physics''' is the branch of mathematics concerned with the development of mathematical methods for application to problems in physics.
Definition:Physics
'''Physics''' is the branch of science which studies basic concepts including motion, energy, force, and the fundamental nature of spacetime. The study of '''physics''' is generally based in mathematical concepts, especially those of calculus.
Definition:Knot Theory
'''Knot theory''' is the branch of topology which studies the embedding of knots in $3$-dimensional space.
Definition:Calculus of Variations
'''Calculus of variations''' is the subfield of analysis concerned maximizing or minimizing real functionals, which are mappings from a set of functions to the real numbers.
Definition:Matrix Algebra
'''Matrix algebra''' is the subfield of algebra whose subjects of manipulation are matrices.
Definition:Graph Theory
'''Graph theory''' is the branch of mathematics concerned with the structure and properties of graphs. As a (graph-theoretical) graph has the same conceptual definition as a relation, it follows that there is considerable overlap between the fields of graph theory and relation theory.
Definition:Relation Theory
'''Relation theory''' is the subfield of set theory concerned with the properties of relations and relational structures. As a relation has the same conceptual definition as a (graph-theoretical) graph, it follows that there is considerable overlap between the fields of relation theory and graph theory.
Definition:Mapping Theory
'''Mapping theory''' is the subfield of set theory concerned with the properties of mappings.
Definition:Measure Theory
'''Measure theory''' is the subfield of analysis concerned with the properties of measures, particularly the Lebesgue measure.
Definition:Linear Algebra
'''Linear algebra''' is the branch of algebra which studies vector spaces and linear transformations between them.
Definition:Commutative Algebra (Mathematical Branch)
'''Commutative algebra''' is the branch of abstract algebra concerned with commutative and unitary rings.
Definition:Hamiltonian Mechanics
'''Hamiltonian mechanics''' was a reformulation of classical mechanics using a more abstract mathematical framework. It laid the groundwork for quantum mechanics.
Definition:Classical Mechanics
'''Classical mechanics''' is the study of the physical laws describing the motion of and forces on bodies under the action of a system of forces.
Definition:Galois Theory
'''Galois theory''' is a subfield of abstract algebra which provides a connection between field theory and group theory.
Definition:Field Theory
'''Field Theory''' is a branch of abstract algebra which studies fields and other related algebraic structures.
Definition:Lattice Theory
'''Lattice theory''' is the branch of abstract algebra concerned with the structure and properties of lattices.
Definition:Geometrical Mechanics
'''Geometrical mechanics''' is the branch of mathematics which uses techniques from geometry to solve problems in mechanics.
Definition:Mechanics
'''Mechanics''' is the branch of applied mathematics concerned with the motion of and forces on bodies.
Definition:Dimensional Analysis
'''Dimensional analysis''' is the branch of applied mathematics which studies the relationships between physical quantities by identifying their fundamental dimensions and units of measurement. As such it can also be considered to be a branch of number theory.
Definition:Applied Mathematics
'''Applied mathematics''' is the branch of mathematics which concerns the solution of real world applications by the construction of mathematical models.
Definition:Astronomy
'''Astronomy''' is the branch of physics which studies the features which can be seen in the sky that are outside the atmosphere of Earth. Certain of its mathematical aspects can be considered a subfield of solid geometry.
Definition:Modulo Arithmetic
'''Modulo arithmetic''' is the branch of abstract algebra which studies the residue class of integers under a modulus. As such it can also be considered to be a branch of number theory.
Definition:Homological Algebra
'''Homological algebra''' is the branch of algebra and topology which studies homology.
Definition:Umbral Calculus
'''Umbral calculus''' is a branch of analysis which studies polynomial spaces by means of linear functionals. The modern approach concerns the study of Sheffer sequences
Definition:Order Theory
'''Order theory''' is the branch of relation theory which studies orderings.
Definition:Category Theory
'''Category theory''' is the branch of abstract algebra which studies categories. It can be described as the '''theory of functors'''.
Definition:Laws of Conservation
The '''laws of conservation''' are laws of physics that states that certain particular measurable properties of isolated physical systems do not change as the system evolves over time.
Definition:Multiplication/Natural Numbers/Addition
Let $+$ denote addition. The binary operation $\times$ is recursively defined on $\N$ as follows: :$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \left({n + 1}\right) & = m \times n + m \end{cases}$ This operation is called '''multiplication'''. Equivalently, '''multiplication''' can be defined as: :$\forall m, n \in \N: m \times n := +^n m$  where $+^n m$ denotes the $n$th power of $m$ under $+$.
Definition:Ideal Theory
'''Ideal Theory''' is a branch of abstract algebra which studies ideals of rings.
Definition:Algebraic Topology
'''Algebraic Topology''' is a branch of topology which uses tools from abstract algebra to study topological spaces.
Definition:Fractional Calculus
'''Fractional calculus''' is the branch of calculus which considers the concept of derivatives of fractional order.
Definition:Approximation Theory
'''Approximation theory''' is a branch of numerical analysis and applied mathematics concerned with the approximation of a function over an interval can be approximated by a simpler function (usually polynomial).
Definition:Module Theory
'''Module Theory''' is the branch of abstract algebra which studies modules.
Definition:Polynomial Theory
'''Polynomial Theory''' is a branch of abstract algebra which studies polynomials.
Definition:Vector Algebra
'''Vector Algebra''' is the branch of mathematics which studies the algebra of vector spaces.
Definition:Discrete Mathematics
'''Discrete mathematics''' is the branch of mathematics which studies mathematical structures that are discrete rather than continuous.
Definition:Discrete (Word)
'''Discrete''' is a word used to define an object which is not continuous.
Definition:Quantum Mechanics
'''Quantum mechanics''' is the branch of physics concerned with the behaviour of matter at the smallest scale accessible to observation.
Definition:Ergodic Theory
'''Ergodic Theory''' is the study of dynamical systems with an invariant measure.
Definition:Economics
'''Economics''' is the branch of applied mathematics which studies the movement of money.
Definition:Mathematical Programming
'''Mathematical programming''' is the branch of applied mathematics which applies techniques from computer science to solve problems in economics.
Definition:Game Theory
'''Game theory''' is the branch of discrete mathematics which studies mathematical models of conflict and cooperation between intelligent rational decision-makers. Problems in '''game theory''' are a special case of those of linear programming.
Definition:Numerical Analysis
'''Numerical analysis''' is the branch of applied mathematics concerned with the processes by which certain kinds of numerical solutions to problems in (mainly) physics.
Definition:Perturbation Theory
'''Perturbation Theory''' is a branch of applied mathematics which studies systems by investigating their behaviour when certain parameters are changed slghtly. That is, it is the study of the effects of small disturbances.
Definition:Model Theory
'''Model theory''' is a sub-branch of mathematical logic which studies mathematical structures by means of formal languages.
Definition:Fourier Analysis
'''Fourier analysis''' is a branch of analysis which seeks to express functions (usually periodic) by means of series of sines and cosines.
Definition:Operator Theory
'''Operator theory''' is the study of linear operators on function spaces, beginning with differential operators and integral operators.
Definition:Non-Euclidean Geometry
'''Non-Euclidean geometry''' is branch of geometry in which {{AuthorRef|Euclid}}'s fifth postulate does not hold.
Definition:Ordered Geometry
'''Ordered geometry''' is a sub-branch of geometry which featured the concept of betweenness but, like projective geometry, omitting the basic notion of distance.
Definition:Hyperbolic Geometry
'''Hyperbolic geometry''' is a branch of non-Euclidean geometry in which there more than one straight line can pass through a point parallel to another straight line.