Definition:Cube/Algebra

The '''cube''' of a quantity is the third power of that quantity.


Definition:Baire Space (Topology)/Definition 1

$T$ is a '''Baire space''' {{iff}} the union of any countable set of closed sets of $T$ whose interiors are empty also has an empty interior.


Definition:Baire Space (Topology)/Definition 2

$T$ is a '''Baire space''' {{iff}} the intersection of any countable set of open sets of $T$ which are everywhere dense is everywhere dense.


Definition:Baire Space (Topology)/Definition 3

$T$ is a '''Baire space''' {{iff}} the interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty.


Definition:Baire Space (Topology)/Definition 4

$T$ is a '''Baire space''' {{iff}}, whenever the union of any countable set of closed sets of $T$ has an interior point, then one of those closed sets must have an interior point.


Definition:Lychrel Number

A '''Lychrel number''' is a natural number which cannot form a palindromic number through repeated iteration of the reverse-and add process.


Definition:Angle/Types

Angles can be divided into categories: === Zero Angle === {{:Definition:Zero Angle}} === Acute Angle === {{:Definition:Acute Angle}} === Right Angle === {{:Definition:Right Angle}} === Obtuse Angle === {{:Definition:Obtuse Angle}} === Straight Angle === {{:Definition:Straight Angle}} === Reflex Angle === {{:Definition:Reflex Angle}} === Full Angle === {{:Definition:Full Angle}} It is possible to consider angles outside the range $\closedint {0 \degrees} {360 \degrees}$, that is, $\closedint 0 {2 \pi}$. However, in geometric contexts it is usually preferable to convert these to angles inside this range by adding or subtracting multiples of a full angle.


Definition:Angle/Directed versus Undirected

The most basic definition of angle is an undirected angle on the interval $\left[{0^\circ  \,.\,.\, 180^\circ}\right]$ or $\left[{0 \,.\,.\, \pi}\right]$. This definition is often insufficient, in cases such as the external angles of a polygon. Therefore, angles are most commonly defined in one of two ways: :$(1): \quad$ Undirected angles on the interval $\left[{0^\circ \,.\,.\, 360^\circ}\right]$ or $\left[{0 \,.\,.\, 2 \pi}\right]$. :$(2): \quad$ Directed angles, with the positive direction being counterclockwise from a given line (or, if no line is specified, from the $x$-axis). :::This definition is more commonly found in applied mathematics, such as in surveying, navigation, or, more colloquially, in a $720^\circ$ degree spin in skateboarding, skiing, etc.


Definition:Vertical Angles

When two straight lines intersect, the angles opposite each other are called '''vertical angles''': :400px In the above diagram: :$\alpha$ and $\beta$ are '''vertical angles''' :$\gamma$ and $\delta$ are '''vertical angles'''.


Definition:Smooth Mapping

Let $M, N$ be smooth manifolds.  Denote $m := \dim M$ and $n := \dim N$.  Let $\phi: M \to N$ be a mapping.  Then $\phi$ is a '''smooth mapping''' {{iff}}: :for every chart $\struct {U, \kappa}$ on $M$ and every chart $\struct {V, \xi}$ on $N$ such that $V \cap \map \phi U \ne \O$, the mapping: ::$\displaystyle \xi \circ \phi \circ \kappa^{-1}: \map \kappa U \subseteq \R^m \to \map \xi {V \cap \map \phi U} \subseteq \R^n$ :is smooth.


Definition:Fiber Bundle

Let $M, E, F$ be topological spaces.  Let $\pi: E \to M$ be a continuous surjection.  Let $\mathcal U := \left\{ {U_\alpha \subseteq M: \alpha \in I} \right\}$ be an open cover of $M$ with index set $I$.  Let $\operatorname{pr}_{1, \alpha} : U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$. Let there exist homeomorphisms: :$\chi_\alpha : \pi^{-1} \left({U_\alpha}\right) \to U_\alpha \times F$  such that for all $\alpha \in I$: :$\pi {\restriction}_{U_\alpha} = \operatorname{pr}_{1, \alpha} \mathop \circ \chi_\alpha$ where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \mathcal U$. Then the ordered tuple $\left({E, M, \pi, F}\right)$ is called a '''fiber bundle over $M$'''.


Definition:Transversal (Geometry)/Interior Angle

An '''interior angle''' of a transversal is an angle which is between the two lines cut by that transversal. In the above figure, the '''interior angles''' with respect to the transversal $EF$ are: :$\angle AHJ$ :$\angle CJH$ :$\angle BHJ$ :$\angle DJH$


Definition:Transversal (Geometry)/Alternate Angles

'''Alternate angles''' are interior angles of a transversal which are on opposite sides and different lines. In the above figure, the pairs of '''alternate angles''' with respect to the transversal $EF$ are: :$\angle AHJ$ and $\angle DJH$ :$\angle CJH$ and $\angle BHJ$


Definition:Transversal (Geometry)/Corresponding Angles

'''Corresponding angles''' are the angles in equivalent positions on the two lines cut by a transversal with respect to that transversal. In the above figure, the '''corresponding angles''' with respect to the transversal $EF$ are: :$\angle AHJ$ and $\angle CJF$ :$\angle AHE$ and $\angle CJH$ :$\angle BHE$ and $\angle DJH$ :$\angle BHJ$ and $\angle DJF$


Definition:Smooth Curve

Let $M$ be a smooth manifold.  Let $I$ be a open real interval, considered as a smooth manifold of dimension $1$. {{explain|We have various pages defining "manifolds" of various types (differentiable, smooth, complex), but not one defining a basic "manifold".  It can be assumed by inference that a manifold is "a second-countable locally Euclidean space of finite integral dimension" but this needs to be rigorously and unambiguously clarified by including that definition at the top level of the page Definition:Topological Manifold.}} Then a smooth mapping $\gamma : I \to M$ is called a '''smooth curve'''. Category:Definitions/Manifolds exaz9pzlswo05mgzfps2k4cek9jd44a


Definition:Proportion/Constant of Proportion

The constant $k$ is known as the '''constant of proportion'''.


Definition:Coordinate Function

Let $M$ be a locally Euclidean space of dimension $n$.  Let $\left({U, \kappa}\right)$ be a coordinate chart.  Let $\operatorname{pr}_i: \R^n \to \R$ be the $i$th projection.  Then the mapping $\kappa_i$, defined as: :$\kappa_i = \operatorname{pr}_i \mathop \circ \kappa: U \to \left({\operatorname{pr}_i \mathop \circ \kappa \left({U}\right)}\right) \subseteq \R$ is called the '''$i$th coordinate function''' on $U$.


Definition:Tangent Space

=== Real Submanifold === Let $M$ be a real submanifold of $\R^n$ of dimension $d$. Let $p\in M$. ==== Using Local Submersions ==== Let $U$ be a open neighborhood of $p$ in $\R^n$ and $\phi : U \to \R^{n-d}$ be a submersion such that: :$M \cap U = \phi^{-1}(0)$. The '''tangent space of $M$ at $p$''' is: :$T_pM = \ker d\phi(p)$ where $d\phi(p)$ is the differential of $\phi$ at $p$. ==== Using Local Embdeddings ==== Let $U$ be a open set of $\R^d$ and $\phi : U \to \R^{n}$ be an embedding such that: :$p\in \phi(U) \subset M$ The '''tangent space of $M$ at $p$''' is: :$T_pM = \operatorname{im} (d\phi)(\phi^{-1}(p))$ where $(d\phi)(\phi^{-1}(p))$ is the differential of $\phi$ at $\phi^{-1}(p)$. ==== Using Local Immersions ==== === Geometric Tangent Space === {{stub}} === Differentiable Manifold === There are various ways to construct the '''tangent space''' of a differentiable manifold. Let $M$ be a smooth manifold of dimension $m$. Let $p \in M$.  ==== Abstract Definition ==== Let $A$ be an atlas of $M$. Let $B = \{(U,\phi) \in A : p\in U\}$. The '''tangent space of $M$ at $p$''' is the real vector space :$\displaystyle \left(\coprod_{b\in B}\{b\} \times \R^m\right)/\sim$ where $\sim$ is the equivalence relation defined by: :$(i,\xi) \sim (j,\eta) \iff d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) = \eta$ where $d(\phi_j \circ \phi_i^{-1})(\phi_i(p))$ is the differential of $\phi_j \circ \phi_i^{-1}$ at $\phi_i(p)$. Addition is defined by: :$[(i,\xi)] + [(j,\eta)] = \left[\left( i, d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) + \eta \right)\right]$ and scalar multiplication by: :$\lambda \cdot [(i,\xi)] = [(i, \lambda\cdot\xi)]$ ==== As a space of germs of curves ==== {{stub}} ==== As a space of derivations ==== The '''tangent space at $p$''', denoted by $T_p M$, is the set of all tangent vectors at $m$. {{refactor|make this explicit}}


Definition:Fiber Bundle/Total Space

The topological space $E$ is called the '''total space of $B$'''.


Definition:Fiber Bundle/Base Space

The topological space $M$ is called the '''base space of $B$'''.


Definition:Fiber Bundle/Base Point

A point $m \in M$ is called a '''base point of $B$'''.


Definition:Fiber Bundle/Bundle Projection

The continuous surjection $\pi: E \to M$ is called the '''bundle projection of $B$'''.


Definition:Section (Topology)

Let $M, E$ be topological spaces.  Let $\pi: E \to M$ be a continuous surjection.  Let $I_M: M \to M$ be the identity mapping on $M$.  Then a '''section''' of $E$ is a continuous mapping $s: M \to E$ such that $\pi \circ s = I_M$.


Definition:Local Trivialization

Let $\left({E, M, \pi, F}\right)$ be a fiber bundle.  Let $\operatorname{pr}_1: M \times F \to M$ be the first projection on $M \times F$. By definition of fiber bundle, for every point $m \in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism:  :$\chi: \pi^{-1} \left({U}\right) \to U \times F$  such that:  :$\pi {\restriction}_U = \operatorname{pr}_1 \mathop \circ \chi$ where $\pi {\restriction}_U$ is the restriction of $\pi$ to $U$. Then the ordered pair $\left({U, \chi}\right)$ is called a '''local trivialization''' of $E$ over $U$.


Definition:Fiber Bundle/Model Fiber

The topological space $F$ is called the '''model fiber of $B$'''.


Definition:Dimension (Topology)

=== Locally Euclidean Space === {{:Definition:Dimension (Topology)/Locally Euclidean Space}} === Hausdorff Dimension === {{:Definition:Dimension (Topology)/Hausdorff Dimension}} === Lebesgue Covering Dimension === {{:Definition:Dimension (Topology)/Lebesgue Covering Dimension}} === Inductive Dimension === {{:Definition:Dimension (Topology)/Inductive Dimension}} Category:Definitions/Topology clt3l1b93bpp2cr7znc3k39qnh3d28d


Definition:Transition Mapping

Let $B = \left({E, M, \pi, F}\right)$ be a fiber bundle.  Let $\left({U, \chi}\right)$, $\left({V, \xi}\right)$ be two local trivializations with $U \cap V \ne \varnothing$. Then the mapping: :$\xi \circ \chi^{-1} : U \cap V \times F \to U \cap V \times F$ is called a '''transition mapping''' from $\left({U, \chi}\right)$ to $\left({V, \xi}\right)$.


Definition:Smooth Fiber Bundle

Let $E, M, F$ be smooth manifolds.  Let $\pi$ be a smooth surjection.  Let $B = \left({ E, M, \pi, F }\right)$ be a fiber bundle with a system of local trivializations $\left\{{ \left({ U_\alpha, \chi_\alpha }\right) : \alpha \in I }\right\}$ such that $\forall \alpha \in I$ :  :$\chi_\alpha : \pi^{-1}\left({ U_\alpha }\right) \to U_\alpha \times F$ is smooth.  Then $B$ is called a '''smooth fiber bundle''' on $M$.


Definition:Fiber Bundle/System of Local Trivializations

The set $\left\{ {\left({U_\alpha, \chi_\alpha}\right): \alpha \in I}\right\}$ is called a '''system of local trivializations''' of $E$ on $M$.


Definition:Vector Bundle

Let $B = \struct {E, M , \pi, F}$ be a fiber bundle.  Then $B$ is a '''vector bundle''' over $M$ if the model fiber $F$ is a topological vector space.


Definition:Smooth Vector Bundle

Let $B = \tuple {E, M, \pi, F}$ be a smooth fiber bundle.  Let $F$ be a vector space.  Let vector addition and scalar multiplication on $F$ be smooth. Then $B$ is a '''smooth vector bundle''' over $M$.


Definition:Borel Sigma-Algebra/Borel Set

The elements of $\map {\mathcal B} {S, \tau}$ are called the '''Borel (measurable) sets''' of $\struct {S, \tau}$.


Definition:Linear Measure/Height/Euclidean Definition

The '''height''' of a polygon is the linear measure going up the page.


Definition:Even Integer/Even-Times Even

Let $n$ be an integer. Then $n$ is '''even-times even''' {{iff}} it has $4$ as a divisor. The first few non-negative '''even-times even''' numbers are: :$0, 4, 8, 12, 16, 20, \ldots$


Definition:Even Integer/Even-Times Odd

Let $n$ be an integer. Then $n$ is '''even-times odd''' {{iff}} it has $2$ as a divisor and also an odd number. The first few non-negative '''even-times odd''' numbers are: :$2, 6, 10, 12, 14, 18, \ldots$


Definition:Odd Integer/Odd-Times Odd

Let $n \in \Z$, i.e. let $n$ be an integer. === Definition 1 === {{:Definition:Odd Integer/Odd-Times Odd/Definition 1}} === Definition 2 === {{:Definition:Odd Integer/Odd-Times Odd/Definition 2}} === Sequence === {{:Definition:Odd Integer/Odd-Times Odd/Sequence}}


Definition:Plane Number/Side

The '''side''' of a plane number is one of the (natural) numbers which are its divisors.


Definition:Solid Number/Side

The '''side''' of a solid number is one of the (natural) numbers which are its divisors.


Definition:Side of Number

=== Side of Plane Number === {{:Definition:Side of Plane Number}} === Side of Solid Number === {{:Definition:Side of Solid Number}} Category:Definitions/Euclidean Number Theory fuwb3abn0ln4di6c41oov518khzuqsh


Definition:Solid Number/Similar Numbers

Let $m$ and $n$ be solid numbers. Let: : $m = p_1 \times p_2 \times p_3$ where $p_1 \le p_2 \le p_3$ : $n = q_1 \times q_2 \times q_3$ where $q_1 \le q_2 \le q_3$ Then $m$ and $n$ are '''similar''' iff: :$p_1 : q_1 = p_2 : q_2 = p_3 : q_3$


Definition:Similar Numbers

=== Similar Plane Numbers === {{:Definition:Plane Number/Similar Numbers}} === Similar Solid Numbers === {{:Definition:Solid Number/Similar Numbers}} {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/21 - Similar Numbers}}'' {{EuclidDefRefNocat|VII|21|Similar Numbers}} Category:Definitions/Euclidean Number Theory bkaggf8650zkdak4px6fy31u10m43om


Definition:Plane Number/Similar Numbers

Let $m$ and $n$ be plane numbers. Let: :$m = p_1 \times p_2$ where $p_1 \le p_2$ :$n = q_1 \times q_2$ where $q_1 \le q_2$ Then $m$ and $n$ are '''similar''' {{iff}}: :$p_1 : q_1 = p_2 : q_2$


Definition:Parenthesization

Let $\circ$ be a product defined on a set $S$. Let $a_1 \circ a_2 \circ \cdots a_i \circ \cdots \circ a_n$ denote a word in $S$ for some $n > 2$. To distinguish all possible products of $a_1, a_2, \dotsc, a_n$, parentheses are inserted into the word. A set of parentheses applied on a product is called a '''parenthesization''' of that word.


Definition:Binomial (Euclidean)/Order

The '''order''' of $a + b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth.


Definition:Apotome/Order

The '''order''' of $a - b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth.


Definition:Divisor (Algebra)/Real Number

Let $\R$ be the set of real numbers. Let $x, y \in \R$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z: y = t \times x$ where $\Z$ is the set of integers.


Definition:Greatest Common Divisor/Real Numbers

Let $a, b \in \R$ be commensurable. Then there exists a greatest element $d \in \R_{>0}$ such that: : $d \divides a$ : $d \divides b$ where $d \divides a$ denotes that $d$ is a divisor of $a$. This is called the '''greatest common divisor of $a$ and $b$''' and denoted $\gcd \set {a, b}$.


Definition:Transversal (Geometry)/Exterior Angle

An '''exterior angle''' of a transversal is an angle which is not between the two lines cut by a transversal. In the above figure, the '''exterior angles''' with respect to the transversal $EF$ are: :$\angle AHE$ :$\angle CJF$ :$\angle BHE$ :$\angle DJF$


Definition:Angle/Containment

The two lines whose intersection forms an angle are said to '''contain''' that angle.


Definition:Essentially Bounded Function

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $L^\infty $ be the associated Lebesgue $\infty$-space. Then any $f \in L^\infty$ is known as an '''essentially bounded function.'''


Definition:Supremum Seminorm

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \R$ be a real-valued function. The '''supremum seminorm''' of $f$, commonly denoted as $\left\Vert{f}\right\Vert_\infty$, is defined as: :$\displaystyle \left\Vert{f}\right\Vert_\infty = \inf_{\substack {N \mathop \in \Sigma \\ \mu \left({N}\right) \mathop = 0}} \sup \ \left\{ {\left\vert{f \left({x}\right)}\right\vert: x \notin N}\right\}$ Observe that for all $M > \Vert f \Vert_\infty$, then: : $\mu \left({ \left\{ {x \in X: \left\vert{f \left({x}\right)}\right\vert \ge M}\right\} }\right) = 0$ and that an essentially bounded function is almost everywhere equal to a bounded function. {{wtd|Move the above into their own proof and definition pages}}


Definition:Conjugate Exponents

Let $p, q \in \R_{\ge 1}$. Then $p$ and $q$ are '''conjugate exponents''' iff: :$\dfrac 1 p + \dfrac 1 q = 1$ === General Definition === {{:Definition:Conjugate Exponents/General Definition}} Category:Definitions/Measure Theory 72hyj5t8tcetr3ftrhw01of3uv9fl6d


Definition:Conjugate Exponents/General Definition

Let $\left\{{p_1, p_2, \ldots, p_n}\right\}$ be such that: : $\forall i \in \left\{{1, 2, \ldots, n}\right\}: p_i \in \R_{\ge 1}$ Then $\left\{{p_1, p_2, \ldots, p_n}\right\}$ are '''conjugate exponents''' iff: :$\displaystyle \frac 1 {p_1} + \frac 1 {p_2} + \cdots + \frac 1 {p_n} = \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$


Definition:Inscribe/Circle in Polygon

A circle is '''inscribed in''' a polygon when it is tangent to each of the sides of that polygon: :300px


Definition:Inscribe/Polygon in Circle

A polygon is '''inscribed in''' a circle when each of its vertices lies on the circumference of the circle: :300px


Definition:Inscribe/Polygon in Polygon

A polygon is '''inscribed in''' another polygon when each of its vertices lies on the corresponding side of the other polygon.


Definition:Circumscribe/Circle around Polygon

A circle is '''circumscribed around''' a polygon when its circumference passes through each of the vertices of that polygon: :300px


Definition:Circumscribe/Polygon around Circle

A polygon is '''circumscribed around''' a circle when each of its sides is tangent to the circumference of the circle: :300px


Definition:Circumscribe/Polygon around Polygon

A polygon is '''circumscribed around '''another polygon when each of its sides passes through each of the vertices of the other polygon.


Definition:Leading Term

Let $P = a \circ b$ be an expression. The term $a$ is known as the '''leading term''' of $P$.


Definition:Following Term

Let $P = a \circ b$ be an expression. The term $b$ is known as the '''following term''' of $P$.


Definition:Unit (One)

A numerical quantity whose cardinality corresponds to the number $1$ (one) is called '''a unit'''. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/1 - Unit}}'' {{EuclidDefRefNocat|VII|1|Unit}}


Definition:Divisor (Algebra)/Real Number/Part

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/1 - Part}}'' {{EuclidDefRefNocat|V|1|Part}}


Definition:Common Divisor/Real Numbers

Let $S$ be a finite set of real numbers, that is: :$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \R}$ Let $c \in \R$ such that $c$ divides all the elements of $S$, that is: :$\forall x \in S: c \divides x$ Then $c$ is a '''common divisor''' of all the elements in $S$.


Definition:Commensurable in Square Only

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''commensurable in square only''' iff: : $\left ({\dfrac a b}\right)^2$ is rational. but: : $\dfrac a b$ is irrational. That is, such that: : $a$ and $b$ are commensurable in square but: : $a$ and $b$ are incommensurable in length. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}} and: :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}


Definition:Rational Line Segment

A '''rational line segment''' is a line segment the square of whose length is a rational number of units of area. In other words, a '''rational line segment''' is a line segment whose length belongs to the set $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/3 - Rational Line Segment}}'' {{EuclidDefRefNocat|X|3|Rational Line Segment}}


Definition:Rational Area

A '''rational area''' is a geometric figure whose area is a rational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/4 - Rational Area}}'' {{EuclidDefRefNocat|X|4|Rational Area}}


Definition:Irrational Line Segment

An '''irrational line segment''' is a line segment such that the square of whose length is an irrational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/3 - Rational Line Segment}}'' {{EuclidDefRefNocat|X|3|Rational Line Segment}}


Definition:Medial

A '''medial''' is a strictly positive real number which is the mean proportional between two rational line segments which are commensurable in square only. Thus a magnitude $a \in \R_{>0}$ is '''medial''' {{iff}} $a$ is of the form: :$a = \rho \sqrt [4] k$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational. {{:Euclid:Proposition/X/21}}


Definition:Irrational Area

An '''irrational area''' is a geometric figure whose area is an irrational number of units of area. {{EuclidSaid}} : ''{{:Definition:Euclid's Definitions - Book X/4 - Rational Area}}'' {{EuclidDefRefNocat|X|4|Rational Area}}


Definition:Successor Mapping

Let $\struct {P, s, 0}$ be a Peano structure. Then the mapping $s: P \to P$ is called the '''successor mapping on $P$'''.


Definition:Non-Successor Element

Let $\struct {P, s, 0}$ be a Peano structure. Then the element $0 \in P$ is called the '''non-successor element'''. This is justified by Axiom $(\text P 4)$, which stipulates that $0$ is not in the image of the successor mapping $s$.


Definition:Medial Area

A '''medial area''' is an area which is equal to the square on a medial straight line. Thus an area $A$ is '''medial''' {{iff}} it is of the form: :$A = \rho^2 \sqrt k$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational. {{:Euclid:Proposition/X/23/Porism}}


Definition:Bimedial/First Bimedial

Let $a, b \in \R_{>0}$ be in the forms: :$a = k^{1/4} \rho$ :$b = k^{3/4} \rho$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational. Then $a + b$ is a '''first bimedial'''. {{:Euclid:Proposition/X/37}}


Definition:Bimedial/Second Bimedial

Let $a, b \in \R_{>0}$ be in the forms: :$a = k^{1/4} \rho$ :$b = \dfrac {\lambda^{1/2} \rho} {k^{1/4} }$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational. : $\lambda$ is a rational number whose square root is irrational. Then $a + b$ is a '''second bimedial'''. {{:Euclid:Proposition/X/38}}


Definition:Major (Euclidean)

Let $a, b \in \R_{>0}$ in the forms: :$a = \dfrac \rho {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$ :$b = \dfrac \rho {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational. Then $a + b$ is a '''major'''. {{:Euclid:Proposition/X/39}}


Definition:Side of Sum of Medial Areas

Let $a, b \in \R_{>0}$ be in the forms: :$a = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$ :$b = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\lambda$ is a rational number whose square root is irrational. Then $a + b$ is the '''side of the sum of (two) medial areas'''. {{:Euclid:Proposition/X/41}}


Definition:Side of Rational plus Medial Area

Let $a, b \in \R_{>0}$ be in the forms: :$a = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} + k}$ :$b = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} - k}$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\lambda$ is a rational number whose square root is irrational. Then $a + b$ is the '''side of the sum of (two) medial areas'''. {{:Euclid:Proposition/X/40}}


Definition:Binomial (Euclidean)/Term

The '''terms''' of $a + b$ are the elements $a$ and $b$.


Definition:Bimedial

=== First Bimedial === {{:Definition:Bimedial/First Bimedial}} === Second Bimedial === {{:Definition:Bimedial/Second Bimedial}}


Definition:Bimedial/Order

The '''order''' of $a + b$ is the name of its classification into one of the two categories: first or second.


Definition:Apotome of Medial/Second Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$ be in the forms: :$a = k^{1/4} \rho$ :$b = \dfrac {\lambda^{1/2} \rho} {k^{1/4} }$ where: :$\rho$ is a rational number :$k$ is a rational number whose square root is irrational. :$\lambda$ is a rational number whose square root is irrational. Then $a - b$ is a '''second apotome of a medial'''. {{:Euclid:Proposition/X/75}}


Definition:Apotome of Medial/First Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$ be in the forms: :$a = k^{1/4} \rho$ :$b = k^{3/4} \rho$ where: :$\rho$ is a rational number :$k$ is a rational number whose square root is irrational. Then $a - b$ is a '''first apotome of a medial'''. {{:Euclid:Proposition/X/74}}


Definition:Apotome of Medial

=== First Apotome of Medial === {{:Definition:Apotome of Medial/First Apotome}} === Second Apotome of Medial === {{:Definition:Apotome of Medial/Second Apotome}}


Definition:Apotome of Medial/Order

The '''order''' of $a - b$ is the name of its classification into one of the two categories: first or second.


Definition:Minor (Euclidean)

Let $a, b \in \R_{>0}$ in the forms: :$a = \dfrac \rho {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$ :$b = \dfrac \rho {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$ where: :$\rho$ is a rational number :$k$ is a rational number whose square root is irrational. Then $a - b$ is a '''minor'''. {{:Euclid:Proposition/X/76}}


Definition:That which produces Medial Whole with Rational Area

Let $a, b \in \R_{>0}$ be in the forms: :$a = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} + k}$ :$b = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} - k}$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\lambda$ is a rational number whose square root is irrational. Then $a - b$ is '''that which produces a medial whole with a rational area'''. {{:Euclid:Proposition/X/77}}


Definition:That which produces Medial Whole with Medial Area

Let $a, b \in \R_{>0}$ be in the forms: :$a = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$ :$b = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\lambda$ is a rational number whose square root is irrational. Then $a - b$ is '''that which produces a medial whole with a medial area'''. {{:Euclid:Proposition/X/78}}


Definition:Apotome/Whole

The real number $a$ is called the '''whole''' of the apotome.


Definition:Apotome/Annex

The real number $b$ is called the '''annex''' of the apotome.


Definition:Apotome/Terms

The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome/Whole}} === Annex === {{Definition:Apotome/Annex}}


Definition:Common Section

Let $A$ and $B$ be planes. The '''common section''' of $A$ and $B$ is the intersection of $A$ and $B$.


Definition:Similar Planes

'''Similar planes''' are plane figures which are similar. Category:Definitions/Euclidean Geometry 6img9a3vyysc0t9nhmogs6mxs2ohfp6


Definition:Containment of Solid Figure

A solid geometric figure is said to be '''contained''' by the surfaces that form its boundary.


Definition:Similar Solid Figures

{{EuclidDefinition|book=XI|def=9|name=Similar Solid Figures}} {{stub}} Category:Definitions/Solid Geometry g03otrtjq5ra8g4g45yp2bg50wx7o1a


Definition:Similar Equal Solid Figures

{{EuclidDefinition|book=XI|def=10|name=Similar Equal Solid Figures}} {{stub}} Category:Definitions/Solid Geometry 6jdhgei61uzyv7y7ax6v7tmsrgojcam


Definition:Solid Angle

{{EuclidDefinition|book=XI|def=11|name=Solid Angle}} === Containment of Solid Angle === {{:Definition:Solid Angle/Containment}} === Vertex of Solid Angle === {{:Definition:Solid Angle/Vertex}} Category:Definitions/Solid Geometry hyb7wr54ukev6qia0sl0294rd2x8opi


Definition:Plane of Reference

The '''plane of reference''' is a conceptual plane in a spatial context which is arbitrarily distinguished from the others. Category:Definitions/Solid Geometry obrrchumu5siktshe431w8i0996p6rj


Definition:Plane Surface/Side

From the definition of surface, it follows that a plane locally separates space into two '''sides'''. Thus the '''sides''' of a plane are the parts of that space into which the plane separates it.


Definition:Solid Angle/Containment

The three plane angles which together form a solid angle are said to '''contain''' that solid angle.


Definition:Parallelepiped

A '''parallelepiped''' is a polyhedron formed by three pairs of parallel planes: :500px In the above example, the pairs of parallel planes are: :$ABCD$ and $HGFE$ :$ADEH$ and $BCFG$ :$ABGH$ and $DCFE$


Definition:Geometric Figure/Three-Dimensional Figure/Face

The '''faces''' of a three-dimensional figure are the surfaces which form its extremities.


Definition:Parallelepiped/Base

One of the faces of the parallelepiped is chosen arbitrarily, distinguished from the others and called the '''base of the parallelepiped'''. It is usual to choose the '''base''' to be the one which is conceptually on the bottom. In the above, $ABCD$ would conventionally be identified as being the '''base'''


Definition:Octahedron

An '''octahedron''' is a polyhedron which has $8$ faces.


Definition:Tetrahedron

A '''tetrahedron''' is a polyhedron which has $4$ (triangular) faces.


Definition:Dodecahedron

A '''dodecahedron''' is a polyhedron which has $12$ faces.


Definition:Hexahedron

A '''hexahedron''' is a polyhedron which has $6$ faces. === Cube === {{:Definition:Hexahedron/Regular}}


Definition:Icosahedron

An '''icosahedron''' is a polyhedron which has $20$ faces.


Definition:Polyhedron/Edge

The '''edges''' of a polyhedron are the sides of the polygons which constitute its faces.


Definition:Polyhedron/Vertex

The '''vertices''' of a polyhedron are the vertices of the polygons which constitute its faces.


Definition:Elevated

=== Elevated Point === {{:Definition:Elevated/Point}} === Elevated Line === {{:Definition:Elevated/Line}} Category:Definitions/Solid Geometry 347p9uj8vd4weqm5lcblp1pwasx7zp0


Definition:Angle/Vertex

The point at which the lines containing an angle meet is known as the '''vertex''' of that angle.


Definition:Solid Angle/Vertex

The common vertex of the angles containing a solid angle is known as the '''vertex''' of that solid angle.


Definition:Prism/Base

The '''bases''' of a prism are the two parallel polygons which form the faces at either end of the prism. In the above diagram, the faces $ABCDE$ and $FGHIJ$ are the '''bases''' of the prism.


Definition:Pyramid/Base

The polygon of a pyramid to whose vertices the apex is joined is called the '''base''' of the pyramid. In the above diagram, $ABCDE$ is the '''base''' of the pyramid $ABCDEQ$.


Definition:Pyramid/Apex

The vertex of a pyramid which is the common vertex of its triangular faces is called the '''apex''' of the pyramid. In the above diagram, $Q$ is the '''apex'''.


Definition:Square Pyramid

:300px A '''square pyramid''' is a pyramid whose base is a square. Category:Definitions/Pyramids oxtzsa3zp635hqh0sj9c2aj1ryid8vh


Definition:Pyramid/Height

The '''height''' of a pyramid is the length of the perpendicular from the plane of the base to its apex. In the above diagram, $h$ is the height.


Definition:Cylinder/Height

The '''height''' of a cylinder is the length of a line segment drawn perpendicular to the base and its opposite plane. In the above diagram, $h$ is the '''height''' of the cylinder $ACBDFE$.


Definition:Height of Solid Figure

The '''height''' of a solid figure is the length of a perpendicular from the base to the point or points most distant from it. === Height of Parallelepiped === {{:Definition:Height of Parallelepiped}} === Height of Prism === :500px {{:Definition:Height of Prism}} === Height of Pyramid === {{:Definition:Height of Pyramid}} === Height of Cone === {{:Definition:Height of Cone}} === Height of Cylinder === {{:Definition:Height of Cylinder}} Category:Definitions/Solid Geometry 14tt10lxi71w6r2j1axuoo08vit7tmd


Definition:Base of Solid Figure

The '''base''' of a solid figure is one of its faces which has been distinguished from the others in some way. The solid figure is usually oriented so that the '''base''' is situated at the bottom. === Base of Parallelepiped === :500px {{:Definition:Base of Parallelepiped}} === Base of Pyramid === :400px {{:Definition:Base of Pyramid}} === Base of Prism === :500px {{:Definition:Base of Prism}} === Base of Cone === :300px {{:Definition:Base of Cone}} ==== Base of Right Circular Cone ==== :300px {{:Definition:Base of Right Circular Cone}} === Base of Cylinder === :300px {{:Definition:Base of Cylinder}}


Definition:Decagon

A '''decagon''' is a polygon with exactly $10$ sides. :300px


Definition:Geometric Sequence/Term

The elements: :$x_n$ for $n = 0, 1, 2, 3, \ldots$ are the '''terms''' of $\sequence {x_n}$.


Definition:Proportion/Inverse

Two real variables $x$ and $y$ are '''inversely proportional''' {{iff}} their product is a constant: :$\forall x, y \in \R: x \propto \dfrac 1 y \iff \exists k \in \R, k \ne 0: x y = k$


Definition:Proportion/Joint

Two real variables $x$ and $y$ are '''jointly proportional''' to a third real variable $z$ {{iff}} the product of $x$ and $y$ is a constant multiple of $z$: :$\forall x, y \in \R: x y \propto z \iff \exists k \in \R, k \ne 0: x y = k z$


Definition:Proportion/Continued

Four magnitudes $a, b, c, d$ are '''in continued proportion''' {{iff}} $a : b = b : c = c : d$.


Definition:Linear Measure/Distance

The '''distance''' between two points $A$ and $B$ in space is defined as the length of a straight line that would be drawn from $A$ to $B$.


Definition:Physical Constant

A '''physical constant''' is a physical quantity that is believed to be unchanging in both space and time.


Definition:Recursive Sequence/Initial Terms

Let $S$ be a recursive sequence. In order for $S$ to be defined, it is necessary to define the '''initial term''' (or terms) explicitly. For example, in the Fibonacci sequence, the '''initial terms''' are defined as: :$F_0 = 0, F_1 = 1$


Definition:Proper Subset/Improper

$S$ is an '''improper subset''' of $T$ {{iff}} $S$ is a subset of $T$ but specifically ''not'' a proper subset of $T$. That is, either: :$S = T$ or: :$S = \O$


Definition:Set Union/Family of Sets/Two Sets

Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. From the definition of the union of $S_i$: :$\displaystyle \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$ it follows that: :$\displaystyle \bigcup \set {S_\alpha, S_\beta} := S_\alpha \cup S_\beta$


Definition:Set Intersection/Family of Sets/Two Sets

Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. From the definition of the intersection of $S_i$: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$ it follows that: :$\displaystyle \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$


Definition:Cartesian Product/Countable

Let $\sequence {S_n}_{n \mathop \in \N}$ be an infinite sequence of sets.  The '''cartesian product''' of $\sequence {S_n}$ is defined as: :$\displaystyle \prod_{k \mathop = 1}^\infty S_k = \set {\tuple {x_1, x_2, \ldots, x_n, \ldots}: \forall k \in \N: x_k \in S_k}$ It defines the concept: :$S_1 \times S_2 \times \cdots \times S_n \times \cdots$ Thus $\displaystyle \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\tuple {x_1, x_2, \ldots, x_n, \ldots}$ with $x_k \in S_k$.


Definition:Composition of Mappings/General Definition

Let $f_1: S_1 \to S_2, f_2: S_2 \to S_3, \ldots, f_n: S_n \to S_{n + 1}$ be mappings such that the domain of $f_k$ is the same set as the codomain of $f_{k - 1}$. Then the '''composite of $f_1, f_2, \ldots, f_n$''' is defined and denoted as: {{begin-eqn}} {{eqn | l = \forall x \in S_1: \map {\paren {f_n \circ \cdots \circ f_2 \circ f_1} } x       | o = :=       | r = \begin {cases} \map {f_1} x & : n = 1 \\ \map {f_n} {\map {\paren {f_{n - 1} \circ \cdots \circ f_2 \circ f_1} } x} : & n > 1 \end {cases}       | c =  }} {{eqn | r = \map {f_n} {\dotsm \map {f_2} {\map {f_1} x} \dotsm}       | c =  }} {{end-eqn}}


Definition:Inverse Mapping/Definition 1

Let $f: S \to T$ be a mapping. Let $f^{-1} \subseteq T \times S$ be the inverse of $f$: :$f^{-1} := \set {\tuple {t, s}: \map f s = t}$ Let $f^{-1}$ itself be a mapping: :$\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$ and :$\forall y \in T: \exists x \in S: \tuple {y, x} \in f$ Then $f^{-1}$ is called the '''inverse mapping of $f$'''.


Definition:Chebyshev Distance

The '''Chebyshev distance''' on $A_1 \times A_2$ is defined as: :$\map {d_\infty} {x, y} := \max \set {\map {d_1} {x_1, y_1}, \map {d_2} {x_2, y_2} }$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_1 \times A_2$.


Definition:Taxicab Metric/General Definition

The '''taxicab metric''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as: : $\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n d_{i'} \left({x_i, y_i}\right)$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


Definition:Chebyshev Distance/General Definition

The '''Chebyshev distance''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ is defined as: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \mathcal A$.


Definition:Euclidean Metric

The '''Euclidean metric''' on $A_{1'} \times A_{2'}$ is defined as: :$\map {d_2} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^2 + \paren {\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.


Definition:Euclidean Metric/General Definition

The '''Euclidean metric''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as: : $\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^2}\right)^{\frac 1 2}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


Definition:Taxicab Metric/Real Vector Space

The '''taxicab metric''' on $\R^n$ is defined as: : $\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n \left\vert {x_i - y_i}\right\vert$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.


Definition:Chebyshev Distance/Real Vector Space

The '''Chebyshev distance''' on $\R^n$ is defined as: :$\displaystyle \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.


Definition:Standard Discrete Metric/Real Number Plane

The '''discrete metric''' on $\R^2$ is defined as: : $\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.


Definition:Euclidean Metric/Real Number Plane

The '''Euclidean metric''' on $\R^2$ is defined as: :$\displaystyle \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.


Definition:Taxicab Metric/Graphical Example

This diagram shows the open $\epsilon$-ball of point $A$ in the $\left({\R^2, d_1}\right)$ metric space where $d_1$ is the taxicab metric. :300px Note that $\epsilon = \epsilon_1 + \epsilon_2$. Neither the boundary lines nor the extreme points are actually part of the open $\epsilon$-ball.


Definition:P-Product Metric/Real Vector Space

The '''$p$-product metric''' on $\R^n$ is defined as: :$\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left\vert{x_i - y_i}\right\vert^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.


Definition:Taxicab Metric/Real Number Plane

The '''taxicab metric''' on $\R^2$ is defined as: :$\map {d_1} {x, y} := \size  {x_1 - y_1} + \size {x_2 - y_2}$ where $x = \tuple {x_1, x_2}$, $y = \tuple {y_1, y_2} \in \R^2$.


Definition:Chebyshev Distance/Real Number Plane

The '''Chebyshev distance''' on $\R^2$ is defined as: :$\map {d_\infty} {x, y}:= \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.


Definition:Euclidean Metric/Ordinary Space

The '''Euclidean metric''' on $\R^3$ is defined as: :$\map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 + \paren {x_3 - y_3}^2}$ where $x = \tuple {x_1, x_2, x_3}, y = \tuple {y_1, y_2, y_3} \in \R^3$.


Definition:Euclidean Metric/Complex Plane

The '''Euclidean metric''' on $\C$ is defined as: :$\displaystyle \forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$ where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.


Definition:Euclidean Metric/Rational Number Plane

The '''Euclidean metric''' on $\Q^2$ is defined as: :$\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \Q^2$.


Definition:Continuous Real Function/Open Interval

Let $f$ be a real function defined on an open interval $\openint a b$. Then $f$ is '''continuous on $\openint a b$''' {{iff}} it is continuous at every point of $\openint a b$.


Definition:Continuous Real Function/Closed Interval/Definition 1

The function $f$ is '''continuous on $\left[{a \,.\,.\, b}\right]$''' {{iff}} it is: :$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$ :$(2): \quad$ continuous on the right at $a$ :$(3): \quad$ continuous on the left at $b$. That is, if $f$ is to be continuous over the ''whole'' of a closed interval, it needs to be continuous at the end points. Because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined.


Definition:Continuous Real Function/Half Open Interval

Let $f$ be a real function defined on a half open interval $\hointl a b$. Then $f$ is '''continuous on $\hointl a b$''' {{iff}} it is: :$(1): \quad$ continuous at every point of $\openint a b$ :$(2): \quad$ continuous on the left at $b$. Let $f$ be a real function defined on a half open interval $\hointr a b$. Then $f$ is '''continuous on $\hointr a b$''' {{iff}} it is: :$(1): \quad$ continuous at every point of $\openint a b$ :$(2): \quad$ continuous on the right at $a$.


Definition:Supremum Metric/Bounded Real Functions on Interval

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $A$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:Supremum Metric

Let $S$ be a set. Let $M = \left({A', d'}\right)$ be a metric space. Let $A$ be the set of all bounded mappings $f: S \to M$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} d' \left({f \left({x}\right), g \left({x}\right)}\right)$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:Supremum Metric/Continuous Real Functions

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $A$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:L1 Metric

=== Closed Real Interval === {{:Definition:L1 Metric/Closed Real Interval}} Category:Definitions/L1 Metric aihu5b2srlf0jq3gj9kmoz3l6zyxaav


Definition:L1 Metric/Closed Real Interval

Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: d_1 \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$ Then $d_1$ is the '''$L^1$ metric''' on $\left[{a \,.\,.\, b}\right]$.


Definition:L2 Metric/Closed Real Interval

Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: \map d {f, g} := \paren {\int_a^b \paren {\map f t - \map g t}^2 \rd t}^{\frac 1 2}$ Then $d$ is the '''$L^2$ metric''' on $\closedint a b$.


Definition:L2 Metric

=== Closed Real Interval === {{:Definition:L2 Metric/Closed Real Interval}} Category:Definitions/L2 Metric omvah450xw3aqgxhw9r8qsidzeb3taw


Definition:Bounded Metric Space/Definition 1

'''$M'$ is bounded (in $M$)''' {{iff}}: :$\exists a \in A, K \in \R: \forall x \in B: \map {d_B} {x, a} \le K$ That is, there exists an element of $A$ within a finite distance of all elements of $B$.


Definition:Bounded Metric Space/Definition 2

$M'$ is '''bounded''' {{iff}}: :$\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$ That is, there exists a finite distance such that all pairs of elements of $B$ are within that distance.


Definition:Hilbert Sequence Space

Let $d_2: A \times A: \to \R$ be the real-valued function defined as: :$\displaystyle \forall x = \sequence {x_i}, y = \sequence {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$ The metric space $\struct {A, d_2}$ is the '''Hilbert sequence space on $\R$''' and is denoted $\ell^2$.


Definition:Supremum Metric/Bounded Real-Valued Functions

Let $S$ be a set. Let $A$ be the set of all bounded real-valued functions $f: S \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:Supremum of Real Sequence

Let $\sequence {x_n}$ be a real sequence. Let $\set {x_n: n \in \N}$ admit a supremum. Then the '''supremum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \sup {\sequence {x_n} } = \map \sup {\set {x_n: n \in \N} }$


Definition:Infimum of Real Sequence

Let $\sequence {x_n}$ be a real sequence. Let $\set {x_n: n \in \N}$ admit an infimum. Then the '''infimum''' of $\sequence {x_n}$) is defined as: :$\map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$


Definition:Supremum Metric/Bounded Real Sequences

Let $A$ be the set of all bounded real sequences. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall \left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle \in A: d \left({\left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle}\right) := \sup_{n \mathop \in \N} \left\vert{x_n - y_n}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:Supremum Metric/Bounded Continuous Mappings

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in A_1} d_2 \left({f \left({x}\right), g \left({x}\right)}\right)$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.


Definition:Supremum Metric/Differentiability Class

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $r \in \N$ be a natural number. Let $\mathscr D^r \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$ which are of differentiability class $r$. Let $d: \mathscr D^r \left[{a \,.\,.\, b}\right] \times \mathscr D^r \left[{a \,.\,.\, b}\right] \to \R$ be the function defined as: :$\displaystyle \forall f, g \in \mathscr D^r \left[{a \,.\,.\, b}\right]: d \left({f, g}\right) := \sup_{\substack {x \mathop \in \left[{a \,.\,.\, b}\right] \\ i \in \left\{ {0, 1, 2, \ldots, r}\right\} } } \left\vert{f^{\left({i}\right)} \left({x}\right) - g^{\left({i}\right)} \left({x}\right)}\right\vert$ where: : $f^{\left({i}\right)}$ denotes the $i$th derivative of $f$ : $f^{\left({0}\right)}$ denotes $f$ : $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $\mathscr D^r \left[{a \,.\,.\, b}\right]$.


Definition:Lp Metric

=== Closed Real Interval === {{:Definition:Lp Metric/Closed Real Interval}} Category:Definitions/Lp Metrics 78tx38phmzq5alt9mzh28ohae55m5yg


Definition:Lp Metric/Closed Real Interval

Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let $p \in \R_{\ge 1}$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: d \left({f, g}\right) := \left({\int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert^p \ \mathrm d t}\right)^{\frac 1 p}$ Then $d$ is the '''$L^p$ metric''' on $\left[{a \,.\,.\, b}\right]$.


Definition:P-Sequence Metric/Real Sequences

Let $d_p: A \times A: \to \R$ be the real-valued function defined as: :$\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in A: d_p \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left\vert{x_k - y_k}\right\vert^p}\right)^{\frac 1 p}$ The metric space $\left({A, d_p}\right)$ is the '''$p$-sequence space on $\R$''' and is denoted $\ell^p$.


Definition:P-Sequence Metric

=== Real Sequences === {{:Definition:P-Sequence Metric/Real Sequences}} Category:Definitions/P-Sequence Metrics Category:Definitions/Examples of Metric Spaces m2zeg7e77318tab4lbkow9vm6dzc8el


Definition:P-adic Valuation/Integers

The '''$p$-adic valuation (on $\Z$)''' is the mapping $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ defined as: :$\map {\nu_p^\Z} n := \begin {cases} +\infty & : n = 0 \\ \sup \set {v \in \N: p^v \divides n} & : n \ne 0 \end{cases}$ where: :$\sup$ denotes supremum :$p^v \divides n$ expresses that $p^v$ divides $n$.


Definition:P-adic Valuation/Rational Numbers

Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ by: :$\nu_p^\Q \left({\dfrac a b}\right) := \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)$ This mapping $\nu_p^\Q$ is called the '''$p$-adic valuation (on $\Q$)''' and is usually denoted $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$.


Definition:Valuation Axioms

{{begin-axiom}} {{axiom | n = \text V 1         | q = \forall a, b \in R         | ml= \map \nu {a \times b}         | mo= =         | mr= \map \nu a + \map \nu b }} {{axiom | n = \text V 2         | q = \forall a \in R         | ml= \map \nu a = +\infty         | mo= \iff         | mr= a = 0_R         | rc= where $0_R$ is the ring zero }} {{axiom | n = \text V 3         | q = \forall a, b \in R         | ml= \map \nu {a + b}         | mo= \ge         | mr= \min \set {\map \nu a, \map \nu b} }} {{end-axiom}}


Definition:Multiplicative Norm Axioms

{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= =         | mr= \norm x \times \norm y }} {{axiom | n = \text N 3         | lc= Triangle Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}


Definition:Norm Axioms (Vector Space)

{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive definiteness:         | q = \forall x \in V         | ml= \norm x = 0         | mo= \iff         | mr= x = \mathbf 0_V }} {{axiom | n = \text N 2         | lc= Positive homogeneity:         | q = \forall x \in V, \lambda \in R         | ml= \norm {\lambda x}         | mo= =         | mr= \norm {\lambda}_R \times \norm x }} {{axiom | n = \text N 3         | lc= Triangle inequality:         | q = \forall x, y \in V         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}


Definition:Topology Induced by Metric/Definition 1

The '''topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$''' is defined as the set $\tau$ of all open sets of $M$.


Definition:Topology Induced by Metric/Definition 2

The '''topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$''' is defined as the topology $\tau$ generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.


Definition:Closed Set/Real Analysis

=== Real Numbers === {{:Definition:Closed Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Closed Set/Real Analysis/Real Euclidean Space}}


Definition:System of Neighborhoods

Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$. Let $\mathcal N_a$ be the set of all neighborhoods of $a$ in $M$. Then $\mathcal N_a$ is the '''system of neighborhoods''' of the point $a$.


Definition:Continuous Mapping (Metric Space)/Point/Definition 4

$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :for each neighborhood $N'$ of $f \left({a}\right)$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \left[{N}\right] \subseteq N'$.


Definition:Basis for Neighborhood System

Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$ be a point in $M$. Let $\mathcal B_a$ be a set of neighborhoods of $a$ in $M$. Then $\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff: :$\forall N_a \subseteq M: \exists B \in \mathcal B_a: B \subseteq N_a$ where $N_a$ denotes a neighborhood of $a$ in $M$. That is, $\mathcal B_a$ is a '''basis for the neighborhood system at $a$''' iff every neighborhood of $a$ contains an element of $\mathcal B_a$ as a subset.


=== Limit at (Positive) Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Positive}} === Limit at Negative Infinity === {{:Definition:Limit of Real Function/Limit at Infinity/Negative}}


Definition:Convergent Mapping/Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $f \left({x}\right)$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''.


Definition:Convergent Mapping/Complex Function

Let $f: \C \to \C$ be a complex function defined everywhere on $\C$ ''except possibly'' at $c$. Let $f \left({z}\right)$ tend to the limit $L$ as $z$ tends to $c$. Then $f$ '''converges to the limit $L$ as $z$ tends to $c$'''.


Definition:Convergent Mapping/Real Function

Let $f: \R \to \R$ be a real function defined everywhere on $A_1$ ''except possibly'' at $c$. Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$. Then $f$ '''converges to the limit $L$ as $x$ tends to $c$'''.


Definition:Convergent Sequence/Metric Space/Definition 1

$\sequence {x_k}$ '''converges to the limit $l \in A$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$


Definition:Convergent Sequence/Metric Space/Definition 2

$\sequence {x_k}$ '''converges to the limit $l \in A$''' {{iff}}: :$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$ where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.


Definition:Convergent Sequence/Metric Space/Definition 3

$\sequence {x_k}$ '''converges to the limit $l \in A$''' {{iff}}: :$\displaystyle \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$


Definition:Almost All/Set Theory

=== Countable === {{:Definition:Almost All/Set Theory/Countable}} === Uncountable === {{:Definition:Almost All/Set Theory/Uncountable}}


Definition:Limit of Sequence/Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ converge to a value $l \in \R$. Then $l$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity'''.


Definition:Convergent Sequence/Real Numbers

Let $\sequence {x_k}$ be a sequence in $\R$. The sequence $\sequence {x_k}$ '''converges to the limit $l \in \R$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$ where $\size x$ denotes the absolute value of $x$.


Definition:Limit of Sequence/Rational Numbers

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\Q$. Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in \R$, where $\R$ denotes the set of real numbers. Then $l$ is a '''limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity'''.


Definition:Convergent Sequence/Rational Numbers

Let $\sequence {x_k}$ be a sequence in $\Q$. $\sequence {x_k}$ '''converges to the limit $l \in \R$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$ where $\size x$ is the absolute value of $x$.


Definition:Limit of Sequence/Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$. Let $\sequence {z_n}$ converge to a value $l \in \C$. Then $l$ is a '''limit of $\sequence {z_n}$ as $n$ tends to infinity'''.


Definition:Convergent Sequence/Complex Numbers/Definition 1

Let $\sequence {z_k}$ be a sequence in $\C$. $\sequence {z_k}$ '''converges to the limit $c \in \C$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$ where $\cmod z$ denotes the modulus of $z$.


Definition:Rational Number/Fraction/Vulgar

A '''vulgar fraction''' is a rational number whose absolute value is less than $1$, expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers.


Definition:Rational Number/Fraction/Improper

An '''improper fraction''' is a rational number whose absolute value is greater than $1$, specifically when expressed in the form $r = \dfrac p q$ where $p$ and $q$ are integers such that $p > q$.


Definition:Rational Number/Fraction/Mixed Number

A '''mixed number''' is a rational number whose absolute value is greater than $1$, expressed in the form $r = n \frac p q$ where: : $p$ and $q$ are integers such that $p < q$ : $r = n + \dfrac p q$


Definition:Absolute Value/Definition 1

Let $x \in \R$ be a real number. The '''absolute value''' of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows: :$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$


Definition:Absolute Value/Definition 2

Let $x \in \R$ be a real number. The '''absolute value''' of $x$ is denoted $\size x$, and is defined as: :$\size x = +\sqrt {x^2}$ where $+\sqrt {x^2}$ is the positive square root of $x^2$.


Definition:Absolute Value/Number Classes

The absolute value function applies to the various number classes as follows: : Natural numbers $\N$: All elements of $\N$ are greater than or equal to zero, so the concept is irrelevant. : Integers $\Z$: As defined here. : Rational numbers $\Q$: As defined here. : Real numbers $\R$: As defined here. : Complex numbers $\C$: As $\C$ is not an ordered set, the definition of the absolute value function based upon whether a complex number is greater than or less than zero cannot be applied. The notation $\cmod z$, where $z \in \C$, is defined as the modulus of $z$ and has a different meaning.


Definition:Open Ball/Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\epsilon$-ball.


Definition:Basis for Open Sets (Metric Space)

Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal B$ be a set of open sets of $M$. Then $\mathcal B$ is a '''basis for the open sets of $M$''' iff: :for each open set $U$ of $M$, $U$ is the union of sets of $\mathcal B$.


Definition:Closure (Metric Space)

Let $M = \struct {A, d}$ be a metric space. Let $H \subseteq A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. The '''closure of $H$ (in $M$)''' is the union of all isolated points of $H$ and all limit points of $H$: :$H^- := H' \cup H^i$


Definition:Closure (Topology)/Notation

The topological closure of $H$ is variously denoted: :$\map \cl H$ :$\map {\mathrm {Cl} } H$ :$\overline H$ :$H^-$ Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement. $\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear. $H^-$ is neat and compact, but has the disadvantage of being relatively obscure. On {{ProofWiki}}, $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.


Definition:Hermitian Conjugate

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$. Then the '''Hermitian conjugate''' of $\mathbf A$ is denoted $\mathbf A^\dagger$ and is defined as: :$\mathbf A^\dagger = \sqbrk \beta_{n m}: \forall i \in \set {1, 2, \ldots, n}, j \in \set {1, 2, \ldots, m}: \beta_{i j} = \overline {\alpha_{j i} }$ where $\overline {\alpha_{j i} }$ denotes the complex conjugate of $\alpha_{j i}$.


Definition:Hermitian Matrix

Let $\mathbf A$ be a square matrix over $\C$. $\mathbf A$ is '''Hermitian''' {{iff}}: :$\mathbf A = \mathbf A^\dagger$ where $\mathbf A^\dagger$ is the Hermitian conjugate of $\mathbf A$.


Definition:Binomial Coefficient/Technical Note

The $\LaTeX$ code to render the binomial coefficient $\dbinom n k$ can be written in the following ways: :\dbinom n k or: :\displaystyle {n \choose k} The \dbinom form is preferred on {{ProofWiki}} because it is simpler. It is in fact an abbreviated form of \displaystyle \binom n k, which is the preferred construction when \displaystyle is required for another entity in the expression. While the form \binom n k is valid $\LaTeX$ syntax, it renders the entity in the reduced size inline style: $\binom n k$ which {{ProofWiki}} does not endorse. To render compound arguments, braces are needed to delimit the parameter when using \dbinom, but (confusingly) not \choose. For example, to render $\dbinom {a + b} {c d}$ the following can be used: :<nowiki>\dbinom {a + b} {c d}</nowiki> or: :<nowiki>\displaystyle {a + b \choose c d}</nowiki> $\displaystyle {a + b \choose c d}$ Again, for consistency across {{ProofWiki}}, the \dbinom form is preferred.


Definition:Unitary Matrix

Let $\mathbf U$ be an invertible square matrix over the complex numbers $\C$. Then $\mathbf U$ is '''unitary''' {{iff}}: :$\mathbf U^{-1} = \mathbf U^\dagger$ where: :$\mathbf U^{-1}$ is the inverse of $\mathbf U$ :$\mathbf U^\dagger$ is the Hermitian conjugate of $\mathbf U$.


Definition:Homeomorphism/Metric Spaces

=== Definition 1 === {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Metric Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Metric Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Metric Spaces/Definition 4}}


Definition:Möbius Transformation

A '''Möbius transformation''' is a mapping $f: \overline \C \to \overline \C$ of the form: :$\map f z = \dfrac {a z + b} {c z + d}$ where: :$\overline C$ denotes the extended complex plane :$a, b, c, d \in \C$ such that $a d - b c \ne 0$ We define: :$\map f {-\dfrac d c} = \infty$ if $c \ne 0$, and: :$\map f \infty = \begin{cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$


Definition:Liouville Number

A real number $x$ is a '''Liouville number''' if for all $n \in \N$, there exist $p, q \in \Z$ (which may depend on $n$) with $q > 1$ such that: :$0 < \size {x - \dfrac p q} < \dfrac 1 {q^n}$


Definition:Bell Number

The '''Bell Numbers''' $B_n$ are a sequence of natural numbers defined as the number of ways a set with $n$ elements can be partitioned.


Definition:Catalan's Constant

'''Catalan's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G       | r = \map \beta 2       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^{\infty} \frac{\paren {-1}^n} {\paren {2 n + 1}^2}       | c =  }} {{eqn | r = \frac 1 {1^2} - \frac 1 {3^2} + \frac 1 {5^2} - \frac 1 {7^2} + \cdots       | c =  }} {{end-eqn}} where $\beta$ is the Dirichlet beta function. Its numerical value is approximately: :$G = 0 \cdotp 91596 \, 55941 \, 77219 \, 01505 \, 46035 \, 14932 \, 38411 \, 0774 \ldots$


Definition:Lipschitz Equivalence/Metrics/Definition 1

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\exists h, k \in \R_{>0}$ such that: : $\forall x, y \in A: h \map {d_2} {x, y} \le \map {d_1} {x, y} \le k \map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''.


Definition:Lipschitz Equivalence/Metrics/Definition 2

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $\exists K_1, K_2 \in \R_{>0}$ such that: : $(1): \quad \forall x, y \in A: \map {d_2} {x, y} \le K_1 \map {d_1} {x, y}$ : $(2): \quad \forall x, y \in A: \map {d_1} {x, y} \le K_2 \map {d_2} {x, y}$ Then $d_1$ and $d_2$ are described as '''Lipschitz equivalent'''.


Definition:Homeomorphism/Metric Spaces/Definition 1

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : $f$ is continuous from $M_1$ to $M_2$ : $f^{-1}$ is continuous from $M_2$ to $M_1$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.


Definition:Homeomorphism/Metric Spaces/Definition 2

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $U \subseteq A_1$, $U$ is an open set of $M_1$ {{iff}} $f \left[{U}\right]$ is an open set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.


Definition:Homeomorphism/Metric Spaces/Definition 3

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $V \subseteq A_1$, $V$ is a closed set of $M_1$ {{iff}} $f \left[{V}\right]$ is a closed set of $M_2$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.


Definition:Homeomorphism/Metric Spaces/Definition 4

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection such that: : for all $a \in A_1$ and $N \subseteq A_1$, $N$ is a neighborhood of $a$ {{iff}} $f \left[{N}\right]$ is a neighborhood of $f \left({a}\right)$. Then: : $f$ is a '''homeomorphism''' : $M_1$ and $M_2$ are '''homeomorphic'''.


Definition:Cardinality of Continuum

The '''Cardinality of the Continuum''' is the cardinality of the set of real numbers, denoted $\left \lvert {\R}\right \rvert$ or $\mathfrak c$. {{refactor|State and prove this in a separate page}} It is an infinite cardinal number.


Definition:Exact Sequence of Groups

Let $\left({G, \circ}\right)$ be a group. Consider the sequence of groups $\left\langle{G_i}\right\rangle$ and group homomorphisms $\phi_i$: :$\displaystyle \cdots \stackrel{\phi_{i-2}}{\longrightarrow} G_{i-1} \stackrel{\phi_{i-1}}{\longrightarrow} G_i \stackrel{\phi_i}{\longrightarrow} G_{i+1} \stackrel{\phi_{i+1}}{\longrightarrow} \cdots$ $\left\langle{G_i}\right\rangle$ is  '''exact''' {{iff}}: :$\forall i: \operatorname{Im} \left({\phi_i}\right) = \ker \left({\phi_{i+1} }\right)$ where: :$\operatorname{Im} \left({\phi_i}\right)$ denotes the image of $\phi_i$ :$\ker \left({\phi_{i+1} }\right)$ denotes the kernel of $\phi_{i+1}$.


Definition:Short Exact Sequence of Groups

Let $\left({G, \cdot}\right)$ be a group. An exact sequence of the form :$1 \longrightarrow K \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta}{\longrightarrow} H \longrightarrow 1$ is called a '''short exact sequence''', where $1$ represents the trivial group. Category:Definitions/Group Theory 82r1xq04tv43yjqdjvkm62opeyg8m27


Definition:Dozen

A '''dozen''' is another name for a set whose cardinality is $12$.


Definition:Gross

A '''gross''' is another name for a set whose cardinality is $12 \times 12$, or $144$.


Definition:Score

A '''score''' is another name for a set whose cardinality is $20$.


Definition:Integral Sign

The symbol: :$\displaystyle \int \ldots \rd \mu$ is called the '''integral sign'''. Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.


Definition:Axis/Positive Direction

Consider a coordinate system whose axes are each aligned with an instance of the real number line $\R$. The direction along an axis in which the corresponding elements of $\R$ are increasing is called the '''positive direction'''.


Definition:Smallest Set by Set Inclusion

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by $\subseteq$ considered as an ordering. Then $T \in \TT$ is the '''smallest set''' of $\TT$ {{iff}} $T$ is the smallest element of $\struct {\TT, \subseteq}$. That is: :$\forall X \in \TT: T \subseteq X$


Definition:Smallest Set by Set Inclusion/Class Theory

Let $A$ be a class. Then a set $m$ is the '''smallest element''' of $A$ (with respect to the inclusion relation) {{iff}}: :$(1): \quad m \in A$ :$(2): \quad \forall S: \paren {S \in A \implies m \subseteq S}$


Definition:Euclidean Space/Euclidean Topology/Real

Let $\R^n$ be an $n$-dimensional real vector space. Let $M = \struct {\R^n, d}$ be a real Euclidean $n$-space. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.


Definition:Euclidean Space/Euclidean Topology/Rational

Let $\Q^n$ be an $n$-dimensional vector space of rational numbers. Let $M = \left({\Q^n, d}\right)$ be a rational Euclidean $n$-space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.


Definition:Euclidean Space/Euclidean Topology/Complex

Let $\C$ be the complex plane. Let $M = \left({\C, d}\right)$ be a complex Euclidean space. The topology induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.


Definition:Neighborhood Filter/Point

Let $x \in S$. Let $\mathcal N_x$ be the set of all neighborhoods of $x$ in $T$. Then $\mathcal N_x$ is the '''neighborhood filter''' of $x$.


{{:Definition:Separated by Neighborhoods/Sets/Neighborhoods}}


{{:Definition:Separated by Neighborhoods/Points/Neighborhoods}}


Definition:Hausdorff Space/Definition 3

$\left({S, \tau}\right)$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}}: :$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$ That is: : for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.


Definition:Separated by Closed Neighborhoods/Points

Let $x, y \in S$ such that: :$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$ where $N_x^-$ and $N_y^-$ are the closures in $T$ of $N_x$ and $N_y$ respectively. That is, that $x$ and $y$ both have neighborhoods in $T$ whose closures are disjoint. Then $x$ and $y$ are described as '''separated by closed neighborhoods'''.


Definition:Neighborhood Space

Let $S$ be a set. For each $x \in S$, let there be given a set $\mathcal N_x$ of subsets of $S$ which satisfy the neighborhood space axioms: {{:Axiom:Neighborhood Space Axioms}} The sets $\mathcal N_x$ are the neighborhoods of $x$ in $S$. Let $\mathcal N$ be the set of open sets of $S$: :$\mathcal N = \left\{{U \subseteq S: U}\right.$ is a neighborhood of each of its elements $\left.{}\right\}$ The set $S$ together with $\mathcal N$ is called a '''neighborhood space''' and is denoted $\left({S, \mathcal N}\right)$.


Definition:Neighborhood Filter/Set

Let $A \subseteq S$ such that $A \ne \O$. Let $\NN_A$ be the set of all neighborhoods of $A$. Then $\NN_A$ is the '''neighborhood filter''' of $A$.


Definition:Neighborhood (Neighborhood Space)

Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $x \in S$. Let $\mathcal N_x$ satisfy the neighborhood space axioms. Then each element $N$ of $\mathcal N_x$ is called a '''neighborhood''' of $x$.


Definition:Open Set (Neighborhood Space)

Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $U \subseteq S$ be a neighborhood of each of its elements. Then $U$ is an '''open set''' of $S$.


Definition:Topological Space Induced by Neighborhood Space

Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $\tau$ be the set of open sets of $\left({S, \mathcal N}\right)$. Then $\left({S, \tau}\right)$ is the '''topological space induced by $\left({S, \mathcal N}\right)$'''.


Definition:Combinatorics

'''Combinatorics''' is that branch of mathematics concerned with counting things. '''Combinatorial''' problems are so named because they are exercises in counting the number of combinations of various objects. It has been stated that it is the core of the discipline of discrete mathematics.


Definition:Binomial Coefficient/Notation

The notation $\dbinom n k$ for the binomial coefficient was introduced by {{AuthorRef|Andreas Freiherr von Ettingshausen}} in his $1826$ work {{BookLink|Die kombinatorische Analysis, als Vorbereitungslehre zum Studium der theoretischen höheren Mathematik|Andreas v. Ettingshausen}}. It appears to have become the ''de facto'' standard in recent years. As a result, $\dbinom n k$ is frequently voiced '''the binomial coefficient $n$ over $k$'''. Other notations include: :$C \left({n, k}\right)$ :${}^n C_k$ :${}_n C_k$ :$C^n_k$ :${C_n}^k$ all of which can cause a certain degree of confusion.


Definition:Binomial Coefficient/Integers/Definition 1

Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''binomial coefficient''' $\dbinom n k$ is defined as: :$\dbinom n k = \begin{cases} \dfrac {n!} {k! \paren {n - k}!} & : 0 \le k \le n \\ & \\ 0 & : \text { otherwise } \end{cases}$ where $n!$ denotes the factorial of $n$.


Definition:Binomial Coefficient/Historical Note

The binomial coefficients have been known about since at least the ancient Greeks and Romans, who were familiar with them for small values of $k$. See the historical note to Pascal's Triangle for further history.


Definition:Binomial Coefficient/Integers/Definition 2

Let $n \in \Z_{\ge 0}$ and $k \in \Z$. The number of different ways $k$ objects can be chosen (irrespective of order) from a set of $n$ objects is denoted: :$\dbinom n k$ This number $\dbinom n k$ is known as a '''binomial coefficient'''.


Definition:Binomial Coefficient/Integers/Definition 3

Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''binomial coefficient''' $\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\paren {a + b}^n$.


Definition:Geometric Mean/Mean Proportional/General Definition

In the language of {{AuthorRef|Euclid}}, the terms of a (finite) geometric sequence of positive integers between (and not including) the first and last terms are called '''mean proportionals'''.


Definition:Geometric Sequence of Integers in Lowest Terms

Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers. Let $r$ be the common ratio of $G_n$. Let $S$ be the set of all such geometric sequence: :$S = \left\{{G: G}\right.$ is a geometric sequence of integers whose common ratio is $\left.{r}\right\}$ Then $G_n$ is in '''lowest terms''' if the absolute values of the terms of $G_n$ are the smallest, term for term, of all the elements of $S$: :$\forall Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n} \in S: \forall j \in \set {0, 1, \ldots, n}: \size {a_j} \le \size {b_j}$


Definition:Geometric Sequence/Finite

A '''finite geometric sequence''' is a geometric sequence with a finite number of terms .


Definition:Geometric Sequence/Finite/Extremes

Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a finite geometric sequence. The '''extremes''' of $G_n$ are the terms $a_0$ and $a_n$ of $G_n$.


Definition:Geometric Sequence/Initial Term

Let $G = \left\langle{a_0, a_1, \ldots}\right\rangle$ be a geometric sequence. The '''initial term''' of $G_n$ is the term $a_0$. The same definition applies to a finite geometric sequence $G_n = \sequence {a_0, a_1, \ldots, a_n}$.


Definition:Geometric Sequence/Finite/Final Term

Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a finite geometric sequence. The '''final term''' of $G_n$ is the term $a_n$.


Definition:Geometric Sequence/Integers

A '''geometric sequence of integers''' is a finite geometric sequence whose terms are all integers. Category:Definitions/Geometric Sequences o8n7wjhsvrr4hhufm061sjzlhrei5zf


Definition:Unit (One)/Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. By Zero Complement is Not Empty, $S^*$ is not empty. Therefore, by axiom $(NO4)$, it has a smallest element for $\preceq$. This smallest element is called '''one''' and denoted $1$.


Definition:Commensurable/Notation

There appears to be no universally acknowledged symbol to denote commensurability. {{AuthorRef|Thomas Little Heath|Thomas L. Heath}} in his edition of {{BookLink|Euclid: The Thirteen Books of The Elements: Volume 3|Sir Thomas L. Heath|ed = 2nd|edpage = Second Edition}} makes the following suggestions: :$(1): \quad$ To denote that $A$ is commensurable or commensurable in length with $B$:  :::$A \mathop{\frown} B$ :$(2): \quad$ To denote that $A$ is commensurable in square with $B$:  :::$A \mathop{\frown\!\!-} B$ :$(3): \quad$ To denote that $A$ is incommensurable or incommensurable in length with $B$:  :::$A \mathop{\smile} B$ :$(4): \quad$ To denote that $A$ is incommensurable in square with $B$:  :::$A \mathop{\smile\!\!-} B$ This convention may be used on {{ProofWiki}} if accompanied by a note which includes a link to this page.


Definition:Apotome of Medial/Whole

The real number $a$ is called the '''whole''' of the apotome of a medial.


Definition:Apotome of Medial/Annex

The real number $b$ is called the '''annex''' of the apotome of a medial.


Definition:Apotome of Medial/Terms

The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome of Medial/Whole}} === Annex === {{Definition:Apotome of Medial/Annex}}


Definition:Minor (Euclidean)/Whole

The real number $a$ is called the '''whole''' of the minor.


Definition:Minor (Euclidean)/Annex

The real number $b$ is called the '''annex''' of the minor.


Definition:Minor (Euclidean)/Terms

The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Minor (Euclidean)/Whole}} === Annex === {{Definition:Minor (Euclidean)/Annex}}


Definition:That which produces Medial Whole with Rational Area/Whole

The real number $a$ is called the '''whole''' of the straight line which produces with a rational area a medial whole.


Definition:That which produces Medial Whole with Rational Area/Annex

The real number $b$ is called the '''annex''' of the straight line which produces with a rational area a medial whole.


Definition:That which produces Medial Whole with Rational Area/Terms

The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Rational Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Rational Area/Annex}}


Definition:That which produces Medial Whole with Medial Area/Whole

The real number $a$ is called the '''whole''' of the straight line which produces with a medial area a medial whole.


Definition:That which produces Medial Whole with Medial Area/Annex

The real number $b$ is called the '''annex''' of the straight line which produces with a medial area a medial whole.


Definition:That which produces Medial Whole with Medial Area/Terms

The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Medial Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Medial Area/Annex}}


Definition:Piecewise Continuously Differentiable Function/Definition 2

$f$ is '''piecewise continuously differentiable''' {{iff}}: :there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$f$ is continuously differentiable on $\closedint {x_{i − 1} } {x_i}$, where the derivative at $x_{i − 1}$ understood as right-handed and the derivative at $x_i$ understood as left-handed, for every $i \in \set {1, \ldots, n}$.


Definition:Right Angle/Perpendicular/Plane

{{EuclidDefinition|book = XI|def = 3|name = Line at Right Angles to Plane}} :400px In the above diagram, the line $AB$ has been constructed so as to be a '''perpendicular''' to the plane containing the straight lines $CD$ and $EF$.


Definition:Right Angle/Perpendicular/Plane to Plane

{{EuclidDefinition|book = XI|def = 4|name = Plane at Right Angles to Plane}} :400px In the above diagram, the two planes have been constructed so as to make lines perpendicular to their common section perpendicular to each other. Thus the two planes are perpendicular to each other.


Definition:Pi/Definition 1

Take a circle in a plane whose circumference is $C$ and whose radius is $r$. Then $\pi$ can be defined as $\pi = \dfrac C {2r}$.


Definition:Pi/Definition 2

The real functions sine and cosine can be shown to be periodic. The number $\pi$ is defined as the real number such that: :the period of both sine and cosine is $2 \pi$.


Definition:Piecewise Continuously Differentiable Function/Definition 1

$f$ is '''piecewise continuously differentiable''' {{iff}}: :$(1): \quad f$ is continuous :$(2): \quad$ there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$(2.1): \quad f$ is continuously differentiable on $\openint {x_{i - 1}} {x_i}$ for every $i \in \set {1, \ldots, n}$ ::$(2.2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i - 1}}^+} \map {f'} x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map {f'} x$ exist for every $i \in \set {1, \ldots, n}$.


Definition:Parallelepiped/Height

:400px The '''height''' of a parallelepiped is the length of the perpendicular from the plane of the base to the plane opposite. In the above diagram, $h$ is the '''height''' of the parallelepiped whose base is $AB$.


Definition:Similar Situation

Two similar solid figures are said to be in a '''similar situation''' {{iff}} corresponding surfaces are similarly inclined and when corresponding edges are parallel.


Definition:Solid Geometry

'''Solid geometry''' is the study of geometric figures in three dimensions. As such it is a misnomer, because the figures themselves under study are strictly speaking the surfaces of such figures.


Definition:Group Theory

'''Group Theory''' is a branch of abstract algebra which studies groups and other related algebraic structures.


Definition:Abstract Algebra

'''Abstract algebra''' is a branch of mathematics which studies algebraic structures and algebraic systems. It can be roughly described as the study of sets equipped with operations.


Definition:Algebra (Mathematical Branch)

'''Algebra''' is the branch of mathematics which studies the techniques of manipulation of objects and expressions.


Definition:Parallelepiped/Opposite Face

The '''opposite face''' of the face $F$ of a parallelepiped $P$ is the face of $P$ which is parallel to $F$. In the above example, the pairs of parallel planes are: :Face $ABCD$ is opposite $HGFE$ :Face $ADEH$ is opposite $BCFG$ :Face $ABGH$ is opposite $DCFE$


Definition:Prism/Height

The '''height''' of a prism is the length of the perpendicular between the bases of the prism. In the above diagram, the distance $h$ is the '''height''' of the prism $AJ$.


Definition:Prism/Opposite

Let $P$ be a prism whose bases are $P_1$ and $P_2$. The '''opposite face''' of the base $P_1$ is the base $P_2$. In the above example, the base $ABCDE$ is opposite $FGHIJ$.


Definition:Prism/Height/Euclidean Variant

:400px Although the height of a prism is generally understood to be the length of the perpendicular joining opposite faces, {{AuthorRef|Euclid}} was inconsistent in his usage in {{BookLink|The Elements|Euclid}}. In his {{EuclidPropLink|book = XI|prop = 39|title = Prisms of equal Height with Parallelogram and Triangle as Base}}, he defines the base of one prism as being one of the opposite parallel faces, but of the other he defines the base as being an arbitrary one of the parallelograms. Having defined the base in this manner, the '''height''' is then defined as being the height of one of the opposite parallel faces whose base is the edge which intersects the base so defined. Using this definition, the distance $h$ in the above diagram is the '''height''' of the prism $PQRSTU$.


Definition:Prism/Lateral Face

The '''lateral faces''' of a prism are the parallelogramic faces which join the bases. In the above diagram, the faces $ABGF, BCHG, CDIH, DEJI$ and $EAFJ$ are the '''lateral faces''' of the prism.


Definition:Triangular Prism

:250px A '''triangular prism''' is a prism whose bases are triangles. Category:Definitions/Prisms g9i3f6e84hppos7p0sans6sphy8prn2


Definition:Natural Numbers/Von Neumann Construction

Let $\omega$ denote the minimal infinite successor set. The '''natural numbers''' can be defined as the elements of $\omega$. Following Definition 2 of $\omega$, this amounts to defining the '''natural numbers''' as the finite ordinals. In terms of the empty set $\O$ and successor sets, we thus define: {{begin-eqn}} {{eqn | l = 0       | o = :=       | r = \O = \set {}       | c =  }} {{eqn | l = 1       | o = :=       | r = 0^+ = 0 \cup \set 0 = \set 0       | c =  }} {{eqn | l = 2       | o = :=       | r = 1^+ = 1 \cup \set 1 = \set {0, 1}       | c =  }} {{eqn | l = 3       | o = :=       | r = 2^+ = 2 \cup \set 2 = \set {0, 1, 2}       | c =  }} {{eqn | o = \vdots       | c =  }} {{eqn | l = n + 1       | o = :=       | r = n^+ = n \cup \set n       | c =  }} {{end-eqn}}


Definition:Piecewise Continuous Function/One-Sided Limits

$f$ is '''piecewise continuous with one-sided limits''' {{iff}}: :there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that, for all $i \in \set {1, 2, \ldots, n}$: ::$(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$ ::$(2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map f x$ exist.


Definition:Piecewise Continuous Function/Bounded

$f$ is a '''bounded piecewise continuous function''' {{iff}}: :there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that: ::$(1): \quad$ for all $i \in \set {1, 2, \ldots, n}$, $f$ is continuous on $\openint {x_{i − 1} } {x_i}$  ::$(2): \quad$ $f$ is bounded on $\closedint a b$.


Definition:Axis of Solid Figure

=== Axis of Cone === :300px {{:Definition:Axis of Cone}} === Axis of Cylinder === :300px {{:Definition:Axis of Cylinder}} === Axis of Sphere === {{:Definition:Axis of Sphere}}


Definition:Apex

The '''apex''' of a geometric figure is the point which is distinguished from the others by dint of it being furthest away from its base. Not all figures have a discernible '''apex'''; for example, parallelograms, prisms and parallelepipeds do not. === Apex of Triangle === :300px {{:Definition:Apex of Triangle}} === Apex of Isosceles Triangle === :300px {{:Definition:Apex of Isosceles Triangle}} === Apex of Cone === :300px {{:Definition:Apex of Cone}} === Apex of Pyramid === :400px {{:Definition:Apex of Pyramid}}


Definition:Base of Geometric Figure

The '''base''' of a geometric figure is a specific part of that figure which is distinguished from the remainder of that figure and placed (actually or figuratively) at the '''bottom''' of a depiction or visualisation. In some cases the '''base''' is truly qualitiatively different from the rest of the figure. In other cases the '''base''' is selected arbitrarily as one of several parts of the figure which may equally well be so chosen.


Definition:Quadrant

A '''quadrant''' is a sector of a circle whose angle is a right angle.


Definition:Minimal Infinite Successor Set/Definition 3

{{Refactor|Based on Takeuti}} The '''minimal infinite successor set''' $\omega$ is defined as: :$\omega := \set {x \in \On: \paren {x \cup \set x} \subseteq K_I}$ where: :$K_I$ is the class of all non-limit ordinals :$\On$ is the class of all ordinals.


Definition:Minimal Infinite Successor Set/Definition 1

Let $S$ be an infinite successor set. The '''minimal infinite successor set''' $\omega$ is the infinite successor set given by: :$\omega := \displaystyle \bigcap \set {S' \subseteq S: \text{$S'$ is an infinite successor set} }$ that is, $\omega$ is the intersection of every infinite successor set which is a subset of $S$.


Definition:Equiangular Geometric Figures

Two geometric figures are '''equiangular (with each other)''' when the angles of each pair of their corresponding vertices are equal. Category:Definitions/Geometry 5fzaybsrt5vfpk414g7y44x8c7d6qw8


Definition:Tetrahedron/Base

One of the faces of a tetrahedron can be chosen arbitrarily, distinguished from the others and identified as the '''base''' of the tetrahedron. In the above diagram, $ABC$ is the '''base''' of the tetrahedron $ABCD$.


Definition:Tetrahedron/Apex

Once the base of a tetrahedron has been identified, the vertex which does not lie on the base is called the '''apex''' of the tetrahedron. In the above diagram, given that the base of the tetrahedron $ABCD$ is the triangle $ABC$, the '''apex''' is $D$.


Definition:Polyhedron/Face

The '''faces''' of a polyhedron are the polygons which contain it.


Definition:Ludolphine Number

The '''Ludolphine number''' is the expression of the value of $\pi$ to $35$ decimal places: :$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \ldots$


Definition:Sign of Number

The '''sign''' of a number is the symbol indicating whether it is: : positive, denoted by the symbol $+$ or: : negative, denoted by the symbol $-$. Hence a number's '''sign''' has evolved to define the fact of the number being positive or negative independently of the symbol itself. Thus: :the '''sign''' of $3.14159$ is positive and :the '''sign''' of $-75$ is negative.


Definition:Piecewise Continuous Function/Improper Integrals

$f$ is '''piecewise continuous with improper integrals''' {{iff}}: :there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \in \left\{ {1, 2, \ldots, n}\right\}$: ::$(1): \quad f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$  ::$(2): \quad$ the improper integrals $\displaystyle \int_{ {x_{i - 1} }^+}^{ {x_i}^-} f \left({x}\right) \rd x$ all exist.


Definition:Commutative/Elements

Let $\circ$ be a binary operation. Two elements $x, y$ are said to '''commute''' {{iff}}: :$x \circ y = y \circ x$


Definition:Commutative/Operation

Let $\struct {S, \circ}$ be an algebraic structure. Then $\circ$ is '''commutative on $S$''' {{iff}}: :$\forall x, y \in S: x \circ y = y \circ x$


Definition:Commutative Division Ring

A '''commutative division ring''' is a division ring $\left({R, +, \circ}\right)$ in which the ring product $\circ$ is commutative. Category:Definitions/Ring Theory awslszg1z8vsnyd5n03j4syolarfftl


Definition:Commutative/Algebraic Structure

Let $\left({S, \circ}\right)$ be an algebraic structure whose operation $\circ$ is a commutative operation. Then $\left({S, \circ}\right)$ is a '''commutative (algebraic) structure'''.


Definition:Operation/Operand

An '''operand''' is one of the objects on which an operation generates its new object.


Definition:Arithmetic

'''Arithmetic''' is the branch of mathematics which concerns the manipulation of numbers, using the operations addition, subtraction, multiplication and division, and the taking of powers.


Definition:Smallest Field

The '''smallest field''' is the set of integers modulo $2$ under modulo addition and modulo multiplication: :$\struct {\Z_2, +_2, \times_2}$ This field has $2$ elements.


Definition:Addition/Modulo Addition/Definition 1

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$


Definition:Addition/Modulo Addition/Definition 2

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$ The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$x +_m y$ equals the remainder after $x + y$ has been divided by $m$.


Definition:Addition/Modulo Addition/Definition 3

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$ The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$x +_m y := x + y - j m$ where $j$ is the largest integer such that $j m \le x + y$.


Definition:Multiplication/Modulo Multiplication/Definition 1

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m \times_m \eqclass b m = \eqclass {a b} m$


Definition:Multiplication/Modulo Multiplication/Definition 3

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$x \cdot_m y := x y - k m$ where $k$ is the largest integer such that $k m \le x y$.


Definition:Multiplication/Modulo Multiplication/Definition 2

Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$x \cdot_m y$ equals the remainder after $x y$ has been divided by $m$.


Definition:Big Dipper Operation

Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$. Let $+_{m, n}$ be the binary operation on $\Z_{>0}$ defined as: :$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying: :$m + k n \le x + y$ The operation $+_{m, n}$ is known as the '''Big Dipper'''. :600px When the stars of the [http://en.wikipedia.org/wiki/Big_Dipper Big Dipper] are numbered as shown, the sequence: :$1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \ldots$ traces out those stars in the order: :first the handle: $\text{Alkaid}, \text{Mizar}, \text{Alioth}$ then: :round the pan indefinitely: $\text{Megrez}, \text{Dubhe}, \text{Merak}, \text{Phecda}, \text{Megrez}, \ldots$ === Cayley Table === {{:Big Dipper Operation/Cayley Table}}


Definition:Parity Ring

The '''parity ring''' is the ring of two elements which defines the nature of the parity of integers under addition and multiplication: :$\struct {\set {\text{even}, \text{odd} }, +, \times}$


Definition:Group of Gaussian Integer Units

Let $i$ be the imaginary unit: $i = \sqrt {-1}$. Let $U_\C$ be the set of complex numbers defined as: :$U_\C = \set {1, i, -1, -i}$ Let $\times$ denote the operation of complex multiplication. The algebraic structure $\struct {U_\C, \times}$ is the '''group of units of the ring of Gaussian integers'''.


Definition:Ring of Gaussian Integers

The '''ring of Gaussian integers''' $\struct {\Z \sqbrk i, +, \times}$ is the algebraic structure formed from: : the set of Gaussian integers $\Z \sqbrk i$ : the operation of complex addition : the operation of complex multiplication.


Definition:Finite Ordinal

Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''finite''' {{iff}} one of the following holds: :$\alpha = \O$ :$\alpha = \beta^+$ for some '''finite ordinal''' $\beta$ where $\O$ denotes the empty set, and $\beta^+$ is the successor ordinal of $\beta$.


Definition:Minimal Infinite Successor Set/Definition 2

The '''minimal infinite successor set''' $\omega$ is defined as the set of all finite ordinals: :$\omega := \set {\alpha : \text{$\alpha$ is a finite ordinal} }$


Definition:Root of Unity/Complex/First

The root $e^{2 i \pi / n}$ is known as the '''first (complex) $n$th root of unity'''.


Definition:Finite Algebraic Structure

A '''finite algebraic structure''' is an algebraic structure whose underlying set is finite. Category:Definitions/Abstract Algebra c9ktkuy8penh33a3t2ls0vtwf8x35ln


Definition:Euclidean Relation/Left-Euclidean

$\mathcal R$ is '''left-Euclidean''' {{iff}}: :$\tuple {x, z} \in \mathcal R \land \tuple {y, z} \in \mathcal R \implies \tuple {x, y} \in \mathcal R$