hcju/proofwikips huggingface dataset:
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''' : :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'''.
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 ===  === 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 ====  ==== 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}}
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'''  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'''  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}}
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}}  :''{{Definition:Euclid's Definitions - Book VII/21 - Similar Numbers}}'' {{EuclidDefRefNocat|VII|21|Similar Numbers}}
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''' : :$p_1 : q_1 = p_2 : q_2$
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}}
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'''.  :''{{:Definition:Euclid's Definitions - Book VII/1 - Unit}}'' {{EuclidDefRefNocat|VII|1|Unit}}
Definition:Divisor (Algebra)/Real Number/Part
 :''{{: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.  :''{{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 $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$.  : ''{{: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.  : ''{{: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.  : ''{{: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'''  $a$ 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.  : ''{{: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'''  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}}