hcju/proofwikips huggingface dataset:
Definition:Tone (Music)
A '''tone''' is a musical interval whose components have frequencies in the ratio $9 : 8$.
Definition:Well-Tempered Scale
A '''well-tempered scale''' is an octave which has been divided into $12$ equal intervals called semitones.
Definition:Semitone
A '''semitone''' is a musical interval whose components have frequencies in the ratio $2^{1/12} : 1$.
Definition:Imperial/Length
The imperial units of length are as follows: === Mil === {{Definition:Imperial/Length/Mil}} === Inch === {{Definition:Imperial/Length/Inch}} === Foot === {{Definition:Imperial/Length/Foot}} === Yard === {{Definition:Imperial/Length/Yard}} === Rod, Pole or Perch === {{Definition:Imperial/Length/Rod, Pole or Perch}} === Chain === {{Definition:Imperial/Length/Chain}} === Furlong === {{Definition:Imperial/Length/Furlong}} === International Mile === {{Definition:Imperial/Length/International Mile}} === League === {{Definition:Imperial/Length/League}}
Definition:Imperial/Length/Inch
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''inch''' }} {{eqn | r = 2 \cdotp 54       | c = centimetres }} {{eqn | r = 0 \cdotp 0254       | c = metres }} {{end-eqn}} This definition is exact, and is how the imperial unit of length is defined.
Definition:Imperial/Length/Foot
The '''foot''' is the FPS base unit of length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''foot''' }} {{eqn | r = 12       | c = inches }} {{eqn | r = 30.48       | c = centimetres }} {{eqn | r = 0 \cdotp 3048       | c = metres }} {{end-eqn}}
Definition:Imperial/Length/Yard
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''yard''' }} {{eqn | r = 3       | c = feet }} {{eqn | r = 36       | c = inches }} {{eqn | r = 91 \cdot 44       | c = centimetres }} {{eqn | r = 0 \cdotp 9144       | c = metres }} {{end-eqn}}
Definition:Imperial/Length/Rod, Pole or Perch
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rod, pole or perch''' }} {{eqn | r = 5 \tfrac 1 2       | c = yards }} {{eqn | r = 16 \tfrac 1 2       | c = feet }} {{eqn | r = 502 \cdot 92       | c = centimetres }} {{eqn | r = 5 \cdotp 0292       | c = metres }} {{end-eqn}}
Definition:Imperial/Length/Chain
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chain''' }} {{eqn | r = 4       | c = rods, poles or perches }} {{eqn | r = 22       | c = yards }} {{eqn | r = 66       | c = feet }} {{eqn | r = 20 \cdotp 1168       | c = metres }} {{end-eqn}}
Definition:Imperial/Length/Furlong
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''furlong''' }} {{eqn | r = 10       | c = chains }} {{eqn | r = 40       | c = rods, poles or perches }} {{eqn | r = 220       | c = yards }} {{eqn | r = 660       | c = feet }} {{eqn | r = 201 \cdotp 168       | c = metres }} {{end-eqn}}
Definition:Imperial/Length/International Mile
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''(international) mile''' }} {{eqn | r = 8       | c = furlongs }} {{eqn | r = 80       | c = chains }} {{eqn | r = 320       | c = rods, poles or perches }} {{eqn | r = 1760       | c = yards }} {{eqn | r = 1609 \cdotp 344       | c = metres (exactly) }} {{end-eqn}}
Definition:Imperial/Length/League
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''league''' }} {{eqn | o =        | r = 3       | c = (international) miles }} {{eqn | r = 5280       | c = yards }} {{eqn | o = \approx       | r = 2       | c = leuca }} {{eqn | o = \approx       | r = 3000       | c = paces }} {{eqn | r = 4828 \cdotp 032       | c = metres (exactly) }} {{end-eqn}}
Definition:Palindromic Number
A '''palindromic number (base $b$)''' is a number which is a palindrome when expressed in number base $b$. That is, it reads the same reversed as it does forwards. When the base is not specified, it is commonplace to understand it as being $10$.
Definition:Palindromic Prime
A '''palindromic prime''' is a prime number which is also a palindromic number.
Definition:Palindrome
A '''palindrome''' is a sequence of objects which stays the same when reversed.
Definition:Ulam Number
The sequence of '''Ulam numbers''' is defined as: :It begins $1, 2, 3$ :Each new term is the next to be uniquely the sum of $2$ previous terms in the sequence.
Definition:Ulam Number/Sequence
The sequence of '''Ulam numbers''' begins as follows: :$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, \ldots{}$
Definition:Lucas Number/Sequence
The '''Lucas sequence''' begins: :$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, \ldots$
Definition:Continuity/Functional
Let $S$ be a set of mappings. Let $y \in S$ be a mapping. Let $J \sqbrk y: S \to \R$ be a functional. Suppose:    :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - y_0} < \delta \implies \size {J \sqbrk y - J \sqbrk {y_0} } < \epsilon$ Then $J \sqbrk y$ is said to be a '''continuous functional''' and is continuous at the point $y_0 \in S$.
Definition:Functional/Real
Let $S$ be a set of mappings. Let $J: S \to \R$ be a mapping from $S$ to the real numbers $\R$: :$\forall y \in S: \exists x \in \R: J \sqbrk y = x$ Then $J: S \to \R$ is known as a '''(real) functional''', denoted by $J \sqbrk y$. That is, a '''(real) functional''' is a real-valued function whose arguments are themselves mappings.
Definition:Ordered Set of All Mappings
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a set. The '''ordered set of all mappings''' from $X$ to $L$ is defined by: :$L^X := \left({S^X, \precsim}\right)$ where :$\forall f, g \in S^X: f \precsim g \iff f \preceq g$ :$\preceq$ denotes the ordering on mappings, :$S^X$ denotes the set of all mappings from $X$ into $S$.
Definition:General Fibonacci Sequence
Let $r, s, t, u$ be numbers, usually integers but not necessarily so limited. Let $\sequence {a_n}$ be the sequence defined as: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$ Then $\sequence {a_n}$ is a '''general Fibonacci sequence'''.
Definition:Lucas Number/Definition 1
The '''Lucas numbers''' are a sequence which is formally defined recursively as: :$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Definition:Lucas Number/Definition 2
The '''Lucas numbers''' are a sequence defined as: :$L_n = F_{n - 1} + F_{n + 1}$ where $F_k$ is the $k$th Fibonacci number.
Definition:Increment/Functional
Let $S$ be a set of mappings. Let $y, h: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R$ be a functional defined on a normed vector space. Consider the following difference: :$ \Delta J \sqbrk {y; h} = J \sqbrk {y + h} - J \sqbrk y$ Then $\Delta J \sqbrk {y; h}$ is known as the '''increment of the functional $J$'''.
Definition:Differentiable Functional
Let $S$ be a normed linear space of mappings. Let $y, h \in S: \R \to \R$ be real functions. Let $J \sqbrk y$, $\phi \sqbrk {y; h}$ be functionals. Let $\Delta J \sqbrk {y; h}$ be an increment of the functional $J$ such that: :$\Delta J \sqbrk {y; h} = \phi \sqbrk {y;h} + \epsilon \norm h$ where $\epsilon = \epsilon \sqbrk {y; h}$ is a functional, and $\norm h$ is the norm of $S$. Suppose $\phi \sqbrk {y; h}$ is a linear  $h$ and: :$\ds \lim_{\norm h \mathop \to 0} \epsilon = 0$ Then the functional $J \sqbrk y $ is said to be '''differentiable'''.
Definition:Ordered Set of Increasing Mappings
Let $L = \left({S_1, \preceq_1}\right)$, $K = \left({S_2, \preceq_2}\right)$ be ordered sets. The '''ordered set of increasing mappings''' from $K$ into $L$ is ordered subset of $L^{S_2} = \left({S_1^{S_2}, \preceq'}\right)$ and is defined by :$\operatorname{Increasing}\left({K, L}\right) = \left({X, \precsim}\right)$ where :$X = \left\{ {f:S_2 \to S_1: f}\right.$ is increasing mapping$\left.\right\}$ :$\mathord\precsim = \mathord\preceq' \cap \left({X \times X}\right)$ :$L^{S_2}$ denotes the ordered set of all mappings from $S_2$ into $L$, :$S_1^{S_2}$ denotes the set of all mappings from $S_2$ into $S_1$. $\operatorname{Increasing}\left({K, L}\right)$ as ordered subset of ordered set is ordered set by Ordered Subset of Ordered Set is Ordered Set.
Definition:Prime Decomposition/Linguistic Note
The UK English spelling of '''prime factorization''' is '''prime factorisation'''.
Definition:Prime Decomposition/Multiplicity
For each $p_j \in \left\{ {p_1, p_2, \ldots, p_r}\right\}$, its power $k_j$ is known as the '''multiplicity of $p_j$'''.
Definition:Ordered Subset
Let $R = \struct{S, \preceq}$ be a relational structure. {{Questionable|A relational substructure of a relational structure is not necessarily ordered. So either: (a) $\struct{S, \preceq}$ needs to be ordered; (b) $\struct{S', \preceq'}$ needs to be ordered; or (c) the page needs to be renamed. The theorem Ordered Subset of Ordered Set is Ordered Set and many links to the page suggests (a) is needed.|Leigh.Samphier}} A '''ordered subset''' of $R$ is a relational structure $\struct{S', \preceq'}$ such that :$S'$ is a subset of $S$, :$\mathord\preceq' = \mathord\preceq \cap \paren{S' \times S'}$ That is :$\forall x, y \in S': x \preceq y \iff x \preceq' y$
Definition:Abundant Number/Definition 3
$n$ is '''abundant'''  it is smaller than its aliquot sum.
Definition:Deficient Number/Definition 3
$n$ is '''deficient'''  it is greater than its aliquot sum.
Definition:Ordered Set of Closure Operators
Let $L = \left({S, \preceq}\right)$ be an ordered set. The '''ordered set of closure operators''' of $L$ is ordered subset of $\operatorname{Increasing} \left({L, L}\right) = \left({X, \preceq'}\right)$ and is defined by :$\operatorname{Closure}\left({L}\right) := \left({Y, \precsim}\right)$ where :$Y = \left\{ {f:S \to S: f}\right.$ is closure operator$\left.\right\}$ :$\mathord\precsim = \mathord\preceq' \cap \left({Y \times Y}\right)$ :$\operatorname{Increasing} \left({L, L}\right)$  denotes the ordered set of increasing mappings from $L$ into $L$. $\operatorname{Closure}\left({L}\right)$ as an ordered subset of an ordered set is an ordered set by Ordered Subset of Ordered Set is Ordered Set.
Definition:Extremum/Functional
Let $S$ be a set of mappings. Let $J \sqbrk y: S \to \R$ be a functional. Let $y, \hat y: \R \to \R$ be real functions. Suppose for $y = \hat y \paren x$ there exists a neighbourhood of the curve $y = \hat y \paren x$ such that the difference $J \sqbrk y - J \sqbrk {\hat y}$ does not change its sign in this neighbourhood. Then $y = \hat y$ is called a ('''relative''') '''extremum''' of the functional $J$. {{Help|Check if "some neighbourhood" could be more precisely defined |user = Julius}}
Definition:Superabundant Number
Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''superabundant''' : :$\forall m \in \Z_{>0}, m < n: \dfrac {\map \sigma m} m < \dfrac {\map \sigma n} n$ where $\map \sigma n$ is the $\sigma$ function of $n$.
Definition:Duodecimal Notation
'''Duodecimal notation''' is the technique of expressing numbers in base $12$.
Definition:Imperial/Mass
There are three imperial systems of measurement of mass: * Avoirdupois * Apothecaries' weights * Troy === Grain === {{Definition:Imperial/Mass/Grain}} == Avoirdupois == {{:Definition:Avoirdupois}} == Avoirdupois: Short Variants == {{:Definition:Avoirdupois/Short}} == Apothecaries' Weights == {{:Definition:Apothecaries' Weights and Measures/Mass}} == Troy == {{:Definition:Troy}}
Definition:Avoirdupois/Ounce
:$1$ '''ounce avoirdupois''' $= 16$ drams $= 437 \cdotp 5$ grains $= 28 \cdotp 34952 \, 3125$ grams. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ounce avoirdupois''' }} {{eqn | r = 16       | c = drams }} {{eqn | r = 437 \cdotp 5       | c = grains }} {{eqn | r = 28 \cdotp 34952 \, 3125       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact.
Definition:Ounce/Linguistic Note
The word '''ounce''' derives from the Latin word '''uncia''', meaning '''$\dfrac 1 {12}$ part''' (of an '''as'''). Despite the gradual migration to the metric system, the word '''ounce''' still lives on as a rhetorical flourish for '''something small''', for example: :''If you only had an '''ounce''' of sense you would understand that ...''
Definition:Apothecaries' Weights and Measures/Mass/Scruple
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''scruple''' }} {{eqn | r = 20       | c = grains }} {{eqn | r = 1 \cdotp 3       | c = grams }} {{end-eqn}}
Definition:Ounce
=== Ounce Avoirdupois === {{:Definition:Avoirdupois/Ounce}} === Apothecaries' Ounce === {{:Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Troy Ounce === {{:Definition:Troy/Ounce}} === Uncia (Roman Ounce) === {{:Definition:Roman Weights and Measures/Mass/Uncia}}
Definition:Apothecaries' Weights and Measures
The '''apothecaries' (system of) weights and measures''' is a system of mass and volume units, derived from the Roman system. They were used by physicians and apothecaries for medical recipes, and also sometimes by scientists. Although in some cases bearing the same names as their imperial counterparts, the actual units themselves were subtly different in size.
Definition:Apothecaries' Weights and Measures/Mass
=== Scruple === {{Definition:Apothecaries' Weights and Measures/Mass/Scruple}} === Drachm === {{Definition:Apothecaries' Weights and Measures/Mass/Drachm}} === Ounce === {{Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Pound === {{Definition:Apothecaries' Weights and Measures/Mass/Pound}}
Definition:System (Order Theory)
Let $L = \left({S, \preceq}\right)$ be an ordered set. The '''system''' of $L$ is an ordered subset of $L$.
Definition:Infima Inheriting
Let $L = \struct {S, \preceq}$ be an ordered set. Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$. Then $R$ '''inherits infima''' of $L$  :for all subsets $X$ of $T$ if $X$ admits an infimum of $L$, then ::$X$ admits an infimum of $R$ and $\inf_R X = \inf_L X$
Definition:Closure System
Let $L$ be an ordered set. Let $S$ be a system of $L$. Then $S$ is '''closure'''  $S$ inherits infima.
Definition:Ordered Set of Closure Systems
Let $L$ be an ordered set. The '''ordered set of closure systems''' of $L$ is a relational structure :$\operatorname{ClSystems}\left({L}\right) = \left({X, \precsim}\right)$ where :$X$ is the set of all closure systems of $L$, :dor all closure systems $S_1 = \left({T_1, \preceq_1}\right), S_2 = \left({T_2, \preceq_2}\right)$ of $L$: $S_1 \precsim S_2 \iff T_1 \subseteq T_2$
Definition:Weak Extremum
Let $S$ be a set of mappings. Let $y, \hat y \in S: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R$ be a functional. Suppose there exists $\epsilon \in \R_{> 0}$ such that for $\norm {y - \hat y}_1 < \epsilon$ the expression $J \sqbrk y - J \sqbrk {\hat y}$ has the same sign for all $y$. Here $\norm{\, \cdot \,}_1$ denotes the norm of in the space $C^1$. Then $y = \hat y$ is a '''weak extremum''' of the functional $J \sqbrk y$.
Definition:Apothecaries' Weights and Measures/Mass/Ounce
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' ounce''' }} {{eqn | r = 8       | c = drachms }} {{eqn | r = 24       | c = scruples }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = troy ounce }} {{eqn | r = 31 \cdotp 1       | c = grams }} {{end-eqn}}
Definition:Troy/Ounce
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy ounce''' }} {{eqn | r = 20       | c = pennyweight }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = apothecaries' ounce }} {{eqn | o = \approx       | r = 31 \cdotp 1       | c = grams }} {{end-eqn}}
Definition:Troy
=== Pennyweight === {{Definition:Troy/Pennyweight}} === Ounce === {{Definition:Troy/Ounce}} === Pound === {{Definition:Troy/Pound}} === Talent === {{Definition:Troy/Talent}}
Definition:Roman Weights and Measures/Mass/Uncia
:$1$ '''uncia''' $= \dfrac 1 {12}$ libra $= 24$ scrupuli. Its value in modern units is estimated to be approximately $423$ grains or $27.4$ grams.
Definition:Roman Weights and Measures
The '''Roman system of weights''' is a system of mass units originating from the time of the Roman empire. This system continued to be used throughout Europe, although it evolved in different directions according to use.
Definition:Roman Weights and Measures/Mass
The Roman units of mass are as follows: === Calcus === {{Definition:Roman Weights and Measures/Mass/Calcus}} === Siliqua === {{Definition:Roman Weights and Measures/Mass/Siliqua}} === Scrupulus === {{Definition:Roman Weights and Measures/Mass/Scrupulus}} === Uncia === {{Definition:Roman Weights and Measures/Mass/Uncia}} === Libra === {{Definition:Roman Weights and Measures/Mass/Libra}}
Definition:Roman Weights and Measures/Mass/Scrupulus
:$1$ '''scrupulus''' $= \dfrac 1 {24}$ uncia $= 8$ calci. Its value in modern units is estimated to be approximately $17.6$ grains, or approximately $0.14$ grams.
Definition:Roman Weights and Measures/Mass/Calcus
:$1$ '''calcus''' $= \dfrac 1 8$ scrupulus. Its value in modern units is estimated to be approximately $2.2$ grains, or approximately $0.14$ grams.
Definition:Operator Generated by Ordered Subset
Let $L = \left({X, \precsim}\right)$ be an ordered set. Let $S = \left({T, \preceq}\right)$ be an ordered subset of $L$. The '''operator generated by ordered set''' $S$ is defined by :$\forall x \in X:\operatorname{operator}\left({S}\right)\left({x}\right) := \inf_L\left({x^\succeq \cap T}\right)$ where $x^\succeq$ denotes the upper closure of $x$.
Definition:Roman Weights and Measures/Mass/Siliqua
:$1$ '''siliqua''' $= \dfrac 1 6$ scrupulus $= \dfrac 1 {144}$ uncia. Its value in modern units is estimated to be approximately $2.9$ grains, or approximately $0.19$ grams. Its modern equivalent is the carat.
Definition:Roman Weights and Measures/Mass/Libra
:$1$ '''libra''' $= 12$ unciae. Its value in modern units is estimated to be approximately $5 \, 076$ grains or $329$ grams.
Definition:Avoirdupois/Dram
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''dram''' }} {{eqn | r = 27 \cdotp 34375       | c = grains }} {{eqn | r = \dfrac 1 {16}       | c = ounce avoirdupois }} {{eqn | o = \approx       | r = 1 \cdotp 77       | c = grams }} {{end-eqn}}
Definition:Imperial/Mass/Grain
The '''grain''' is the imperial unit of mass which is used as the basis of all three of the imperial weight systems. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''grain''' }} {{eqn | r = \dfrac 1 {437 \cdot 5}       | c = ounce avoirdupois }} {{eqn | o = \approx       | r = 64 \cdotp 8       | c = milligrams }} {{end-eqn}}
Definition:Avoirdupois
=== Dram === {{Definition:Avoirdupois/Dram}} === Ounce === {{Definition:Avoirdupois/Ounce}} === Pound === {{Definition:Avoirdupois/Pound}} === Stone === {{Definition:Avoirdupois/Stone}} === Quarter === {{Definition:Avoirdupois/Quarter}} === Hundredweight === {{Definition:Avoirdupois/Hundredweight}} === Ton === {{Definition:Avoirdupois/Ton}}
Definition:Avoirdupois/Pound
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pound avoirdupois''' }} {{eqn | r = 16       | c = ounces avoirdupois }} {{eqn | r = 7 \, 000       | c = grains }} {{eqn | r = 453 \cdotp 59237       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact.
Definition:Avoirdupois/Stone
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''stone''' }} {{eqn | r = 14       | c = pounds avoirdupois }} {{eqn | r = 224       | c = ounces avoirdupois }} {{eqn | o = \approx       | r = 6 \, 350 \cdotp 29318       | c = grams }} {{end-eqn}} The gram equivalent is exact.
Definition:Avoirdupois/Ton
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ton''' }} {{eqn | r = 20       | c = hundredweight }} {{eqn | r = 80       | c = quarters }} {{eqn | r = 160       | c = stone }} {{eqn | r = 2240       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 1016       | c = kilograms }} {{end-eqn}}
Definition:Avoirdupois/Hundredweight
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hundredweight''' }} {{eqn | r = 4       | c = quarters }} {{eqn | r = 8       | c = stone }} {{eqn | r = 112       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 50 \cdotp 8       | c = kilograms }} {{end-eqn}}
Definition:Avoirdupois/Quarter
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quarter''' }} {{eqn | r = 2       | c = stone }} {{eqn | r = 28       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 12 \cdotp 7       | c = kilograms }} {{end-eqn}}
Definition:Avoirdupois/Short
=== Short Hundredweight === {{Definition:Avoirdupois/Short/Hundredweight}} === Short Ton === {{Definition:Avoirdupois/Short/Ton}}
Definition:Avoirdupois/Short/Hundredweight
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short hundredweight''' }} {{eqn | r = 100       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 45 \cdotp 359       | c = kilograms }} {{end-eqn}}
Definition:Avoirdupois/Short/Ton
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short ton''' }} {{eqn | r = 20       | c = short hundredweight }} {{eqn | r = 2000       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 907 \cdot 2       | c = kilograms }} {{end-eqn}}
Definition:Apothecaries' Weights and Measures/Mass/Drachm
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''drachm''' }} {{eqn | r = 3       | c = scruples }} {{eqn | r = 60       | c = grains }} {{eqn | r = 3 \cdotp 89       | c = grams }} {{end-eqn}}
Definition:Apothecaries' Weights and Measures/Mass/Pound
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' pound''' }} {{eqn | r = 12       | c = apothecaries' ounces }} {{eqn | r = 5 \, 760       | c = grains }} {{eqn | r = 373 \cdotp 24       | c = grams }} {{eqn | r = 1       | c = troy pound }} {{end-eqn}}
Definition:Troy/Pennyweight
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pennyweight''' }} {{eqn | r = 24       | c = grains }} {{eqn | o = \approx       | r = 1 \cdotp 56       | c = grams }} {{end-eqn}}
Definition:Troy/Pound
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy pound''' }} {{eqn | r = 12       | c = troy ounces }} {{eqn | r = 5 \, 760       | c = grains }} {{eqn | r = 1       | c = apothecaries' pound }} {{eqn | o = \approx       | r = 373 \cdotp 24       | c = grams }} {{end-eqn}}
Definition:Pound/Symbol
The symbol for the pound is $\text{lb}$. This derives from the libra (Roman pound) from which the pound evolved. It always needs to be distinguished from the pound sterling, whose symbol is $\pounds$.
Definition:Ounce/Symbol
The symbol for the ounce is $\text{oz}$. This derives indirectly from the uncia (Roman ounce) via the $15$th century Italian '''onza'''.
Definition:Avoirdupois/Quarter/Symbol
The symbol for the quarter is $\text{qtr}$.
Definition:Troy/Pennyweight/Symbol
The symbol for the pennyweight is $\text{dwt}$. This derives from the Latin '''denarius''', which evolved into the (old) penny.
Definition:Kilogram/Symbol
The symbol for the '''kilogram''' is $\mathrm {kg}$.
Definition:Milligram
:$1$ '''milligram''' is defined as $= \dfrac 1 {1 \, 000}$ of a gram.
Definition:Milligram/Symbol
The symbol for the '''milligram''' is $\mathrm {mg}$.
Definition:Sexagesimal Notation
'''Sexagesimal notation''' is the technique of expressing numbers in base $60$.
Definition:Long Hundred
A '''long hundred''' is another term for the number $120$.
Definition:Short Hundred
A '''short hundred''' is another term for the number $100$.
Definition:Rhombic Dodecahedron
The '''rhombic dodecahedron''' is a Catalan polyhedron with $12$ congruent faces each in the shape of a rhombus.
Definition:Reversal
Let $m = \sqbrk {a_n a_{n - 1} a_{n - 2} \ldots a_2 a_1 a_0}$ be an integer expressed in base $10$. That is: :$m = \displaystyle \sum_{k \mathop = 0}^n a_k 10^k$ Its '''reversal''' $m'$ is the integer created by writing the digits of $m$ in the opposite order: :$m' = \sqbrk {a_0 a_1 a_2 \ldots a_{n - 2} a_{n - 1} a_n}$ That is: :$m' = \displaystyle \sum_{k \mathop = 0}^n a_{n - k} 10^k$
Definition:Emirp
An '''emirp''' is a prime number whose reversal is a different prime number.