hcju/proofwikips huggingface dataset:
Combination Theorem for Continuous Mappings/Topological Semigroup
Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{G, *, \tau_{_G}}$ be a topological semigroup. Let $\lambda \in G$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule}}
Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule
:$f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule
:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Volume of Solid of Revolution/Parametric Form
Let $x: \R \to \R$ and $y: \R \to \R$ be real functions defined on the interval $\closedint a b$. Let $y$ be integrable on the (closed) interval $\closedint a b$. Let $x$ be differentiable on the (open) interval $\openint a b$. Let the points be defined: :$A = \tuple {\map x a, \map y a}$ :$B = \tuple {\map x b, \map y b}$ :$C = \tuple {\map x b, 0}$ :$D = \tuple {\map x a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by: :$\set {\tuple {\map x t, \map y t}: a \le t \le b}$ Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\displaystyle V = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t$
Combination Theorem for Continuous Mappings/Topological Ring
Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \tau_{_R}}$ be a topological ring. Let $\lambda \in R$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule}}
Combination Theorem for Continuous Mappings/Topological Division Ring
Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \tau_{_R}}$ be a topological division ring. Let $\lambda \in R$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings. Let $U = S \setminus \set{x : \map g x = 0}$ Let $g^{-1} : U \to R$ denote the mapping defined by: :$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$ Let $\tau_{_U}$ be the subspace topology on $U$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule}}
Weight of Body at Earth's Surface
Let $B$ be a body of mass $m$ situated at (or near) the surface of Earth. Then the weight of $B$ is given by: :$W = m g$ where $g$ is the value of the acceleration due to gravity at the surface of Earth.
Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule
:$-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule
:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Ring/Product Rule
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule
:$\lambda * f + \mu * g: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping :$f * \lambda + g * \mu: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.
Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Division Ring/Multiple Rule
:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule
:$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule
:$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Pointwise Operation is Composite of Operation with Mapping to Cartesian Product
Let $S$ be a set. Let $\struct {T, *}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let the algebraic structure $\struct {T^S, \oplus}$ be the algebraic structure on $T^S$ induced by $*$. Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings. Let $f \times g : S \to T \times T$ be the mapping from $S$ to the cartesian product $T \times T$ Defined by: :$\forall x \in S : \map {\paren {f \times g}} x = \tuple {\map f x, \map g x}$ Then: :$f \oplus g = * \circ \paren {f \times g}$ That is, $f \oplus g$ is the composition of the binary operation $*$ with the mapping $f \times g : S \to T \times T$.
Integral Representation of Bessel Function of the First Kind
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.
Integral Representation of Bessel Function of the First Kind/Integer Order
Let $n \in \Z$ be an integer. Then: :$\displaystyle \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$
Integral Representation of Bessel Function of the First Kind/Non-Integer Order
Let $n \in \Z$ be an integer. Then: :$\displaystyle \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$
Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule
:$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule
:$- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.
Legendre Transform of Strictly Convex Real Function is Strictly Convex
Let $\map f x$ be a strictly convex real function. Then the function $\map {f^*} p$ acquired through the Legendre Transform is also strictly convex.
Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1
:$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$
Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2
:$x * I_{_R}$ is a bijection and $x^{-1} * I_{_R}$ is the inverse of $x * I_{_R}$ :$I_{_R} * x$ is a bijection and $I_{_R} * x^{-1}$ is the inverse of $I_{_R} * x$
Open Balls of P-adic Number
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {B_{p^{-n} } } a = a + p^{n + 1} \Z_p$
Local Basis of P-adic Number
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Then the set of open balls $\set {\map {B_{p^{-n} } } a : n \in Z}$ is a local basis of $a$ consisting of clopen sets.
Closed Ball of P-adic Number
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ denote the $p$-adic integers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {B^-_{p^{-n} } } a = a + p^n \Z_p$ where $a + p^n \Z_p$ denotes the left coset of the principal ideal $p^n \Z_p$ containing $a$ in the subring $\Z_p$. That is, the closed ball $\map { {B_\epsilon}^-} a$ is the set: :$a + p^n \Z_p = \set{a + p^n z : z \in \Z_p}$
Closed Ball is Disjoint Union of Open Balls in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$: :let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  :let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.  Then: :$(1): \quad \forall n \in Z : \map {B^-_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 0}^{p - 1} \map {B_{p^{-n} } } {a + i p^n}$ :$(2): \quad \forall n \in Z : \set {\map {B_{p^{-n} } } {a + i p^n} : i = 0, \dotsc, p - 1}$ is a set of pairwise disjoint open balls.
Sphere is Disjoint Union of Open Balls in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$: :let $\map {S_\epsilon} a$ denote the sphere of $a$ of radius $\epsilon$.  :let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {S_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 1}^{p - 1} \map {B_{p^{-n} } } {a + i p^n}$
Null Sequence induces Local Basis in Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\sequence{x_n}$ be a real null sequence such that: :$\forall n \in N: x_n > 0$ Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ in $M$. Then: :$\mathcal B_{\sequence{x_n}} = \set{\map {B_{x_n}} a : n \in \N}$ is a local basis at $a$.
Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals
:$\mathcal B = \set {\map {B_{1/n}} a : n \in \N}$ is a local basis at $a$.
Consecutive Integers which are Powers of 2 or 3
The only pairs of consecutive positive integers which are powers of $2$ or $3$ are: :$\tuple {1, 2}$, $\tuple {2, 3}$, $\tuple {3, 4}$, $\tuple {8, 9}$
Altitudes of Triangle Bisect Angles of Orthic Triangle
Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be its orthic triangle. The altitudes of $\triangle ABC$ are the angle bisectors of $\triangle DEF$.
Open Ball in Normed Division Ring is Open Ball in Induced Metric
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the open ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {B_\epsilon} {a; d }$ denote the open ball in the metric space $\struct {R, d}$. Then: :$\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {B_\epsilon} {a; d }$
Closed Ball in Normed Division Ring is Closed Ball in Induced Metric
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the closed ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {{B_\epsilon}^-} {a; d }$ denote the closed ball in the metric space $\struct {R, d}$. Then: :$\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ = $\map {{B_\epsilon}^-} {a; d }$
Sphere in Normed Division Ring is Sphere in Induced Metric
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the sphere in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {S_\epsilon} {a; d }$ denote the sphere in the metric space $\struct {R, d}$. Then: :$\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {S_\epsilon} {a; d }$
Sphere is Set Difference of Closed Ball with Open Ball
Let $M = \struct{A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; d}$ denote the $\epsilon$-closed ball of $a$ in $M$. Let $\map {B_\epsilon} {a; d}$ denote the $\epsilon$-open ball of $a$ in $M$. Let $\map {S_\epsilon} {a; d}$ denote the $\epsilon$-sphere of $a$ in $M$. Then: :$\map {S_\epsilon} {a; d} = \map {{B_\epsilon}^-} {a; d} \setminus \map {B_\epsilon} {a; d}$
Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-closed ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-open ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-sphere of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Then: :$\map {S_\epsilon} {a; \norm {\,\cdot\,} } = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} } \setminus \map {B_\epsilon} {a; \norm {\,\cdot\,} }$
Conservation of Momentum
Let $P$ be a physical system. Let it have the action $S$: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ does not depend on one of the coordinates explicitly: :$\dfrac {\partial L} {\partial x_j} = 0$ Then the total momentum of $P$ along the axis $x_j$ is conserved.
Conservation of Angular Momentum (Lagrangian Mechanics)
Let $P$ be a physical system composed of finite number of particles. Let it have the action $S$: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ is invariant  rotation around $z$-axis. Then the total angular momentum of $P$ along $z$-axis is conserved.
Leibniz's Law for Sets
Let $S$ be an arbitrary set. Then: :$x = y \dashv \vdash x \in S \iff y \in S$ for all $S$ in the universe of discourse. This is therefore the justification behind the notion of the definition of set equality.
Area of Annulus
Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\map \Area A = \pi \paren {R^2 - r^2}$
Area of Annulus as Area of Rectangle
Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\map \Area A = 2 \pi \paren {r + \dfrac w 2} \times w$ where $w$ denotes the width of $A$. That is, it is the area of the rectangle contained by: :the width of $A$ :the circle midway in radius between the inner radius and outer radius of $A$.
Characterization of Closed Ball in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {{B_\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\epsilon$.  Let $x, y \in \Q_p$. Let $n \in Z$.  ::$(1)\quad x \in \map {B^{\,-}_{p^{-n}}} y$ ::$(2)\quad \norm{x -y}_p \le p^{-n}$ ::$(3)\quad \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$ ::$(4)\quad x - y \in p^n \Z_p$ ::$(5)\quad x + p^n \Z_p = y + p^n \Z_p$
Characterization of Open Ball in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the closed ball of center $a$ of radius $\epsilon$.  Let $n \in Z$. Let $x, y \in \Q_p$. : :$(1): \quad x \in \map {B_{p^{-n} } } y$ :$(2): \quad \norm{x - y}_p < p^{-n}$ :$(3): \quad \map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$ :$(4): \quad x - y \in p^{n + 1} \Z_p$ :$(5): \quad x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$
Complete Archimedean Valued Field is Real or Complex Numbers
Let $\struct{k, \norm{\,\cdot\,}}$ be a complete valued field with Archimedean norm $\norm{\,\cdot\,}$. Then either: :$k$ is isomorphic to the real numbers $\R$ and $\norm{\,\cdot\,}$ is equivalent to the absolute value $\size{\,\cdot\,}$ on $\R$ or :$k$ is isomorphic to the complex numbers $\C$ and $\norm{\,\cdot\,}$ is equivalent to the complex modulus $\size{\,\cdot\,}$ on $\C$
Equivalence of Definitions of Convergent P-adic Sequence
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$.  === Definition 1 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 3}} === Definition 4 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 4}}
Equivalence of Definitions of Convergence in Normed Division Rings
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. {{TFAE|def = Convergent Sequence in Normed Division Ring}} === Definition 1 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 3}}
Finite Complement Topology is not Metrizable
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a metrizable space.
Poisson Brackets of Classical Particle in Radial Potential on Plane
Let $P$ be a classical particle embedded in a 2-dimensional Euclidean manifold. Let the real-valued functions $\map r t$, $\map \theta t$ denote the position of $P$ in polar coordinates, where $t$ is time. Suppose, the potential energy of $P$ depends only on $r$. Then $P$ has the following Poisson brackets: {{begin-eqn}} {{eqn| l = \sqbrk {r, p_r}       | r = 1 }} {{eqn| l = \sqbrk {\theta, p_\theta}      | r = 1 }} {{eqn| l = \sqbrk {r, H}      | r = \dfrac {p_r} m }} {{eqn| l = \sqbrk {\theta, H}       | r = \dfrac {p_\theta} {m r^2} }} {{eqn| l = \sqbrk {p_r, H}       | r = -\dfrac {\partial U} {\partial r} }} {{eqn| l = \sqbrk {p_\theta, H}       | r = 0 }} {{end-eqn}}
Multiplication by Power of 10 by Moving Decimal Point
Let $n \in \R$ be a real number. Let $n$ be expressed in decimal notation. Let $10^d$ denote a power of $10$ for some integer $d$ Then $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $d$ places to the right. Thus, if $d$ is negative, and so $10^d = 10^{-e}$ for some $e \in \Z_{>0}$, $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $e$ places to the left.
Number of Significant Figures in Result of Multiplication
Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $m \times n$ can have is $\min \set {d_m, d_n}$.
Number of Significant Figures in Result of Division
Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $\dfrac m n$ can have is $\min \set {d_m, d_n}$.
Number of Significant Figures in Result of Square Root
Let $m$ be a numbers which is presented to $d$ significant figures. Then the most significant figures that $\sqrt m$ can have is also $d$.
Number of Significant Figures in Result of Addition or Subtraction
Let $m$ and $n$ be numbers. Let $d_m$ and $d_n$ be the position of the least significant digit of $m$ and $n$ respectively. Then the least significant digit in either $m + n$ or $m - n$ is in the position corresponding to the greater significant digit of $d_m$ and $d_n$.
Range of Common Logarithm of Number between 1 and 10
Let $x \in \R$ be a real number such that: :$1 \le x < 10$ Then: :$0 \le \log_{10} x \le 1$ where $\log_{10}$ denotes the common logarithm function.
Common Logarithm of Number in Scientific Notation
Let $n$ be a positive real number which is presented (possibly approximated) in scientific notation as: :$n = a \times 10^d$ where: :$1 \le a < 10$ :$d \in \Z$ is an integer. Then: :$\log_{10} n = \log_{10} a + d$ where: :$0 \le \log_{10} a < 1$
Characteristic of Common Logarithm of Number Greater than 1
Let $x \in \R_{>1}$ be a (strictly) positive real number greater than $1$. The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of digits to the left of the decimal point of $x$.
Characteristic of Common Logarithm of Number Less than 1
Let $x \in \R_{>0}$ be a (strictly) positive real number such that $x < 1$. The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of zero digits to the immediate right of the decimal point of $x$.
P-adic Norm satisfies Non-Archimedean Norm Axioms
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \times \Q_p \to \R_{\ge 0}$. Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms: {{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in \Q_p         | ml= \norm x_p = 0         | mo= \iff         | mr= x = 0 }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in \Q_p         | ml= \norm {x \cdot y}_p         | mo= =         | mr= \norm x_p \times \norm y_p }} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in Q_p         | ml= \norm {x + y}_p         | mo= \le         | mr= \max \set {\norm x_p, \norm y_p} }} {{end-axiom}}
Circles with Same Poles are Parallel
Let $S$ be a sphere. Let $C$ and $D$ be circles on $S$ (either great circles or small circles). Let $C$ and $D$ both have the same pair of poles. Then $C$ and $D$ are parallel.
Three Points on Sphere in Same Hemisphere
Let $S$ be a sphere. Let $A$, $B$ and $C$ be points on $S$ which do not all lie on the same great circle. Then it is possible to divide $S$ into two hemispheres such that $A$, $B$ and $C$ all lie on the same hemisphere.
Side of Spherical Triangle is Less than 2 Right Angles
Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is less than $2$ right angles.
Center is Element of Closed Ball
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $M$. Then: :$a \in \map {{B_\epsilon}^-} a$