hcju/proofwikips huggingface dataset:
Irrational Number Space is Complete Metric Space
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is a complete metric space.
Rational Number Space is Meager
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is meager.
Rational Number Space is Separable
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is separable.
Underlying Set of Topological Space is Everywhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Then the underlying set $S$ of $T$ is everywhere dense in $T$.
Irrational Number Space is Separable
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is separable.
Irrational Number Space is Second-Countable
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is second-countable.
Compact Set of Rational Numbers is Nowhere Dense
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $S \subseteq \Q$ be a compact set of $\Q$. Then $S$ is nowhere dense in $\Q$.
Compact Set of Irrational Numbers is Nowhere Dense
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Let $S \subseteq \R \setminus \Q$ be a compact set of $\R \setminus \Q$. Then $S$ is nowhere dense in $\R \setminus \Q$.
Rational Number Space is not Locally Compact Hausdorff Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is not a locally compact Hausdorff Space.
Irrational Number Space is not Locally Compact Hausdorff Space
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is not a locally compact Hausdorff Space.
Rationals plus Irrational are Everywhere Dense in Irrationals
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Let $x \in \R \setminus \Q$ be an arbitrary irrational number. Let $S_x$ be the set defined as: :$S_x := \set {x + q: q \in \Q}$ Then $S_x$ is everywhere dense in $\struct {\R \setminus \Q, \tau_d}$.
Rational Number Space is Totally Separated
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is totally separated.
Irrational Number Space is Totally Separated
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is totally separated.
Rational Number Space is Dense-in-itself
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is dense-in-itself.
Irrational Number Space is Dense-in-itself
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is dense-in-itself.
Irrational Number Space is not Scattered
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is not scattered.
Rational Number Space is Zero Dimensional
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is zero dimensional.
Irrational Number Space is Zero Dimensional
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is zero dimensional.
Open Unit Interval on Rational Number Space is Bounded but not Compact
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then: :$\openint 0 1 \cap \Q$ is totally bounded but not compact where $\openint 0 1$ is the open unit interval.
Closure of Integer Reciprocal Space
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then: :$A^- = A \cup \set 0$ where $A^-$ denotes the closure of $A$ in $\R$.
Zero is Limit Point of Integer Reciprocal Space
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $0$ is the only limit point of $A$ in $\R$.
Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \cup \closedint 2 3$ where $\closedint 2 3$ is a closed interval of $\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is an $\omega$-accumulation point of $B$ in $\R$.
Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \cup \closedint 2 3$ where $\closedint 2 3$ is a closed interval of $\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is not a condensation point of $B$ in $\R$.
Integer Reciprocal Space contains Cauchy Sequence with no Limit Point
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $A$ has a Cauchy sequence which has no limit point in $A$.
Integer Reciprocal Space with Zero is not Locally Connected
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not locally connected.
Discrete Space is Locally Connected
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is locally connected.
Integer Reciprocal Space with Zero is Totally Separated
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is totally separated.
Quasicomponents of Integer Reciprocal Space with Zero are Single Points
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the quasicomponents of $A$ are singletons.
Interior of Union of Adjacent Open Intervals
Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A = A^\circ$ where $A^\circ$ is the interior of $A$.
Interior of Closure of Interior of Union of Adjacent Open Intervals
Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{\circ - \circ} = A^{- \circ} = \openint a c$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.
Closure of Interior of Closure of Union of Adjacent Open Intervals
Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{- \circ -} = A^{\circ -} = A^- = \closedint a c$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.
Exterior of Exterior of Union of Adjacent Open Intervals
Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{ee} = \openint a c$ where $A^e$ is the exterior of $A$.
Open Real Interval is Regular Open
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\openint a b$ be an open interval of $\R$. Then $\openint a b$ is regular open in $\struct {\R, \tau_d}$.
Closure of Open Real Interval is Closed Real Interval
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\openint a b$ be an open interval of $\R$. Then the closure of $\openint a b$ is the closed interval $\closedint a b$.
Limit Points of Open Real Interval
Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $\openint a b$ be an open interval of $\R$. Then the limit points of $\openint a b$ are: :all the points in $\openint a b$ as well as: :the points $a$ and $b$.
Interior of Closed Real Interval is Open Real Interval
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\closedint a b$ be a closed interval of $\R$. Then the interior of $\closedint a b$ is the open interval $\left({a, b}\right)$.
Union of Regular Open Sets is not necessarily Regular Open
Let $T = \struct {S, \tau}$ be a topological space. Let $U$ and $V$ be regular open sets of $T$. Then $U \cup V$ is not also necessarily a regular open set of $T$.
Closure of Intersection may not equal Intersection of Closures
Let $T = \struct {S, \tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$
Closure of Empty Set is Empty Set
Let $T = \left({S, \tau}\right)$ be a topological space. Then the closure of the empty set $\varnothing$ in $T$ is $\varnothing$.
Closed Real Interval is Regular Closed
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\closedint a b$ be a closed interval of $\R$. Then $\closedint a b$ is regular closed in $\struct {\R, \tau_d}$.
Closed Real Interval is Closed in Real Number Line
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\closedint a b$ be a closed interval of $\R$. Then $\closedint a b$ is closed (in the topological sense) in $\struct {\R, \tau_d}$.
Intersection of Regular Closed Sets is not necessarily Regular Closed
Let $T = \struct {S, \tau}$ be a topological space. Let $U$ and $V$ be regular closed sets of $T$. Then $U \cap V$ is not also necessarily a regular closed set of $T$.
Interior of Union is not necessarily Union of Interiors
Let $T = \struct {S, \tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^\circ$ and ${H_2}^\circ$ denote the interiors of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\left({H_1 \cup H_2}\right)^\circ = {H_1}^\circ \cup {H_2}^\circ$
Expectation of Exponential Distribution
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Then the expectation of $X$ is given by: :$\expect X = \beta$
Von Mangoldt Equivalence
For $n \in \N_{>0}$, let $\Lambda \left({n}\right)$ be the von Mangoldt function. Then: :$\displaystyle \lim_{N \to \infty} \frac 1 N \sum_{n \mathop = 1}^N \Lambda \left({n}\right) = 1$ is logically equivalent to the Prime Number Theorem.
Zeta Equivalence to Prime Number Theorem
Let $\map \zeta z$ be the Riemann $\zeta$ function. The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity. {{explain|What does $z$ range over, and what does it mean by "first $N$ coefficients" of $\dfrac {\zeta'} {\zeta}$?}}
Substitution Rule for Matrices
Let $\mathbf A$ be a square matrix of order $n$. Then: :$(1): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{j k} = a_{i k}$ :$(2): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{k j} = a_{k i}$ where: :$\delta_{i j}$ is the Kronecker delta :$a_{j k}$ is element $\tuple {j, k}$ of $\mathbf A$.
Matrix is Invertible iff Determinant has Multiplicative Inverse/Necessary Condition
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be an invertible square matrix of order $n$. Let $\mathbf B = \mathbf A^{-1}$ be the inverse of $\mathbf A$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\map \det {\mathbf B} = \dfrac 1 {\map \det {\mathbf A} }$
Product of Orthogonal Matrix with Transpose is Identity
Let $\mathbf Q$ be an orthogonal matrix. Then: :$\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$ where: :$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$ :$\mathbf I$ is a unit (identity) matrix
Determinant of Orthogonal Matrix is Plus or Minus One
Let $\mathbf Q$ be an orthogonal matrix. Then: :$\det \mathbf Q = \pm 1$ where $\det \mathbf Q$ is the determinant of $\mathbf Q$.
Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix
Let $\mathbf P$ and $\mathbf Q$ be proper orthogonal matrices. Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$. Then $\mathbf P \mathbf Q$ is a proper orthogonal matrix.
Product of Orthogonal Matrices is Orthogonal Matrix
Let $\mathbf P$ and $\mathbf Q$ be orthogonal matrices. Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$. Then $\mathbf P \mathbf Q$ is an orthogonal matrix.
Internal Group Direct Product is Injective/General Result
Let $G$ be a group whose identity is $e$. Let $\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\displaystyle \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a mapping defined by: :$\displaystyle \map {\phi_n} {h_1, h_2, \ldots, h_n} = \prod_{j \mathop = 1}^n h_j$ Then $\phi_n$ is injective : :$\forall i, j \in \set {1, 2, \ldots, n}: i \ne j \implies H_i \cap H_j = \set e$ That is,  $\sequence {H_n}$ is a sequence of independent subgroups.
Trace of Matrix Product
Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$. Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$. Then: :$\ds \map \tr {\mathbf A \mathbf B} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n a_{i j} b_{j i}$ where $\map \tr {\mathbf A \mathbf B}$ denotes the trace of $\mathbf A \mathbf B$. Using the Einstein summation convention, this can be expressed as: :$\map \tr {\mathbf A \mathbf B} = a_{i j} b_{j i}$
Trace of Matrix Product/General Result
Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$ be square matrices of order $n$. Let $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$. Then: :$(1): \quad \ds \map \tr {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m} = \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \cdots \map {a_{m - 1} } {i_{m - 1}, i_m} \map {a_m} {i_m, i_1}$ where: :$\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\mathbf A_1$ whose indices are $i_1$ and $i_2$ :$\map \tr {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m}$ denotes the trace of $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$. In $(1)$, the Einstein summation convention is used, with the implicit understanding that a summation is performed over each of the indices $i_1$ to $i_m$.
Product of Finite Sequence of Matrices
Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$ be matrices. Let the order of $\mathbf A_j$ be $d_j \times d_{j + 1}$. Let $\displaystyle \mathbf C := \prod_{j \mathop = 1}^n \mathbf A_j = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_n$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$. Then: :$(1): \quad \displaystyle \map c {i_1, i_{n + 1} } = \sum_{i_n \mathop = 1}^{d_n} \dotsm \sum_{i_3 \mathop = 1}^{d_3} \sum_{i_2 \mathop = 1}^{d_2} \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \dotsm \map {a_{n - 1} } {i_{n - 1}, i_n} \map {a_n} {i_n, i_{n + 1} }$ where: :$\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\mathbf A_1$ whose indices are $i_1$ and $i_2$ :the order of $\mathbf C$ is $d_1 \times d_{n+1}$.
Characteristic of Galois Field is Prime
Let $\GF$ be a Galois field.  Then the characteristic of $\GF$ is a prime number.
Finite Ring with No Proper Zero Divisors is Field
Let $\struct {R, +, \circ}$ be a finite non-null ring with no proper zero divisors. Then $R$ is a field.
Equivalence of Definitions of Dot Product
{{TFAE|def = Dot Product}} Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.
Determinant of Kronecker Delta Elements
Let $\lambda$ and $\pi$ be permutations on $\left\{{1, 2, 3}\right\}$. Let: : $\left({i, j, k}\right) = \left({\lambda \left({1}\right), \lambda \left({2}\right), \lambda \left({3}\right)}\right)$ : $\left({r, s, t}\right) = \left({\pi \left({1}\right), \pi \left({2}\right), \pi \left({3}\right)}\right)$ Then: :$\begin{vmatrix}   \delta_{ir} & \delta_{is} & \delta_{it} \\   \delta_{jr} & \delta_{js} & \delta_{jt} \\   \delta_{kr} & \delta_{ks} & \delta_{kt} \end{vmatrix} = \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right)$ where: : $\delta_{ir}$ denotes the Kronecker delta : $\begin{vmatrix} \cdot \end{vmatrix}$ denotes a determinant : $\operatorname{sgn} \left({i, j, k}\right)$ is the sign of the permutation $\left({i, j, k}\right)$ of the set $\left\{{1, 2, 3}\right\}$.
Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary
:$\set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \map {\mathrm {ref} } {\mathbf A} \mathbf x = \mathbf 0}$ where $\map {\mathrm {ref} } {\mathbf A}$ is the reduced echelon form of $\mathbf A$.
Continuous Mapping is Sequentially Continuous/Corollary
Let $f$ be continuous (everywhere) on $X$. Then $f$ is sequentially continuous on $X$.
Product of Complex Numbers in Polar Form
Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form. Then: :$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$
Subtraction of Complex Numbers
Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. The subtraction operation on $z_1$ and $z_2$ is: :$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$
Division of Complex Numbers
Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers such that $z_2 \ne 0$. The operation of division is performed on $z_1$ by $z_2$ as follows: :$\dfrac {z_1} {z_2} = \dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \dfrac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}$
De Moivre's Formula/Proof 1
Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \, \map \sin {\omega x}$
Division of Complex Numbers in Polar Form
Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form, such that $z_2 \ne 0$. Then: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \map \sin {\theta_1 - \theta_2} }$ or: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \map \cis {\theta_1 - \theta_2}$
Roots of Complex Number
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \set {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }: k \in \set {0, 1, 2, \ldots, n - 1} }$ There are $n$ distinct such $n$th roots.
Zeroth Power of Real Number equals One
Let $a \in \R_{>0}$ be a (strictly) positive real number. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^0 = 1$
Exponential of One
: $\exp 1 = e$ where $e$ is Euler's number: $e = 2.718281828\ldots$
Exponential of Zero
: $\exp 0 = 1$
Difference of Logarithms
:$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$
Change of Base of Logarithm/Base 10 to Base e
==== Form 1 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 2}}
Change of Base of Logarithm/Base e to Base 10
==== Form 1 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 2}}
Power of Identity is Identity
Let $\struct {M, \circ}$ be a monoid whose identity element is $e$. Then: :$\forall n \in \Z: e^n = e$
Tangent Exponential Formulation/Formulation 2
:$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$
Cotangent Exponential Formulation
:$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$
Cosecant Exponential Formulation
Let $z$ be a complex number. Let $\csc z$ denote the cosecant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$
Secant Exponential Formulation
Let $z$ be a complex number. Let $\sec z$ denote the secant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$
Periodicity of Complex Exponential Function
For all $k \in \Z$: :$e^{i \paren {\theta + 2 k \pi} } = e^{i \theta}$
Product of Complex Numbers in Exponential Form
Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form. Then: :$z_1 z_2 = r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }$
Division of Complex Numbers in Exponential Form
Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form. Then: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$
De Moivre's Formula/Exponential Form
:$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$
Z/(m)-Module Associated with Ring of Characteristic m
Let $\left({R,+,*}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $m$. Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo $m$. Let $\circ$ be the mapping from $\Z_m \times R$ to $R$ defined as: :$\forall \left[\!\left[a\right]\!\right]_m \in \Z_m: \forall x \in R: \left[\!\left[a\right]\!\right]_m \circ x = a \cdot x$ where $\left[\!\left[a\right]\!\right]_m$ is the residue class of $a$ modulo $m$ and $a \cdot x$ is the $a$th power of $x$. Then $\left({R, +, \circ}\right)_{\Z_m}$ is a unitary $\Z_m$-module.
Sum of Squares of Hyperbolic Secant and Tangent
:$\sech^2 x + \tanh^2 x = 1$ where $\sech$ and $\tanh$ are hyperbolic secant and hyperbolic tangent.
Difference of Squares of Hyperbolic Cotangent and Cosecant
:$\coth^2 x - \csch^2 x = 1$ where $\coth$ and $\csch$ are hyperbolic cotangent and hyperbolic cosecant.
Hyperbolic Cosecant Function is Odd
Let $\csch: \C \to \C$ be the hyperbolic cosecant function on the set of complex numbers. Then $\csch$ is odd: :$\map \csch {-x} = -\csch x$
Hyperbolic Secant Function is Even
:$\map \sech {-x} = \sech x$
Hyperbolic Cotangent Function is Odd
Let $\coth: \C \to \C$ be the hyperbolic cotangent function on the set of complex numbers. Then $\coth$ is odd: :$\map \coth {-x} = -\coth x$
Hyperbolic Sine of Sum
:$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$
Hyperbolic Cosine of Sum
:$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$
Hyperbolic Tangent of Sum
:$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$
Hyperbolic Cotangent of Sum
:$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$
Double Angle Formulas/Hyperbolic Sine
:$\sinh 2 x = 2 \sinh x \cosh x$
Double Angle Formulas/Hyperbolic Cosine
:$\cosh 2 x = \cosh^2 x + \sinh^2 x$
Double Angle Formulas/Hyperbolic Tangent
: $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$