hcju/proofwikips huggingface dataset:
Half Angle Formulas/Hyperbolic Sine
{{begin-eqn}} {{eqn | l = \sinh \frac x 2       | r = +\sqrt {\frac {\cosh x - 1} 2}       | c = for $x \ge 0$ }} {{eqn | l = \sinh \frac x 2       | r = -\sqrt {\dfrac {\cosh x - 1} 2}       | c = for $x \le 0$ }} {{end-eqn}}
Half Angle Formulas/Hyperbolic Cosine
:$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$
Half Angle Formulas/Hyperbolic Tangent
{{begin-eqn}} {{eqn | l = \tanh \frac x 2       | r = +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }       | c = for $x \ge 1$ }} {{eqn | l = \tanh \frac x 2       | r = -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }       | c = for $x \le 1$ }} {{end-eqn}}
Half Angle Formulas/Hyperbolic Tangent/Corollary 2
For $x \ne 0$: :$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$
Triple Angle Formulas/Hyperbolic Sine
:$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$
Triple Angle Formulas/Hyperbolic Cosine
: $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$
Triple Angle Formulas/Hyperbolic Tangent
: $\tanh \left({3 x}\right) = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$
Quadruple Angle Formulas/Hyperbolic Sine
: $\sinh \left({4 x}\right) = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$
Quadruple Angle Formulas/Hyperbolic Cosine
: $\cosh \left({4 x}\right) = 8 \cosh^4 x - 8 \cosh^2 x + 1$
Quadruple Angle Formulas/Hyperbolic Tangent
: $\tanh \left({4 x}\right) = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$
Power Reduction Formulas/Hyperbolic Sine Squared
:$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$
Power Reduction Formulas/Hyperbolic Cosine Squared
:$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$
Power Reduction Formulas/Hyperbolic Sine to 4th
:$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$
Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine
:$\sinh x + \sinh y = 2 \sinh \left({\dfrac {x + y} 2}\right) \cosh \left({\dfrac {x - y} 2}\right)$
Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine
:$\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$
Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine
:$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$
Simpson's Formulas/Hyperbolic Sine by Hyperbolic Sine
:$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$
Simpson's Formulas/Hyperbolic Cosine by Hyperbolic Cosine
:$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$
Simpson's Formulas/Hyperbolic Sine by Hyperbolic Cosine
:$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$
Graph of Hyperbolic Cosine Function
:600px
Inverse Hyperbolic Sine of Reciprocal equals Inverse Hyperbolic Cosecant
Everywhere that the function is defined: :$\map {\sinh^{-1} } {\dfrac 1 x} = \csch^{-1} x$ where $\sinh^{-1}$ and $\csch^{-1}$ denote inverse hyperbolic sine and inverse hyperbolic cosecant respectively.
Inverse Hyperbolic Cosine of Reciprocal equals Inverse Hyperbolic Secant
Everywhere that the function is defined: :$\map {\cosh^{-1} } {\dfrac 1 x} = \sech^{-1} x$ where $\sinh^{-1}$ and $\csch^{-1}$ denote inverse hyperbolic cosine and inverse hyperbolic secant respectively.
Inverse Hyperbolic Tangent of Reciprocal equals Inverse Hyperbolic Cotangent
Everywhere that the function is defined: :$\map {\tanh^{-1} } {\dfrac 1 x} = \coth^{-1} x$ where $\tanh^{-1}$ and $\coth^{-1}$ denote inverse hyperbolic tangent and inverse hyperbolic cotangent respectively.
Inverse Hyperbolic Sine is Odd Function
Let $x \in \R$. Then: :$\map {\sinh^{-1} } {-x} = -\sinh^{-1} x$ where $\map {\sinh^{-1} } {-x}$ denotes the inverse hyperbolic sine function.
Inverse Hyperbolic Tangent is Odd Function
:$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$
Inverse Hyperbolic Cosecant is Odd Function
Let $x \in \R$. Then: :$\map {\csch^{-1} } {-x} = -\csch^{-1} x$ where $\map {\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function.
Hyperbolic Cosecant in terms of Cosecant
Let $z \in \C$ be a complex number. Then: :$i \csch z = -\csc \paren {i z}$ where: : $\csc$ denotes the cosecant function : $\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$.
Hyperbolic Secant in terms of Secant
Let $z \in \C$ be a complex number. Then: :$\sech z = \sec \paren {i z}$ where: : $\sec$ denotes the secant function : $\sech$ denotes the hyperbolic secant : $i$ is the imaginary unit: $i^2 = -1$.
Hyperbolic Cotangent in terms of Cotangent
Let $z \in \C$ be a complex number. Then: :$\coth z = -\cot \paren {i z}$ where: : $\cot$ denotes the cotangent function : $\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$.
Sine in terms of Hyperbolic Sine
Let $z \in \C$ be a complex number. Then: :$i \sin z = \map \sinh {i z}$ where: :$\sin$ denotes the complex sine :$\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$.
Cosine in terms of Hyperbolic Cosine
Let $z \in \C$ be a complex number. Then: :$\cos z = \map \cosh {i z}$ where: :$\cos$ denotes the complex cosine :$\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$.
Tangent in terms of Hyperbolic Tangent
Let $z \in \C$ be a complex number. Then: :$i \tan z = \map \tanh {i z}$ where: :$\tan$ denotes the tangent function :$\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$.
Cosecant in terms of Hyperbolic Cosecant
Let $z \in \C$ be a complex number. Then: :$i \csc = -\csch \paren {i z}$ where: : $\csc$ denotes the cosecant function : $\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$.
Secant in terms of Hyperbolic Secant
Let $z \in \C$ be a complex number. Then: :$\sec z = \map \sech {i z}$ where: :$\sec$ denotes the secant function :$\sech$ denotes the  hyperbolic secant :$i$ is the imaginary unit: $i^2 = -1$.
Cotangent in terms of Hyperbolic Cotangent
Let $z \in \C$ be a complex number. Then: :$i \cot z = -\coth \paren {i z}$ where: : $\cot$ denotes the cotangent function : $\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$.
General Periodicity Property
Let $f: X \to X$ be a periodic function, where $X$ is either the set of real numbers $\R$ or the set of complex numbers $\C$. Let $L$ be a periodic element of $f$. Then: :$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$ That is, after every distance $L$, the function $f$ repeats itself.
Periodicity of Hyperbolic Sine
Let $k \in \Z$. Then: :$\map \sinh {x + 2 k \pi i} = \sinh x$
Periodicity of Hyperbolic Cosine
Let $k \in \Z$. Then: :$\cosh \left({x + 2 k \pi i}\right) = \cosh x$
Periodicity of Hyperbolic Tangent
Let $k \in \Z$. Then: :$\tanh \left({x + 2 k \pi i}\right) = \tanh x$
Periodicity of Hyperbolic Cosecant
Let $k \in \Z$. Then: :$\operatorname{csch} \left({x + 2 k \pi i}\right) = \operatorname{csch} x$
Periodicity of Hyperbolic Secant
Let $k \in \Z$. Then: :$\map \sech {x + 2 k \pi i} = \sech x$
Periodicity of Hyperbolic Cotangent
Let $k \in \Z$. Then: :$\map \coth {x + 2 k \pi i} = \coth x$
Inverse Sine of Imaginary Number
:$\map {\sin^{-1} } {i x} = i \sinh^{-1} x$
Inverse Hyperbolic Sine of Imaginary Number
:$\sinh^{-1} \left({i x}\right) = i \sin^{-1} x$
Inverse Hyperbolic Tangent of Imaginary Number
:$\map {\tanh^{-1} } {i x} = i \tan^{-1} x$
Inverse Hyperbolic Cotangent of Imaginary Number
:$\map {\coth^{-1} } {i x} = i \cot^{-1} x$
Inverse Cosecant of Imaginary Number
:$\map {\csc^{-1} } {i x} = i \csch^{-1} x$
Inverse Hyperbolic Cosecant of Imaginary Number
:$\map {\csch^{-1} } {i x} = -i \csc^{-1} x$
Arccosecant of Reciprocal equals Arcsine
:$\map \arccsc {\dfrac 1 x} = \arcsin x$
Arcsecant of Reciprocal equals Arccosine
:$\map \arcsec {\dfrac 1 x} = \arccos x$
Arccotangent of Reciprocal equals Arctangent
:$\map \arccot {\dfrac 1 x} = \arctan x$
Solution to Quadratic Equation/Real Coefficients
Let $a, b, c \in \R$. The quadratic equation $a x^2 + b x + c = 0$ has: :Two real solutions if $b^2 - 4 a c > 0$ :One real solution if $b^2 - 4 a c = 0$ :Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates.
Sum of Roots of Quadratic Equation
Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\alpha$ and $\beta$ be the roots of $P$. Then: :$\alpha + \beta = -\dfrac b a$
Product of Roots of Quadratic Equation
Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\alpha$ and $\beta$ be the roots of $P$. Then: :$\alpha \beta = \dfrac c a$
Definition:Discriminant of Polynomial/Cubic Equation
Let: : $Q = \dfrac {3 a c - b^2} {9 a^2}$ : $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ The '''discriminant''' of the cubic equation is given by: : $D := Q^3 + R^2$
Ring is Module over Itself
Let $\struct {R, +, \circ}$ be a ring. Then $\struct {R, +, \circ}_R$ is an $R$-module.  If $\struct {R, +, \circ}$ has a unity, then $\struct {R, +, \circ}_R$ is unitary.
Rational Division is Closed
The set of rational numbers less zero is closed under division: :$\forall a, b \in \Q_{\ne 0}: a / b \in \Q_{\ne 0}$
Complex Modulus of Product of Complex Numbers/General Result
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: : $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$
Complex Modulus of Reciprocal of Complex Number
Let $z \in \C$ be a complex number such that $z \ne 0$. Let $\cmod z$ denote the complex modulus of $z$. Then: :$\cmod {\dfrac 1 z} = \dfrac 1 {\cmod z}$
Triangle Inequality/Complex Numbers/Corollary 1
Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: : $\cmod {z_1 + z_2} \ge \cmod {z_1} - \cmod {z_2}$
Complex Modulus of Additive Inverse
Let $z \in \C$ be a complex number. Let $-z$ be the negative of $z$: :$z + \paren {-z} = 0$ Then: :$\cmod z = \cmod {\paren {-z} }$ where $\cmod z$ denotes the modulus of $z$.
Even Power of Negative Real Number
Let $x \in \R$ be a real number. Let $n \in \Z$ be an even integer. Then: :$\paren {-x}^n = x^n$
Complex Addition is Closed/Proof 1
The set of complex numbers $\C$ is closed under addition: :$\forall z, w \in \C: z + w \in \C$
Complex Addition is Closed/Proof 2
The set of complex numbers $\C$ is closed under addition: :$\forall z, w \in \C: z + w \in \C$
Cardinality of Cartesian Product/General Result
Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be the cartesian product of a (finite) sequence of sets $\sequence {S_n}$. Then: :$\displaystyle \card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k}$
Cardinality of Cartesian Product/General Result/Corollary
Let $S^n$ be a cartesian space. Then: :$\card {S^n} = \card S^n$
Product of Complex Numbers in Polar Form/General Result
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $z_j = \polar {r_j, \theta_j}$ be $z_j$ expressed in polar form for each $j \in \set {1, 2, \ldots, n}$. Then: :$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle
Let $n \in \Z$ be an integer such that $n \ge 3$. Let $z \in \C$ be a complex number such that $z^n = 1$. Let $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ be the set of $n$th roots of unity. Let $U_n$ be plotted on the complex plane. Then the elements of $U_n$ are located at the vertices of a regular $n$-sided polygon $P$, such that: :$(1):\quad$ $P$ is circumscribed by a unit circle whose center is at $\tuple {0, 0}$ :$(2):\quad$ one of those vertices is at $\tuple {1, 0}$.
Equation of Unit Circle in Complex Plane
Consider the unit circle $C$ whose center is at $\tuple {0, 0}$ on the complex plane. Its equation is given by: :$\cmod z = 1$ where $\cmod z$ denotes the complex modulus of $z$.
Equation of Unit Circle
Let the unit circle have its center at the origin of the Cartesian plane. Its equation is given by: :$x^2 + y^2 = 1$ {{expand|Present it in polar coordinates as well}}
Tautological Consequent
:$p \implies \top \dashv \vdash \top$
Tautological Antecedent
:$\top \implies p \dashv \vdash p$
Contradictory Consequent
:$p \implies \bot \dashv \vdash \neg p$
Contradictory Antecedent
:$\bot \implies p \dashv \vdash \top$
Biconditional with Tautology
:$p \iff \top \dashv \vdash p$
Unique Representation of Complex Number in Spherical Form
Let $\mathcal P$ be the complex plane. Let $\mathbb S$ be the unit sphere which is tangent to $\mathcal P$ at $\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\mathbb S$ perpendicular to $\mathcal P$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. :900px Let $A$ be a point on $P$. Let the line $NA$ be constructed. Then $NA$ passes through exactly one point $A'$ on the surface of $\mathbb S$ apart from $N$. Similarly, let $A'$ be a point on the surface of $\mathbb S$ apart from $N$. Let the line $NA'$ be constructed. Then $NA'$ passes through exactly one point $A$ on $P$.
Exclusive Or with Contradiction
:$p \oplus \bot \dashv \vdash p$
Product of Complex Number with Conjugate by Dot and Cross Product
Let $z_1$ and $z_2$ be complex numbers. Then: :$\overline {z_1} z_2 = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$ :$z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$.
Product of Complex Number with Conjugate in Exponential Form
Let $z_1$ and $z_2$ be complex numbers. Then: :$\overline {z_1} z_2 = \cmod {z_1} \, \cmod {z_2} e^{i \theta}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.
Complex Numbers are Perpendicular iff Dot Product is Zero
Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$. Then $z_1$ and $z_2$ are perpendicular : :$z_1 \circ z_2 = 0$ where $z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$.
Complex Numbers are Parallel iff Cross Product is Zero
Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$. Then $z_1$ and $z_2$ are parallel : :$z_1 \times z_2 = 0$ where $z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$.
Tautology iff Negation is Unsatisfiable
Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a tautology iff its negation $\neg \mathbf A$ is unsatisfiable.
Satisfiable iff Negation is Falsifiable
Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is satisfiable iff its negation $\neg \mathbf A$ is falsifiable.
Subset of Satisfiable Set is Satisfiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$. Let $\mathcal F'$ be a subset of $\mathcal F$. Then $\mathcal F'$ is also $\mathscr M$-satisfiable.
Satisfiable Set Union Tautology is Satisfiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$. Let $\phi$ be a tautology for $\mathscr M$. Then $\mathcal F \cup \left\{{\phi}\right\}$ is also $\mathscr M$-satisfiable.
Unsatisfiable Set Union Formula is Unsatisfiable
Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$. Let $\phi$ be a logical formula. Then $\FF \cup \set \phi$ is also $\mathscr M$-unsatisfiable.
Superset of Unsatisfiable Set is Unsatisfiable
Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$. Let $\FF'$ be a superset of $\FF$. Then $\FF'$ is also $\mathscr M$-unsatisfiable.
Unsatisfiable Set minus Tautology is Unsatisfiable
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-unsatisfiable set of formulas from $\mathcal L$. Let $\phi \in \mathcal F$ be a tautology. Then $\mathcal F \setminus \set {\phi}$ is also $\mathscr M$-unsatisfiable.
Semantic Consequence of Set Union Formula
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of logical formulas from $\mathcal L$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\mathcal F$. Let $\psi$ be another logical formula. Then: :$\mathcal F \cup \left\{{\psi}\right\} \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\mathcal F \cup \left\{{\psi}\right\}$.
Semantic Consequence of Superset
Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be a set of logical formulas from $\LL$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$. Let $\FF'$ be another set of logical formulas. Then: :$\FF \cup \FF' \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\FF \cup \FF'$.
Semantic Consequence of Set minus Tautology
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of logical formulas from $\mathcal L$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\mathcal F$. Let $\psi \in \mathcal F$ be a tautology. Then: :$\mathcal F \setminus \left\{{\psi}\right\} \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\mathcal F \setminus \left\{{\psi}\right\}$.
Set of Literals Satisfiable iff No Complementary Pairs
Let $S$ be a set of literals. Then $S$ is satisfiable  it contains no complementary pairs.
Equivalence of Definitions of Exterior Point (Complex Analysis)
{{TFAE|def = Exterior Point (Complex Analysis)|view = exterior point|context = Complex Analysis}} Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$.
Quintuple Angle Formulas/Sine/Corollary
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ :$\dfrac {\sin 5 \theta} {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$ where $\sin$ denotes sine and $\cos$ denotes cosine.
Complex Multiplication as Geometrical Transformation
Let $z_1 = \left\langle{r_1, \theta_1}\right\rangle$ and $z_2 = \left\langle{r_2, \theta_2}\right\rangle$ be complex numbers expressed in polar form. Let $z_1$ and $z_2$ be represented on the complex plane $\C$ in vector form. Let $z = z_1 z_2$ be the product of $z_1$ and $z_2$. Then $z$ can be interpreted as the result of: : rotating $z_1$ about the origin of $\C$ by $\theta_2$ in the positive direction : multiplying the modulus of $z_1$ by $r_2$.
Argument of Quotient equals Difference of Arguments
Let $z_1$ and $z_2$ be complex numbers. Then: :$\map \arg {\dfrac {z_1} {z_2} } = \map \arg {z_1} - \map \arg {z_1} + 2 k \pi$ where: :$\arg$ denotes the argument of a complex number :$k$ can be $0$, $1$ or $-1$.
Cardinality of Finite Vector Space
Let $V$ be a $K$-vector space. Let $K$ be finite. Let the dimension of $V$ be finite. Then: :$\size V = \size K^{\map \dim V}$
Sum of Roots of Polynomial
Let $P$ be the polynomial equation: : $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$ such that $a_n \ne 0$. The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$.
Product of Roots of Polynomial
Let $P$ be the polynomial equation: :$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$ such that $a_n \ne 0$. The product of the roots of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$.
Sum of Sines of Fractions of Pi
Let $n \in \Z$ such that $n > 1$. Then: :$\displaystyle \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n = 0$
Area of Parallelogram in Complex Plane
Let $z_1$ and $z_2$ be complex numbers expressed as vectors. Let $ABCD$ be the parallelogram formed by letting $AD = z_1$ and $AB = z_2$. Then the area $\AA$ of $ABCD$ is given by: :$\AA = z_1 \times z_2$ where $z_1 \times z_2$ denotes the cross product of $z_1$ and $z_2$.
Area of Triangle in Determinant Form
Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane. The area $\mathcal A$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by: :$\mathcal A = \dfrac 1 2 \size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } }$
Equation of Circle in Complex Plane/Formulation 2
Let $\C$ be the complex plane. Let $C$ be a circle in $\C$. Then $C$ may be written as: :$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$ where: :$\alpha \in \R_{\ne 0}$ is real and non-zero :$\gamma \in \R$ is real :$\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$. The curve $C$ is a straight line  $\alpha = 0$ and $\beta \ne 0$.
Equation of Line in Complex Plane/Formulation 1
Let $\C$ be the complex plane. Let $L$ be a straight line in $\C$. Then $L$ may be written as: :$\beta z + \overline \beta \overline z  + \gamma = 0$ where $\gamma$ is real and $\beta$ may be complex.
Vertices of Equilateral Triangle in Complex Plane
Let $z_1$, $z_2$ and $z_3$ be complex numbers. Then: : $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle : :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Product of Sines of Fractions of Pi
Let $m \in \Z$ such that $m > 1$. Then: :$\displaystyle \prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} m = \frac m {2^{m - 1} }$
Product of Complex Conjugates/General Result
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $\overline z$ be the complex conjugate of the complex number $z$. Then: :$\displaystyle \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
Quotient of Complex Conjugates
Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ be the complex conjugate of the complex number $z$. Then: :$\overline {\paren {\dfrac {z_1} {z_2} } } = \dfrac {\paren {\overline {z_1} } } {\paren {\overline {z_2} } }$ for $z_2 \ne 0$.
Condition for Points in Complex Plane to form Parallelogram
Let $A = z_1$, $B = z_2$, $C = z_3$ and $D = z_4$ represent on the complex plane the vertices of a quadrilateral. Then $ABCD$ is a parallelogram : :$z_1 - z_2 + z_3 - z_4 = 0$
Quadrilateral with Bisecting Diagonals is Parallelogram
Let $ABCD$ be a quadrilateral. Let the diagonals of $ABCD$ bisect each other. Then $ABCD$ is a parallelogram.
Medians of Triangle Meet at Point
Let $\triangle ABC$ be a triangle. Then the medians of $\triangle ABC$ meet at a single point. This point is called the centroid of $\triangle ABC$. {{expand|State where that point is, that is $1/3$ the way along each median}}
Midpoints of Sides of Quadrilateral form Parallelogram
Let $\Box ABCD$ be a quadrilateral. Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively. Then $\Box EFGH$ is a parallelogram.
Line from Bisector of Side of Parallelogram to Vertex Trisects Diagonal
Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AD$. Then the point at which the line $BE$ meets $AC$ trisects $AC$.
Sum 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. Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$. Then: :$r_3 = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)}$ :$\theta_3 = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2} }$
De Moivre's Formula/Positive Integer Index
Let $z \in \C$ be a complex number expressed in polar form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$
De Moivre's Formula/Integer Index
Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$