Distance from Subset of Real Numbers

Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. Then:


Limit of Sequence to Zero Distance Point

Let $S$ be a non-empty subset of $\R$. Let the distance $\map d {\xi, S} = 0$ for some $\xi \in \R$. Then there exists a sequence $\sequence {x_n}$ in $S$ such that $\displaystyle \lim_{n \mathop \to \infty} x_n = \xi$.


Image of Closed Real Interval is Bounded

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ is bounded on $\closedint a b$.


Convergent Subsequence in Closed Interval

Let $\closedint a b$ be a closed real interval. Then every sequence of points of $\closedint a b$ contains a subsequence which converges to a point in $\closedint a b$.


Max and Min of Function on Closed Real Interval

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ reaches a maximum and a minimum on $\closedint a b$.


Continuous Image of Closed Interval is Closed Interval

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Then the image of $\left[{a \,.\,.\, b}\right]$ under $f$ is also a closed interval.


Retraction Theorem

Let $M$ be a compact manifold with boundary $\partial M$. Then there is no smooth mapping $f: M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the identity.


Classification of Compact One-Manifolds

Every compact one-dimensional manifold is diffeomorphic to either a circle or a closed interval.


Differentiable Function is Continuous

Let $f$ be a real function defined on an interval $I$. Let $x_0 \in I$ such that $f$ is differentiable at $x_0$. Then $f$ is continuous at $x_0$.


Quotient Rule for Derivatives

Let $\map j x, \map k x$ be real functions defined on the open interval $I$. Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Define the real function $f$ on $I$ by: :$\displaystyle \map f x = \begin{cases} \dfrac {\map j x} {\map k x} & : \map k x \ne 0 \\ 0 & : \text{otherwise} \end{cases}$ Then, if $\map k \xi \ne 0$, $f$ is differentiable at $\xi$, and furthermore: :$\map {f'} \xi = \dfrac {\map {j'} \xi \map k \xi - \map j \xi \map {k'} \xi} {\paren {\map k \xi}^2}$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\displaystyle \forall x \in I: \map k x \ne 0 \implies \map {f'} x = \frac {\map {j'} x \map k x - \map j x \map {k'} x} {\paren {\map k x}^2}$


Preimage Theorem

Let $y$ be a regular value of a smooth submersion $f:X \to Y$. Then the preimage $f^{-1}(y)$ is a smooth submanifold of $X$, with $\dim f^{-1}(y) = \dim X - \dim Y$.


Backwards Induction

Let $P$ be a propositional function on the natural numbers $\N$. Suppose that: :$(1): \quad \forall n \in \N: \map P {2^n}$ holds. :$(2): \quad \map P n \implies \map P {n - 1}$. Then $\map P n$ holds for all $\forall n \in \N$. The proof technique based on this result is called '''backwards induction'''.


Inequalities Concerning Roots

Let $\closedint X Y$ be a closed real interval such that $0 < X \le Y$. Let $x, y \in \closedint X Y$. Then: :$\forall n \in \N_{> 0}: X Y^{1/n} \size {x - y} \le n X Y \size {x^{1/n} - y^{1/n} } \le Y X^{1/n} \size {x - y}$


Continuity of Root Function

Let $n \in \N_{>0}$ be a non-zero natural number. Let $f: \hointr 0 \infty \to \R$ be the real function defined by $\map f x = x^{1/n}$. Then $f$ is continuous at each $\xi > 0$ and continuous on the right at $\xi = 0$.


P-adic Norm is Norm

The $p$-adic norm forms a norm on the rational numbers $\Q$.


Fundamental Theorem of Algebra

Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.


Extendability Theorem for Intersection Numbers

Let $X = \partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$. Let $Y$ be a smooth manifold, $Z$ be a closed smooth submanifold of $Y$, and $f: X \to Y$ a smooth map. Let there exist a smooth map $g: W \to Y$ such that $g \restriction_X = f$. Then: :$I \left({f, Z}\right) = 0$ where $I \left({f, Z}\right)$ is the intersection number. {{explain|what $I \left({f, Z}\right)$ is the intersection number of: presumably the words will go something like "... the intersection number of $f$ with respect to $Z$", or something.}}


Homotopy Group is Group

The set of all homotopy classes of continuous mappings: :$c: \closedint 0 1^n \to X$ satisfying: :$\map c {\partial \closedint 0 1^n} = x_0$ in a space $X$ at a base point $x_0$, under the operation of concatenation on class members, forms a group. {{finish|This operation needs to be shown well-defined; probably a nasty proof}} This group is called the $n$th homotopy group.


Derivative at Maximum or Minimum

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Let $f$ have a local minimum or local maximum at $\xi \in \openint a b$. Then: :$\map {f'} \xi = 0$


Behaviour of Function Near Limit

Let $f$ be a real function. Let $f \left({x}\right) \to l$ as $x \to \xi$. Then: * If $l > 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: f \left({x}\right) > 0$ * If $l < 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: f \left({x}\right) < 0$


Fundamental Group is Independent of Base Point for Path-Connected Space

Let $X$ be a path-connected space. For $x \in X$ let $\pi_1(X,x)$ denote the fundamental group. For $x, y \in X$, there is an isomorphism: :$\phi: \pi_1 \left({X, x}\right) \to \pi_1 \left({X, y}\right)$


List of Fundamental Groups for 2-Manifolds

For the following two-manifolds, the fundamental group for any point in $X$, written $\pi_1 \left({X}\right)$ is isomorphic to the listed group: :$\pi_1 \left({\Bbb S^1 \times \left[{0\,.\,.\,1}\right]}\right) = \Z$ :$\pi_1 \left({\Bbb S^2}\right) = \left\{{e}\right\}$, the trivial group. :$\pi_1 \left({\Bbb T^2}\right) = \pi_1 \left({\Bbb S^1 \times \Bbb S^1}\right) = \Z \times \Z$ :$\pi_1 \left({\Bbb {RP}^2}\right) = \Z_2$


Rolle's Theorem

Let $f$ be a real function which is: :continuous on the closed interval $\closedint a b$ and: :differentiable on the open interval $\openint a b$. Let $\map f a = \map f b$. Then: :$\exists \xi \in \openint a b: \map {f'} \xi = 0$


Mean Value Theorem

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Then: :$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$


Zero Derivative implies Constant Function

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Suppose that: :$\forall x \in \left({a \,.\,.\, b}\right): f' \left({x}\right) = 0$ Then $f$ is constant on $\left[{a \,.\,.\, b}\right]$.


Ostrowski's Theorem

Every non-trivial norm on the rational numbers $\Q$ is equivalent to either: :the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$ or: :the absolute value, $\size {\, \cdot \,}$.


Cauchy Mean Value Theorem

Let $f$ and $g$ be a real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Suppose that: :$\forall x \in \left({a \,.\,.\, b}\right): g' \left({x}\right) \ne 0$ Then: :$\exists \xi \in \left({a \,.\,.\, b}\right): \dfrac {f' \left({\xi}\right)} {g' \left({\xi}\right)} = \dfrac {f \left({b}\right) - f \left({a}\right)} {g \left({b}\right) - g \left({a}\right)}$


Homology Group is Group

The $p^{th}$ singular homology group of a space $X$ is a group.


Limit of Absolute Value

Let $x, \xi \in \R$ be real numbers. Then: :$\left\vert{x - \xi}\right\vert \to 0$ as $x \to \xi$ where $\left\vert{x - \xi}\right\vert$ denotes the Absolute Value.


Limit of Function in Interval

Let $f$ be a real function which is defined on the open interval $\openint a b$. Let $\xi \in \openint a b$ Suppose that, $\forall x \in \openint a b$, either: :$\xi \le \map f x \le x$ or: :$x \le \map f x \le \xi$ Then $\map f x \to \xi$ as $x \to \xi$.


Derivative of Monotone Function

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.


Strictly Monotone Real Function is Bijective

Let $f$ be a real function which is defined on $I \subseteq \R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f: I \to J$ is a bijection.


Inverse of Strictly Monotone Function

Let $f$ be a real function which is defined on $I \subseteq \R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f$ always has an inverse function $f^{-1}$ and: : if $f$ is strictly increasing then so is $f^{-1}$ : if $f$ is strictly decreasing then so is $f^{-1}$.


Rokhlin's Theorem (Intersection Forms)

Let $M$ be a smooth 4-manifold. Then: :$\map {\omega_2} {\map T M} = 0 \implies \operatorname {sign} Q_M = 0 \pmod {16}$ where: :$Q_M$ is the intersection form :$\map T M$ is the tangent bundle :$\omega_2$ is the second Stiefel-Whitney class.


Convex Real Function is Continuous

Let $f$ be a real function which is convex on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$.


Convex Real Function is Left-Hand and Right-Hand Differentiable

Let $f$ be a real function which is either convex on the open interval $\left({a \,.\,.\, b}\right)$. Then the left-hand derivative $f'_- \left({x}\right)$ and right-hand derivative $f'_+ \left({x}\right)$ both exist for all $x \in \left({a \,.\,.\, b}\right)$.


Real Function is Convex iff Derivative is Increasing

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then: :$f$ is convex on $\openint a b$ {{iff}}: :its derivative $f'$ is increasing on $\openint a b$. Thus the intuitive result that a convex function "gets steeper".


Inverse of Strictly Increasing Convex Real Function is Concave

Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is concave on $J$.


Upper and Lower Bounds of Integral

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $\displaystyle \int_a^b \map f x \rd x$ be the definite integral of $\map f x$ over $\closedint a b$. Then: :$\displaystyle m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$ where: :$M$ is the maximum of $f$ :$m$ is the minimum of $f$ on $\closedint a b$.


Wilson's Theorem

A (strictly) positive integer $p$ is a prime {{iff}}: :$\paren {p - 1}! \equiv -1 \pmod p$


Integral of Constant

Let $c$ be a constant.


Sum of Integrals on Adjacent Intervals for Continuous Functions

Let $f$ be a real function which is continuous on any closed interval $I$. Let $a, b, c \in I$. Then: :$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$


Primitives which Differ by Constant

Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$. Let $G$ be a real function defined on $\closedint a b$. Then $G$ is a primitive for $f$ on $\closedint a b$ {{iff}}: :$\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$ That is, {{iff}} $F$ and $G$ differ by a constant on the whole interval.


Definite Integral of Function plus Constant

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $c$ be a constant. Then: :$\ds \int_a^b \paren {\map f t + c} \rd t = \int_a^b \map f t \rd t + c \paren {b - a}$


Continuous Real Function is Darboux Integrable

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.


Definite Integral on Zero Interval

Let $f$ be a real function which is defined on the closed interval $\Bbb I := \closedint a b$, where $a < b$. Then: :$\displaystyle \forall c \in \Bbb I: \int_c^c \map f t \rd t = 0$


Linear Combination of Derivatives

Let $f \left({x}\right), g \left({x}\right)$ be real functions defined on the open interval $I$. Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are differentiable. Then: :$D \left({\lambda f + \mu g}\right) = \lambda D f + \mu D g$ at the point $\xi$. It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then: :$\forall x \in I: D \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda D f \left({x}\right) + \mu D g \left({x}\right)$


Linear Combination of Integrals

Let $f$ and $g$ be real functions which are integrable on the closed interval $\closedint a b$. Let $\lambda$ and $\mu$ be real numbers. Then the following results hold:


Integration by Parts

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$. Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$. Then:


Integration by Substitution

Let $\phi$ be a real function which has a derivative on the closed interval $\closedint a b$. Let $I$ be an open interval which contains the image of $\closedint a b$ under $\phi$. Let $f$ be a real function which is continuous on $I$.


Relative Sizes of Definite Integrals

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$, where $a < b$. If: :$\forall t \in \closedint a b: \map f t \le \map g t$ then: :$\displaystyle \int_a^b \map f t \rd t \le \int_a^b \map g t \rd t$ Similarly, if: :$\forall t \in \closedint a b: \map f t < \map g t$ then: :$\displaystyle \int_a^b \map f t \rd t < \int_a^b \map g t \rd t$


Absolute Value of Definite Integral

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$\displaystyle \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$


Integral Test

Let $f$ be a real function which is continuous, positive and decreasing on the interval $\hointr 1 {+\infty}$. Let the sequence $\sequence {\Delta_n}$ be defined as: :$\displaystyle \Delta_n = \sum_{k \mathop = 1}^n \map f k - \int_1^n \map f x \rd x$ Then $\sequence {\Delta_n} $ is decreasing and bounded below by zero. Hence it converges.


Properties of Natural Logarithm

Let $x \in \R$ be a real number such that $x > 0$. Let $\ln x$ be the natural logarithm of $x$. Then:


Harmonic Series is Divergent

The harmonic series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n$  diverges.


Sum of Reciprocals of Primes is Divergent

Let $n \in \N: n \ge 1$. There exists a (strictly) positive real number $C \in \R_{>0}$ such that: :$(1): \quad \displaystyle \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$ where $\Bbb P$ is the set of all prime numbers. :$(2): \quad \displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = +\infty$


Measure of Interval is Length

Let $I$ be a real interval whose endpoints are $a$ and $b$. Then $I$ is Lebesgue measurable, and the value of the measure is the length of the interval $b - a$.


Measurable Sets form Algebra of Sets

Let $\mu^*$ be an outer measure on a set $X$. Then the set of $\mu^*$-measurable sets is an algebra of sets.


Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry

Every closed, orientable, path connected $3$-dimensional Riemannian manifold which supports a geometry of zero curvature is homeomorphic to one of the following: * Torus $\mathbb T^3$ * Half-Twist Cube * Quarter-Twist Cube * Hantschze-Wendt Manifold * $\frac 1 6$-Twist Hexagonal Prism * $\frac 1 3$-Twist Hexagonal Prism The $3$-torus is described on the torus page. The other manifolds can be described using quotient spaces on familiar prisms, with the equivalence relations described below. <gallery perRow="5"> File:Halftwistcube.JPG|The Half-Twist Cube File:Hantschzewendt.JPG|The Hantschze-Wendt Manifold File:Quartertwistcube.JPG|The Quarter-Twist Cube File:Sixthtwisthexagon.JPG|The $\frac 1 6$-Twist Hexagonal Prism File:Thirdtwisthexagon.JPG|The $\frac 1 3$-Twist Hexagonal Prism </gallery>


Sum of Logarithms

{{:Sum of Logarithms/General Logarithm}}


Heine-Borel Theorem/Real Line

Let $\R$ be the real number line considered as a Euclidean space. Let $C \subseteq \R$. Then $C$ is closed and bounded in $\R$ {{iff}} $C$ is compact.


Measurable Image

Let $\mathfrak M$ be the set of measurable sets of $\R$. For any extended real-valued function $f: \R \to \R \cup \left\{{-\infty \,.\,.\, +\infty}\right\}$ whose domain is measurable, the following statements are equivalent: : $(1): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) > \alpha}\right\} \in \mathfrak M$ : $(2): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) \ge \alpha}\right\} \in \mathfrak M$ : $(3): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) < \alpha}\right\} \in \mathfrak M$ : $(4): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) \le \alpha}\right\} \in \mathfrak M$ These statements imply: : $(5): \quad \forall \alpha \in \R \cup \left\{{-\infty \,.\,.\, +\infty}\right\}: \left\{{x:f \left({x}\right) = \alpha}\right\} \in \mathfrak M$ {{refactor}} {{proofread}}


Lebesgue Integral is Extension of Darboux Integral

Let $f: \closedint a b \to \R$ be a Darboux integrable function. Then it is also Lebesgue integrable, and furthermore: :$\displaystyle R \int_a^b \map f x \rd x = \int_{\closedint a b} f \rd \lambda$ where $\displaystyle R \int_a^b$ is the Darboux integral and $\displaystyle \int_{\closedint a b}$ is the Lebesgue integral.


Properties of Exponential Function

Let $x \in \R$ be a real number. Let $\exp x$ be the exponential of $x$. Then:


Euclidean Metric on Real Vector Space is Metric

The Euclidean metric on a real vector space $\R^n$ is a metric.


Metric Induces Topology

Let $M = \struct {A, d}$ be a metric space. Then the topology $\tau$ induced by the metric $d$ is a topology on $M$.


Properties of Algebras of Sets

Let $X$ be a set. Let $\mathfrak A$ be an algebra of sets on $X$. Then the following hold: :$(1): \quad$ The intersection of two sets in $\mathfrak A$ is in $\mathfrak A$. :$(2): \quad$ The difference of two sets in $\mathfrak A$ is in $\mathfrak A$. :$(3): \quad$ $X \in \mathfrak A$. :$(4): \quad$ The empty set $\O$ is in $\mathfrak A$.


Countable Sets Have Measure Zero

Let $S$ be a countable set. {{explain|Is it assumed that $S \subseteq \R$?}} Then the measure of $S$ is $\map m S = 0$.


Mean Value of Convex Real Function

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Let $f$ be convex on $\openint a b$. Then: :$\forall \xi \in \openint a b: \map f x - \map f \xi \ge \map {f'} \xi \paren {x - \xi}$


Upper Bound of Natural Logarithm

Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$. Then: :$\ln y \le y - 1$


Exponential of Sum/Real Numbers

Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$


Exponential of Product

Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x y} = \paren {\exp y}^x$


Exponent Combination Laws

Let $a, b \in \R_{>0}$ be strictly positive real numbers. Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then:


Derivative of Exponential at Zero

Let $\exp x$ be the exponential of $x$ for real $x$. Then: : $\displaystyle \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$


Classification of Groups of Order up to 15

Up to isomorphism, every group of order $\order G \le 15$ is one of the below: {| class="sortable wikitable" |- bgcolor="#ececec" ! Order !! Abelian !! Non-Abelian |- |1 || $\Z_1$ ||  |- |2 || $\Z_2$ || |- |3 || $\Z_3$ || |- |4 || $\Z_4, \Z_2 \oplus \Z_2$ ||  |- |5 || $\Z_5$ || |- |6 || $\Z_6$ || $D_3 = S_3$ |-  |7 || $\Z_7$ || |- |8 || $\Z_8, \Z_4 \oplus \Z_2, \Z_2 \oplus \Z_2 \oplus \Z_2$ || $D_4, \Dic 2$ |- |9 || $\Z_9, \Z_3 \oplus \Z_3$ ||  |- |10 || $\Z_{10}$ || $D_5$ |- |11 || $\Z_{11}$ || |- |12 || $\Z_{12}, \Z_6 \oplus \Z_2$ || $D_6, A_4, \Dic 3$ |- |13 || $\Z_{13}$ || |- |14 || $\Z_{14}$ || $D_7$ |- |15 || $\Z_{15}$ ||  |} where: : $D_n$ is the dihedral group of order $2 n$ : $S_n$ is the $n$th symmetric group : $A_n$ is the alternating group on $n$ points : $\Dic 2$ is the dicyclic group of order $4 n$.


Cyclic Groups of Order p q

Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$. Let $G$ be a group of order $p q$. Then $G$ is cyclic.


Derivative of Logarithm at One

Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$. Then: :$\displaystyle \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$


Dicyclic Group is Group

The dicyclic group $Q_n$ is a non-abelian group on two generators.


Existence of Interval of Convergence of Power Series

Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Then the interval of convergence of $\map S x$ is a real interval whose midpoint is $\xi$.


Power Series is Differentiable on Interval of Convergence

Let $\xi \in \R$ be a real number. Let $\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Let $\map f x$ have an interval of convergence $I$. Then $\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints. Also: :$\displaystyle \map {D_x} {\map f x} = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1}$


Power Series Expansion for Exponential Function

Let $\exp x$ be the exponential function. Then: {{begin-eqn}} {{eqn | ll= \forall x \in \R:       | l = \exp x       | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}       | c =  }} {{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots       | c =  }} {{end-eqn}}


Properties of Real Sine Function

Let $x \in \R$ be a real number. Let $\sin x$ be the sine of $x$. Then:


Properties of Real Cosine Function

Let $x \in \R$ be a real number. Let $\cos x$ be the cosine of $x$. Then:


Derivative of Cosine Function

:$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$


Limit of Sine of X over X

:$\displaystyle \lim_{x \mathop \to 0} \frac {\sin x} x = 1$


Sum of Squares of Sine and Cosine

:$\cos^2 x + \sin^2 x = 1$


Cosine of Sum

:$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$


Equivalence of Definitions of Sine and Cosine

The definitions for sine and cosine are equivalent. That is: : $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} \iff \sin x = \frac {\text{Opposite}} {\text{Hypotenuse}}$ : $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} \iff \cos x = \frac {\text{Adjacent}} {\text{Hypotenuse}}$


Real Cosine Function is Bounded

: $\size {\cos x} \le 1$


Boundedness of Sine X over X

Let $x \in \R$. Then: :$\size {\dfrac {\sin x} x} \le 1$


Sine and Cosine are Periodic on Reals

The sine and cosine functions are periodic on the set of real numbers $\R$: :$(1): \quad \map \cos {x + 2 \pi} = \cos x$ :$(2): \quad \map \sin {x + 2 \pi} = \sin x$ :800px


Differentiable Bounded Convex Real Function is Constant

Let $f$ be a real function which is: :$(1): \quad$ Differentiable on $\R$ :$(2): \quad$ Bounded on $\R$ :$(3): \quad$ Convex on $\R$. Then $f$ is constant.


Banach-Tarski Paradox

The unit ball $\mathbb D^3 \subset \R^3$ is equidecomposable to the union of two unit balls.


Equidecomposability is Equivalence Relation

The property of being equidecomposable is an equivalence relation on the power set $\mathcal P \left({\R^n}\right)$.


Equidecomposability Unaffected by Union

{{explain|I guess the sets should be disjoint, in the sense that $S_i \cap S_j$ is empty for $i \neq j$. Similar for $T$.}} Let $\left\{{S_1, \ldots, S_m}\right\}, \left\{{T_1, \ldots, T_m }\right\}$ be sets of sets in $\R^n$ such that: : for each $k \in \left\{{1, \dots, m}\right\}, S_k$ and $T_k$ are equidecomposable.  Then the set $\displaystyle S = \bigcup_{i \mathop = 1}^m S_i$ is equidecomposable with $\displaystyle T = \bigcup_{i \mathop = 1}^m T_i$.


Subsets of Equidecomposable Subsets are Equidecomposable

Let $A, B \subseteq \R^n$ be equidecomposable. Let $S \subseteq A$. Then there exists $T \subseteq B$ such that $S$ and $T$ are equidecomposable.


Shape of Cosine Function

The cosine function is: :$(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$ :$(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$ :$(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$ :$(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$


Tangent Function is Periodic on Reals

The tangent function is periodic on the set of real numbers $\R$ with period $\pi$. This can be written: :$\tan x = \tan \left({x \bmod \pi}\right)$ where $x \bmod \pi$ denotes the modulo operation.


Derivative of Tangent Function

:$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$ when $\cos x \ne 0$.


Stirling's Formula

The factorial function can be approximated by the formula: :$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$ where $\sim$ denotes asymptotically equal.


Derivative of Cotangent Function

:$\map {\dfrac \d {\d x} } {\cot x} = -\csc^2 x = \dfrac {-1} {\sin^2 x}$ where $\sin x \ne 0$.


Shape of Sine Function

The sine function is: :$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$ :$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$ :$(3): \quad$ concave on the interval $\closedint 0 \pi$ :$(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$


Derivative of Secant Function

:$\map {\dfrac \d {\d x} } {\sec x} = \sec x \tan x$ where $\cos x \ne 0$.


Derivative of Cosecant Function

:$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$ where $\sin x \ne 0$.


Shape of Tangent Function

The nature of the tangent function on the set of real numbers $\R$ is as follows: : $\tan x$ is continuous and strictly increasing on the interval $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$ : $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$ : $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$ : $\tan x$ is not defined on $\forall n \in \Z: x = \left({n + \dfrac 1 2}\right) \pi$, at which points it is discontinuous : $\forall n \in \Z: \tan \left({n \pi}\right) = 0$.


Derivative of Arcsine Function

:$\dfrac {\map \d {\arcsin x} } {\d x} = \dfrac 1 {\sqrt {1 - x^2} }$


Derivative of Arccosine Function

:$\map {D_x} {\arccos x} = \dfrac {-1} {\sqrt {1 - x^2}}$


Sum of Arcsine and Arccosine

Let $x \in \R$ be a real number such that $-1 \le x \le 1$. Then: : $\arcsin x + \arccos x = \dfrac \pi 2$ where $\arcsin$ and $\arccos$ denote arcsine and arccosine respectively.


Derivative of Arctangent Function

:$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$


Complex Numbers form Field

Consider the algebraic structure $\struct {\C, +, \times}$, where: :$\C$ is the set of all complex numbers :$+$ is the operation of complex addition :$\times$ is the operation of complex multiplication Then $\struct {\C, +, \times}$ forms a field.


Complex Multiplication is Commutative

The operation of multiplication on the set of complex numbers $\C$ is commutative: :$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$


Complex Multiplication Distributes over Addition

The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition. :$\forall z_1, z_2, z_3 \in \C:$ ::$z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$ ::$\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$


Complex Modulus of Product of Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$


Sum of Complex Conjugates

Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}$


Product of Complex Conjugates

Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$


Sum of Complex Number with Conjugate

Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Let $\map \Re z$ be the real part of $z$. Then: :$z + \overline z = 2 \, \map \Re z$


Difference of Complex Number with Conjugate

Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Let $\map \Im z$ be the imaginary part of $z$. Then :$z - \overline z = 2 i \, \map \Im z$


Complex Number equals Conjugate iff Wholly Real

Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Then $z = \overline z$ {{iff}} $z$ is wholly real.


Complex Numbers cannot be Ordered Compatibly with Ring Structure

Let $\struct {\C, +, \times}$ be the field of complex numbers. There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$.


Complex Plane is Metric Space

Let $\C$ be the set of all complex numbers. Let $d: \C \times \C \to \R$ be the function defined as: :$\map d {z_1, z_2} = \size {z_1 - z_2}$ where $\size z$ is the modulus of $z$. Then $d$ is a metric on $\C$ and so $\struct {\C, d}$ is a metric space.


Real Number Line is Metric Space

Let $\R$ be the real number line. Let $d: \R \times \R \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\R$ and so $\struct {\R, d}$ is a metric space.


P-Product Metric on Real Vector Space is Metric

Let $\R^n$ be an $n$-dimensional real vector space. Let $p \in \R_{\ge 1}$. Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$: : $\displaystyle d_p \left({x, y}\right) :=  \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$. Then $d_p$ is a metric.


Taxicab Metric is Metric

The taxicab metric is a metric.


Zero and One are the only Consecutive Perfect Squares

If $n$ is a perfect square other than $0$, then $n+1$ is not a perfect square.


ProofWiki:Jokes

If I cannot open these cans of food, I will die.


P-Product Metric is Metric

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$. Let $p \in \R_{\ge 1}$. Let $d_p: \mathcal A \times \mathcal A \to \R$ be the $p$-product metric on $\mathcal A$: : $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. Then $d_p$ is a metric.


Euler Triangle Formula

Let $d$ be the distance between the incenter and the circumcenter of a triangle. Then: :$d^2 = R \left({R - 2 \rho}\right)$ where: :$R$ is the circumradius :$\rho$ is the inradius.


Open Ball of Point Inside Open Ball

Let $M = \left({A, d}\right)$ be a metric space. Let $B_\epsilon \left({x}\right)$ be an open $\epsilon$-ball in $M = \left({A, d}\right)$. Let $y \in B_\epsilon \left({x}\right)$. Then: : $\exists \delta \in \R: B_\delta \left({y}\right) \subseteq B_\epsilon \left({x}\right)$ That is, for every point in an open $\epsilon$-ball in a metric space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.


Open Sets in Metric Space

Let $M = \struct {A, d}$ be a metric space. Then $\O$ and $A$ are both open in $M$.


Equivalence of Definitions of Continuity on Metric Spaces

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. {{TFAE|def = Continuous on Metric Space|view = continuity|context = Metric Space|contextview = metric spaces}}


Finite Intersection of Open Sets of Metric Space is Open

Let $M = \struct {A, d}$ be a metric space. Let $U_1, U_2, \ldots, U_n$ be open in $M$. Then $\ds \bigcap_{i \mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open.


Union of Open Sets of Metric Space is Open

Let $M = \struct {A, d}$ be a metric space. The union of a set of open sets of $M$ is open in $M$.


Lipschitz Equivalent Metric Spaces are Homeomorphic

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $M_1$ and $M_2$ be Lipschitz equivalent. Then $M_1$ and $M_2$ are homeomorphic.


P-Product Metrics on Real Vector Space are Topologically Equivalent

For $n \in \N$, let $\R^n$ be an Euclidean space. Let $p \in \R_{\ge 1}$. Let $d_p$ be the $p$-product metric on $\R^n$. Let $d_\infty$ be the Chebyshev distance on $\R^n$. Then $d_p$ and $d_\infty$ are topologically equivalent.


Derivative of Function of Constant Multiple

Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: :$\map {D_x} {\map f {c x} } = c \map {D_{c x} } {\map f {c x} }$


Cotangent Function is Periodic on Reals

The cotangent function is periodic on the set of real numbers $\R$ with period $\pi$.


Shape of Cotangent Function

The nature of the cotangent function on the set of real numbers $\R$ is as follows: : $\cot x$ is continuous and strictly decreasing on the interval $\left({0 \,.\,.\, \pi}\right)$ : $\cot x \to + \infty$ as $x \to 0^+$ : $\cot x \to - \infty$ as $x \to \pi^-$ : $\cot x$ is not defined on $\forall n \in \Z: x = n \pi$, at which points it is discontinuous : $\forall n \in \Z: \cot \left({n + \dfrac 1 2}\right) \pi = 0$


Change of Base of Logarithm

:$\log_b x = \dfrac {\log_a x} {\log_a b}$


Derivative of Function to Power of Function

Let $\map u x, \map v x$ be real functions which are differentiable on $\R$. Then: :$\map {\dfrac \d {\d x} } {u^v} = v u^{v - 1} \map {\dfrac \d {\d x} } u + u^v \paren {\ln u} \map {\dfrac \d {\d x} } v$


Liouville's Theorem (Complex Analysis)

Let $f: \C \to \C$ be a bounded entire function. Then $f$ is constant.


Riemann Removable Singularities Theorem

Let $U \subset \C$ be a domain, let $z_0 \in U$, and let $f: U \setminus \left\{ {z_0}\right\} \to \C$ be holomorphic. Then the following are equivalent: :$(1): \quad f$ extends to a holomorphic function $f: U \to \C$. :$(2): \quad f$ extends to a continuous function $f: U \to \C$. :$(3): \quad f$ is bounded in a neighborhood of $z_0$. :$(4): \quad f \left({z}\right) = o \left({\dfrac 1 {\left|{z - z_0}\right|} }\right)$ as $z \to z_0$.


Gaussian Integral

:$\displaystyle \int_{-\infty}^\infty e^{-x^2} \rd x = \sqrt \pi$


Indiscrete Topology is not Metrizable

Let $S$ be a set with more than one element. The indiscrete topology on $S$ is not metrizable.


Composite of Continuous Mappings is Continuous

Let $T_1, T_2, T_3$ be topological spaces. Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be continuous mappings. Then the composite mapping $g \circ f: T_1 \to T_3$ is continuous.


Continuous Mapping is Continuous on Induced Topological Spaces

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $\tau_{d_1}$ and $\tau_{d_2}$ be the topologies induced by the metrics $d_1$ and $d_2$. Let $T_1 = \left({A_1, \tau_{d_1}}\right)$ and $T_2 = \left({A_2, \tau_{d_2}}\right)$ be the resulting topological spaces. Let $f: A_1 \to A_2$ be a mapping. Then $f$ is $\left({d_1, d_2}\right)$-continuous {{iff}} $f$ is $\left({\tau_{d_1}, \tau_{d_2}}\right)$-continuous.


Synthetic Basis and Analytic Basis are Compatible

Let $\struct {S, \tau}$ be a topological space. Then $\BB$ is an analytic basis for $\tau$ {{iff}} $\tau$ is the topology on $S$ generated by the synthetic basis $\BB$.


Picard's Existence Theorem

Let $\map f {x, y} : \R^2 \to \R$ be continuous in a region $D \subseteq \R^2$. Let $\exists M \in \R: \forall x, y \in D: \size {\map f {x, y} } < M$. Let $\map f {x, y}$ satisfy in $D$ the Lipschitz condition in $y$: :$\size{\map f {x, y_1} - \map f {x, y_2} } \le A \size {y_1 - y_2}$ where $A$ is independent of $x, y_1, y_2$. Let the rectangle $R$ be defined as $\set {\tuple {x, y} \in \R^2: \size {x - a} \le h, \size {y - b} \le k}$ such that $M h \le k$. Let $R \subseteq D$. Then $\forall x \in \R: \size {x - a} \le h$, the first order ordinary differential equation: :$y' = \map f {x, y}$ has one and only one solution $y = \map y x$ for which $b = \map y a$.


Continuity Test using Basis

Let $\left({X_1, \tau_1}\right)$ and $\left({X_2, \tau_2}\right)$ be topological spaces. Let $f: X_1 \to X_2$ be a mapping. Let $\mathcal B$ be an analytic basis for $\tau_2$. Suppose that: :$\forall B \in \mathcal B: f^{-1} \left({B}\right) \in \tau_1$ where $f^{-1} \left({B}\right)$ denotes the preimage of $B$ under $f$. Then $f$ is continuous.


Sub-Basis for Real Number Line

Let the real number line $\R$ be considered as a topology under the usual (Euclidean) metric. Then: :$\mathcal B := \left\{{\left({-\infty \,.\,.\, a}\right), \left({b \,.\,.\, \infty}\right): a, b \in \R}\right\}$ is a sub-basis for $\R$.


Synthetic Basis formed from Synthetic Sub-Basis

Let $X$ be a set. Let $\SS$ be a synthetic sub-basis on $X$. Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \text{$\FF$ is finite} }$ Then $\BB$ is a synthetic basis on $X$.


Epimorphism Preserves Distributivity

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be algebraic structures. Let $\phi: R_1 \to R_2$ be an epimorphism. :If $\circ_1$ is left distributive over $+_1$, then $\circ_2$ is left distributive over $+_2$. :If $\circ_1$ is right distributive over $+_1$, then $\circ_2$ is right distributive over $+_2$. Consequently, if $\circ_1$ is distributive over $+_1$, then $\circ_2$ is distributive over $+_2$. That is, epimorphism preserves distributivity.


Epimorphism Preserves Groups

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism. Let $\struct {S, \circ}$ be a group. Then $\struct {T, *}$ is also a group.


Continuity of Composite with Inclusion

Let $T = \left({A, \tau}\right)$ and $T' = \left({A', \tau'}\right)$ be topological spaces. Let $H \subseteq A$. Let $T_H = \left({H, \tau_H}\right)$ be a topological subspace of $T$. Let $i: H \to A$ be the inclusion mapping. Let $f: A \to A'$ and $g: A' \to H$ be mappings. Then the following apply:


Totally Bounded Metric Space is Bounded

Let $M = \struct {A, d}$ be a totally bounded metric space. Then $M$ is bounded.


Convergent Subsequence of Cauchy Sequence

Let $\left({A, d}\right)$ be a metric space. Let $\left\langle{x_n}\right\rangle_{n \in \N}$ be a Cauchy sequence in $A$.  Let $x \in A$. Then $\left\langle{x_n}\right\rangle$ converges to $x$ {{iff}} it has a subsequence that converges to $x$.


Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets

Let $X$ be a set. Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$. Then $\map \sigma {\mathcal G}$, the $\sigma$-algebra generated by $\mathcal G$, exists and is unique.


Projection from Product Topology is Continuous

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are are continuous.


Continuous Mapping to Topological Product

Let $T = T_1 \times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Let $T'$ be a topological space. Let $f: T' \to T$ be a mapping. Then $f$ is continuous {{iff}} $\pr_1 \circ f$ and $\pr_2 \circ f$ are continuous.


Open and Closed Sets in Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Then $S$ and $\O$ are both both open and closed in $T$.


Absolutely Continuous Real Function is Continuous

Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $f$ is continuous.


Closed Set in Topological Subspace

Let $T$ be a topological space. Let $T' \subseteq T$ be a subspace of $T$. Then $V \subseteq T'$ is closed in $T'$ {{iff}} $V = T' \cap W$ for some $W$ closed in $T$.


Continuity Defined from Closed Sets

Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \to T_2$ be a mapping. Then $f$ is continuous {{iff}} for all $V$ closed in $T_2$, $f^{-1} \sqbrk V$ is closed in $T_1$.


Real Numbers are Uncountable

The set of real numbers $\R$ is uncountably infinite.


Condition for Point being in Closure

Let $T$ be a topological space. Let $H \subseteq T$. Let $x \in T$. Then $x \in H^-$ {{iff}} every open set of $T$ which contains $x$ contains a point in $H$.


Topological Closure of Subset is Subset of Topological Closure

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq K$ and $K \subseteq S$. Then: :$\map \cl H \subseteq \map \cl K$ where $\map \cl H$ denotes the closure of $H$.


Closure of Topological Closure equals Closure

Let $T$ be a topological space. Let $H \subseteq T$. Then: : $\left({H^-}\right)^- = H^-$ where $H^-$ denotes the closure of $H$.


Topological Closure is Closed

Let $T$ be a topological space. Let $H \subseteq T$. Then the closure $\map \cl H$ of $H$ is closed in $T$.


Equivalence of Definitions of Closure of Topological Subspace

{{TFAE|def = Closure (Topology)|view = Closure|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$.


Closure of Intersection is Subset of Intersection of Closures

Let $T$ be a topological space. Let $I$ be an indexing set. Let $\forall i \in I: H_i \subseteq T$. Then: : $\displaystyle \left({\bigcap_I H_i}\right)^- \subseteq \bigcap_I H_i^-$ where $H_i^-$ denotes the closure of $H_i$.


Closure of Finite Union equals Union of Closures

Let $T$ be a topological space. Let $n \in \N$. Let: :$\forall i \in \set {1, 2, \ldots, n}: H_i \subseteq T$ Then: :$\displaystyle \bigcup_{i \mathop = 1}^n \map \cl {H_i} = \map \cl {\bigcup_{i \mathop = 1}^n H_i}$


Closure of Open Ball in Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M$. Let $y \in \map \cl {\map {B_\epsilon} x}$, where $\cl$ denotes the closure of $\map {B_\epsilon} x$. Then $\map d {x, y} \le \epsilon$.


Nowhere Dense iff Complement of Closure is Everywhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Then $H$ is nowhere dense in $T$ {{iff}} $S \setminus H^-$ is everywhere dense in $T$, where $H^-$ denotes the closure of $H$.


Totally Bounded Metric Space is Separable

A totally bounded metric space is separable.


Convergent Sequence in Hausdorff Space has Unique Limit

Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $\sequence {x_n}$ be a convergent sequence in $T$. Then $\sequence {x_n}$ has exactly one limit.


Metric Space is Hausdorff

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a Hausdorff space.


Convergent Real Sequence has Unique Limit

Let $\sequence {s_n}$ be a real sequence. Then $\sequence {s_n}$ can have at most one limit.


Distinct Points in Metric Space have Disjoint Open Balls

Let $M = \left({A, d}\right)$ be a metric space. Let $x, y \in M: x \ne y$. Then there exist disjoint open $\epsilon$-balls $B_\epsilon \left({x}\right)$ and $B_\epsilon \left({y}\right)$ containing $x$ and $y$ respectively.


Sequentially Compact Metric Space is Lindelöf

Let $M$ be a metric space. Let $M$ be sequentially compact. Then $M$ is also a Lindelöf space. That is, from every open cover of $M$, it is possible to extract a countable subcover.


Sequentially Compact Metric Space is Compact

A sequentially compact metric space is compact.


Compact Subspace of Metric Space is Bounded

Let $M = \left({A, d}\right)$ be a metric space. Let $C$ be a subspace of $M$. If $C$ is compact, then it is bounded.


Compact Subspace of Hausdorff Space is Closed

Let $H = \struct {A, \tau}$ be a Hausdorff space. Let $C$ be a compact subspace of $H$. Then $C$ is closed in $H$.


Continuous Image of Compact Space is Compact

Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \to T_2$ be a continuous mapping. If $T_1$ is compact then so is its image $f \sqbrk {T_1}$ under $f$. That is, compactness is a continuous invariant.


Closed Subspace of Compact Space is Compact

A closed subspace of a compact space is compact. That is, the property of being compact is weakly hereditary.


Closure of Real Interval is Closed Real Interval

Let $I$ be a non-empty real interval such that one of these holds: : $I = \left({a \,.\,.\, b}\right)$ : $I = \left[{a \,.\,.\, b}\right)$ : $I = \left({a \,.\,.\, b}\right]$ : $I = \left[{a \,.\,.\, b}\right]$ Let $I^-$ denote the closure of $I$. Then $I^-$ is the closed real interval $\left[{a \,.\,.\, b}\right]$.


Topological Product of Compact Spaces

Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Then $T_1 \times T_2$ is compact {{iff}} both $T_1$ and $T_2$ are compact.


Heine-Cantor Theorem

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $M_1$ be compact. Let $f: A_1 \to A_2$ be a continuous mapping. Then $f$ is uniformly continuous.


Continuous Function on Closed Interval is Uniformly Continuous

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a continuous function. Then $f$ is uniformly continuous on $\closedint a b$.


Sine of Complement equals Cosine

:$\sin \left({\dfrac \pi 2 - \theta}\right) = \cos \theta$


Cosine of Complement equals Sine

:$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$


Continuous Bijection from Compact to Hausdorff is Homeomorphism

Let $T_1$ be a compact space. Let $T_2$ be a Hausdorff space. Let $f: T_1 \to T_2$ be a continuous bijection. Then $f$ is a homeomorphism.


Indirect Proof

Let $P$ be a proposition whose truth value is to be proved (either true or false). There are two aspects to this:


Continuous Mapping of Separation

Let $T$ and $T'$ be topological spaces. Let $A \mid B$ be a separation of $T$. Let $f: T \to T'$ be a mapping such that the restrictions $f \restriction_A$ and $f \restriction_B$ are both continuous. Then $f$ is continuous on the whole of $T$.


Continuity from Union of Restrictions

Let $T_1$ and $T_2$ be topological spaces. Then $f: T_1 \to T_2$ is continuous if either: : $\displaystyle T_1 = \bigcup_{i \mathop = 1}^n V_i$ where: :: each $V_i$ is closed in $T_1$, and  :: the restriction $f \restriction_{V_i}$ is continuous for each $V_i$ or: : $\displaystyle T_1 = \bigcup_{i \mathop \in I} U_i$ where: :: $I$ is any (possibly infinite) indexing set; :: each $U_i$ is open in $T_1$, and :: the restriction $f \restriction_{U_i}$ is continuous for each $U_i$.


Fermat's Last Theorem

:$\forall a, b, c, n \in \Z_{>0}, \; n > 2$, the equation $a^n + b^n = c^n$ has no solutions.


Subset of Real Numbers is Interval iff Connected

Let the real number line $\R$ be considered as a topological space. Let $S$ be a subspace of $\R$. Then $S$ is connected {{iff}} $S$ is an interval of $\R$. That is, the only subspaces of $\R$ that are connected are intervals.


Continuous Image of Connected Space is Connected

Let $T_1$ and $T_2$ be topological spaces. Let $S_1 \subseteq T_1$ be connected. Let $f: T_1 \to T_2$ be a continuous mapping. Then the image $f \left({S_1}\right)$ is connected.


Union of Connected Sets with Non-Empty Intersections is Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $I$ be an indexing set. Let $\AA = \family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $\AA$ be such that no two of its elements are disjoint: :$\forall B, C \in \AA: B \cap C \ne \O$ Then $\displaystyle \bigcup \AA$ is itself connected.


Finite Product Space is Connected iff Factors are Connected/Basis for the Induction

Let $T_1$ and $T_2$ be topological spaces. Then the product space $T_1 \times T_2$ is connected {{iff}} $T_1$ and $T_2$ are connected.


Set between Connected Set and Closure is Connected

Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H \subseteq K \subseteq H^-$, where $H^-$ denotes the closure of $H$.  Then $K$ is connected.


Path-Connected Space is Connected

Let $T$ be a topological space which is path-connected. Then $T$ is connected.


Hausdorff Paradox

There is a disjoint decomposition of the sphere $\mathbb S^2$ into four sets $A, B, C, D$ such that $A, B, C, B \cup C$ are all congruent and $D$ is countable.


Closed Topologist's Sine Curve is not Path-Connected

Let $G$ be the graph of the function $y = \sin \left({\dfrac 1 x}\right)$ for $x > 0$. Let $J$ be the line segment joining the points $\left({0, -1}\right)$ and $\left({0, 1}\right)$ in $\R^2$. Then while $G \cup J$ is connected, it is '''not''' path-connected.


Sum of Two Sides of Triangle Greater than Third Side

Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side. {{:Euclid:Proposition/I/20}}


Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides

Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle. {{:Euclid:Proposition/I/21}}


Construction of Triangle from Given Lengths

Given three straight lines such that the sum of the lengths of any two of the lines is greater than the length of the third line, it is possible to construct a triangle having the lengths of these lines as its side lengths. {{:Euclid:Proposition/I/22}}


Connected Open Subset of Euclidean Space is Path-Connected

Let $\R^n$ be a Euclidean $n$-space. Let $U$ be a connected open subset of $\R^n$. Then $U$ is path-connected.


Joining Paths makes Another Path

Let $T$ be a topological space. Let $I \subseteq \R$ be the closed real interval $\closedint 0 1$. Let $f, g: I \to T$ be paths in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: I \to T$ be the mapping given by: $\map h x = \begin{cases} \map f {2x} & : x \in \closedint 0 {\dfrac 1 2} \\ \map g {2x - 1} & : x \in \closedint {\dfrac 1 2} 1 \end{cases}$ Then $h$ is a path in $T$.


Hinge Theorem

If two triangles have two pairs of sides which are the same length, the triangle with the larger included angle also has the larger third side. {{:Euclid:Proposition/I/24}}


Equivalence of Definitions of Component

{{TFAE|def = Component (Topology)|view = Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$.


Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed

Let $M$ be a metric space. Let $C \subseteq M$ be a subspace of $M$ which is sequentially compact in $M$. Then $C$ is sequentially compact in itself {{iff}} $C$ is closed in $M$.


Real Number Line is Complete Metric Space

The real number line $\R$ with the usual (Euclidean) metric forms a complete metric space.


Series Law for Extremal Length

Let $X$ be a Riemann surface. Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every $\gamma \in \Gamma$ contain a $\gamma_1 \in \Gamma_1$ and a $\gamma_2 \in \Gamma_2$ such that $\gamma_1 \cap \gamma_2 = \varnothing$. Then the extremal lengths of $\Gamma_1$, $\Gamma_2$ and $\Gamma$ satisfy: :$\lambda \left({\Gamma}\right) \ge \lambda \left({\Gamma_1}\right) + \lambda \left({\Gamma_2}\right)$


Extremal Length of Union

Let $X$ be a Riemann surface. Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.  Then the extremal length of their union satisfies: :$\dfrac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)} \le \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$ Suppose that additionally $\Gamma_1$ and $\Gamma_2$ are disjoint in the following sense: there exist disjoint Borel subsets: :$A_1, A_2 \subseteq X$ such that $\displaystyle \bigcup \Gamma_1 \subset A_1$ and $\displaystyle \bigcup \Gamma_2 \subset A_2$ Then :$\dfrac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)} = \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$


Comparison Principle for Extremal Length

Let $X$ be a Riemann surface. Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every element of $\Gamma_1$ contain some element of $\Gamma_2$. Then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by: :$\lambda \left({\Gamma_1}\right) \ge \lambda \left({\Gamma_2}\right)$ More precisely, for every conformal metric $\rho$ as in the definition of extremal length, we have: :$L \left({\Gamma_1, \rho}\right) \ge L \left({\Gamma_2, \rho}\right)$


Reverse Triangle Inequality

Let $M = \struct {X, d}$ be a metric space. Then: :$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$ === Normed Division Ring === {{:Reverse Triangle Inequality/Normed Division Ring}} === Normed Vector Space === {{:Reverse Triangle Inequality/Normed Vector Space}} === Real and Complex Numbers === {{:Reverse Triangle Inequality/Real and Complex Fields}}


Inverse Completion of Integral Domain Exists

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Then an inverse completion of $\struct {D, \circ}$ can be constructed.


Zero of Inverse Completion of Integral Domain

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$ as defined in Inverse Completion of Integral Domain Exists. Let $x \in K: x = \dfrac p q$ such that $p = 0_D$. Then $x$ is equal to the zero of $K$. That is, ''any'' element of $K$ of the form $\dfrac {0_D} q$ acts as the zero of $K$.


Inverse Completion Less Zero of Integral Domain is Closed

Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\left({K, \circ}\right)$ be the inverse completion of $\left({D, \circ}\right)$. Then $\left({K^*, \circ}\right)$ is closed, where $K^* = K \setminus \left\{{0_K}\right\}$.


Composition of Mappings is Associative

The composition of mappings is an associative binary operation: :$\paren {f_3 \circ f_2} \circ f_1 = f_3 \circ \paren {f_2 \circ f_1}$ where $f_1, f_2, f_3$ are arbitrary mappings which fulfil the conditions for the relevant compositions to be defined.


Group Abelian iff Cross Cancellation Property

Let $G$ be a group. Then the following are equivalent: :$(1): \quad G$ is abelian :$(2): \quad G$ has the cross cancellation property


Group Abelian iff Middle Cancellation Property

Let $G$ be a group. Then the following are equivalent: :$(1): \quad G$ is abelian :$(2): \quad G$ satisfies the middle cancellation property


Rational Numbers form Ordered Field

The set of rational numbers $\Q$ forms an ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$.


Rational Numbers are Close Packed

Let $a, b \in \Q$ such that $a < b$. Then $\exists c \in \Q: a < c < b$. That is, the set of rational numbers is close packed.


Real Addition is Closed

The set of real numbers $\R$ is closed under addition: :$\forall x, y \in \R: x + y \in \R$


Real Multiplication is Well-Defined

The operation of multiplication on the set of real numbers $\R$ is well-defined.


Real Multiplication is Closed

The operation of multiplication on the set of real numbers $\R$ is closed: :$\forall x, y \in \R: x \times y \in \R$


Real Addition is Associative

The operation of addition on the set of real numbers $\R$ is associative: :$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$


Real Multiplication is Associative

The operation of multiplication on the set of real numbers $\R$ is associative: :$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$


Real Addition is Commutative

The operation of addition on the set of real numbers $\R$ is commutative: :$\forall x, y \in \R: x + y = y + x$


Real Multiplication is Commutative

The operation of multiplication on the set of real numbers $\R$ is commutative: :$\forall x, y \in \R: x \times y = y \times x$


Equidecomposable Nested Sets

Let $A, B, C$ be sets such that $A$ and $C$ are equidecomposable and $A \subseteq B \subseteq C$. Then $B$ and $C$ are equidecomposable.


Rational Numbers form Metric Space

Let $\Q$ be the set of all rational numbers. Let $d: \Q \times \Q \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\Q$ and so $\struct {\Q, d}$ is a metric space.


Rational Numbers form Vector Space

Let $\Q$ be the set of rational numbers. Then the $\Q$-module $\Q^n$ is a vector space. It follows directly, by setting $n = 1$, that the $\Q$-module $\Q$ itself can also be regarded as a vector space.


Five Color Theorem

A planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.


Limit of (Cosine (X) - 1) over X

:$\displaystyle \lim_{x \mathop \to 0} \frac {\map \cos x - 1} x = 0$


Real Multiplication Distributes over Addition

The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition: :$\forall x, y, z \in \R:$ ::$x \times \paren {y + z} = x \times y + x \times z$ ::$\paren {y + z} \times x = y \times x + z \times x$


Cardano's Formula

Let $P$ be the cubic equation: :$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$ Then $P$ has solutions: :$x_1 = S + T - \dfrac b {3 a}$ :$x_2 = - \dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}$ :$x_3 = - \dfrac {S + T} 2 - \dfrac b {3 a} - \dfrac {i \sqrt 3} 2 \paren {S - T}$ where: :$S = \sqrt [3] {R + \sqrt {Q^3 + R^2} }$ :$T = \sqrt [3] {R - \sqrt {Q^3 + R^2} }$ where: :$Q = \dfrac {3 a c - b^2} {9 a^2}$ :$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$


Ferrari's Method

Let $P$ be the quartic equation: :$a x^4 + b x^3 + c x^2 + d x + e = 0$ such that $a \ne 0$. Then $P$ has solutions: :$x = \dfrac {-p \pm \sqrt {p^2 - 8 q} } 4$ where: :$p = \dfrac b a \pm \sqrt {\dfrac {b^2} {a^2} - \dfrac {4 c} a + 4 y_1}$ :$q = y_1 \mp \sqrt {y_1^2 - \dfrac {4 e} a}$ where $y_1$ is a real solution to the cubic: :$y^3 - \dfrac c a y^2 + \paren {\dfrac {b d} {a^2} - \dfrac {4 e} a} y + \paren {\dfrac {4 c e} {a^2} - \dfrac {b^2 e} {a^3} - \dfrac {d^2} {a^2} } = 0$ '''Ferrari's method''' is a technique for solving this quartic.


Properties of Binomial Coefficients

This page gathers together some of the simpler and more common identities concerning binomial coefficients.


Prime Number Theorem

The prime-counting function $\pi \left({n}\right)$, that is, the number of primes less than $n$, satisfies: :$\displaystyle \lim_{n \mathop \to \infty} \map \pi n \frac {\map \ln n} n = 1$ or equivalently:  :$\map \pi n \sim \dfrac n {\map \ln n}$ where $\sim$ denotes asymptotic equivalence.


Induced Structure Inverse

Let $\struct {G, \oplus}$ be a group whose identity is $e_G$. Let $S$ be a set. Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$. Let $f \in G^S$. Let $f^* \in G^S$ be defined as follows: :$\forall f \in G^S: \forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$ Then $f^*$ is the inverse of $f$ for the pointwise operation induced on $G^S$ by $\oplus$.


Quaternion Group is Hamiltonian

The quaternion group $Q$ is Hamiltonian.


Finite Subset of Metric Space has no Limit Points

Let $M = \left({A, d}\right)$ be a metric space. Let $X \subseteq A$ such that $X$ is finite. Then $X$ has no limit points.


Point in Finite Metric Space is Isolated

Let $M = \struct {A, d}$ be a metric space. Let $X \subseteq A$ such that $X$ is finite. Let $x \in X$. Then $x$ is isolated in $X$.


Point in Finite Hausdorff Space is Isolated

Let $T = \left({S, \tau}\right)$ be a Hausdorff space. Let $X \subseteq S$ such that $X$ is finite. Let $x \in X$. Then $x$ is isolated in $X$.


Subsequence of Sequence in Metric Space with Limit

Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Let $x$ be a limit point of $S = \set {x_n: n \in \N}$, the set of members of $\sequence {x_n}$. Then $\sequence {x_n}$ has a subsequence which converges to $x$.


Primes of form Power Less One

Let $m, n \in \N_{>0}$ be natural numbers. Let $m^n - 1$ be prime. Then $m = 2$ and $n$ is prime.


Lebesgue's Number Lemma

Let $M = \struct {A, d}$ be a metric space. Let $M$ be sequentially compact. Then there exists a Lebesgue number for every open cover of $M$.


Second Subsequence Rule

Let $M = \left({A, d}\right)$ be a metric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$. Suppose $\left \langle {x_n} \right \rangle$ has a subsequence which is unbounded. Then $\left \langle {x_n} \right \rangle$ is divergent.


Open Set Less One Point is Open

Let $M = \left({A, d}\right)$ be a metric space. Let $U \subseteq M$ be an open set of $M$. Let $\alpha \in U$. Then $U \setminus \left\{{\alpha}\right\}$ is open in $M$.


Finite Subspace of Dense-in-itself Metric Space is not Open

Let $M = \struct {A, d}$ be a metric space that is dense-in-itself. Let $U$ be a finite subset of $A$. Then $U$ is not an open set of $M$.


Region Less One Point is Region

Let $M = \left({A, d}\right)$ be a dense-in-itself metric space. Let $R \subseteq M$ be a region of $M$. Let $x \in R$. Then $R \setminus \left\{ {x}\right\}$ is also a region of $M$.


Theorem of Even Perfect Numbers

Let $a \in \N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \paren {2^n - 1}$ where $2^n - 1$ is prime. Similarly, if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect.


Lucas-Lehmer Test

Let $q$ be an odd prime. Let $\sequence {L_n}_{n \mathop \in \N}$ be the recursive sequence in $\Z / \paren {2^q - 1} \Z$ defined by: :$L_0 = 4, L_{n + 1} = L_n^2 - 2 \pmod {2^q - 1}$ Then $2^q - 1$ is prime {{iff}} $L_{q - 2} = 0 \pmod {2^q - 1}$.


Sum Over Divisors Equals Sum Over Quotients

Let $n$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be a mapping on the positive integers. Let $\displaystyle \sum_{d \mathop \divides n} \map f d$ be the sum of $\map f d$ over the divisors of $n$. Then: :$\displaystyle \sum_{d \mathop \divides n} \map f d = \sum_{d \mathop \divides n} \map f {\frac n d}$.


Sum of Reciprocals of Divisors equals Abundancy Index

Let $n$ be a positive integer. Let $\map \sigma n$ denote the sigma function of $n$. Then: :$\displaystyle \sum_{d \mathop \divides n} \frac 1 d = \frac {\map \sigma n} n$ where $\dfrac {\map \sigma n} n$ is the abundancy index of $n$.


Sigma Function of Prime Number

Let $n$ be a positive integer. Let $\map \sigma n$ be the sigma function of $n$. Then $\map \sigma n = n + 1$ {{iff}} $n$ is prime.


Sigma Function of Power of Prime

Let $n = p^k$ be the power of a prime number $p$. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Then: :$\map \sigma n = \dfrac {p^{k + 1} - 1} {p - 1}$


Tau of Power of Prime

Let $n = p^k$ be the power of a prime number $p$. Let $\tau \left({n}\right)$ be the $\tau$ function of $n$. That is, let $\tau \left({n}\right)$ be the number of positive divisors of $n$. Then $\tau \left({n}\right) = k + 1$.


Sum Over Divisors of Multiplicative Function

Let $f: \Z_{>0} \to \Z_{>0}$ be a multiplicative function. Let $n \in \Z_{>0}$. Let $\displaystyle \sum_{d \mathop \divides n} \map f d$ be the sum over the divisors of $n$. Then $\displaystyle \map F n = \sum_{d \mathop \divides n} \map f d$ is also a multiplicative function.


Divisors of Product of Coprime Integers

Let $a \divides b c$, where $b \perp c$. Then $\tuple {r, s}$ satisfying: :$a = r s$, where $r \divides b$ and $s \divides c$ is unique up to absolute value with: :$\size r = \gcd \set {a, b}$ :$\size s = \gcd \set {a, c}$


Unity Function is Completely Multiplicative

Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the constant function: :$\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$ Then $f_1$ is completely multiplicative.


Identity Function is Completely Multiplicative

Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function: :$\forall n \in \Z_{>0}: I_{\Z_{>0}} \left({n}\right) = n$ Then $I_{\Z_{>0}}$ is completely multiplicative.


Tau Function is Multiplicative

The divisor counting function: :$\displaystyle \tau: \Z_{>0} \to \Z_{>0}: \map \tau n = \sum_{d \mathop \divides n} 1$ is multiplicative.


Sigma Function is Multiplicative

The sigma function: :$\displaystyle \sigma: \Z_{>0} \to \Z_{>0}: \sigma \left({n}\right) = \sum_{d \mathop \backslash n} d$ is multiplicative.


Primes of form Power of Two plus One

Let $n \in \N$ be a natural number. Let $2^n + 1$ be prime. Then $n = 2^k$ for some natural number $k$.


Integer has Multiplicative Order Modulo n iff Coprime to n

Let $a$ and $n$ be integers. Let the multiplicative order of $a$ modulo $n$ exist. Then $a \perp n$, that is, $a$ and $n$ are coprime.


Integer to Power of Multiple of Order

Let $a$ and $n$ be integers. Let $a \perp n$, that is, let $a$ and $b$ be coprime. Let $c \in \Z_{>0}$ be the multiplicative order of $a$ modulo $n$. Then $a^k \equiv 1 \pmod n$ {{iff}} $k$ is a multiple of $c$.


Sum of Infinite Geometric Sequence

Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$. Let $z \in S$. Let $\size z < 1$, where $\size z$ denotes: :the absolute value of $z$, for real and rational $z$ :the complex modulus of $z$ for complex $z$. Then $\displaystyle \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$.


Square of Riemann Zeta Function

:$\displaystyle \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map d k} {k^z}$ where: :$\zeta$ is the Riemann zeta function :$d$ is the divisor function.


Derivative of Riemann Zeta Function

The derivative of the Riemann zeta function is: :$\displaystyle \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$


Order of Divisor Function

For all $x \ge 1$: :$\displaystyle \sum_{n \mathop \le x} \map d n = x \log x + \paren {2 \gamma - 1} x + \map \OO {\sqrt x}$ where $\gamma$ is the Euler-Mascheroni constant and $\map d n$ is the divisor function. {{explain|Which particular divisor function? Suggest that it might be $\tau$ function that is being referred to.}}


Order of Möbius Function

Let $\mu$ denote the Möbius function . Then: :$\displaystyle \sum_{n \mathop \le N} \map \mu n = \map o N$ where $o$ denotes little-o notation.


Ingham's Theorem on Convergent Dirichlet Series

Let $\left\vert{a_n}\right\vert \le 1$ {{explain|What exactly is $a_n$ in this context?}} For a complex number $z \in \C$, let $\Re \left({z}\right)$ denote the real part of $z$.  Form the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-z}$ which converges to an analytic function $F \left({z}\right)$ for $\Re \left({z}\right) > 1$. {{explain|We have "$\Re \left({z}\right) > 1$" used here and below to mean $\left\{ {z \in \C: \Re \left({z}\right) > 1}\right\}$? In which case, rather than just call it "$\Re \left({z}\right) > 1$", which is unwieldy and suboptimal, it might be better and clearer to give it a single-symbol identifier.}} Let $F \left({z}\right)$ be analytic throughout $\Re \left({z}\right) \ge 1$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-z}$ converges throughout $\Re \left({z}\right) \ge 1$.


Alexander Polynomial is a Knot Invariant

The Alexander polynomial of an elementary knot $K$ is invariant under Reidemeister moves.


Product Form of Sum on Completely Multiplicative Function

Let $f$ be a completely multiplicative arithmetic function. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \map f n$ be absolutely convergent. Then: :$\displaystyle \sum_{n \mathop = 1}^\infty \map f n = \prod_p \frac 1 {1 - \map f p}$ where the infinite product ranges over the primes.


Euler Phi Function of Product with Prime

Let $p$ be prime and $n \in \Z: n \ge 1$. Then: :$\phi \left({p n}\right) = \begin{cases} \left({p - 1}\right) \phi \left({n}\right) & : p \nmid n \\ p \phi \left({n}\right) & : p \mathrel \backslash n  \end{cases}$ where $\phi \left({n}\right)$ denotes the Euler $\phi$ function of $n$. Thus for all $n \ge 1$ and for any prime $p$, we have that $\phi \left({n}\right)$ divides $\phi \left({p n}\right)$.


Euler Phi Function is Even for Argument greater than 2

Let $n \in \Z: n \ge 1$. Let $\map \phi n$ be the Euler $\phi$ function of $n$. Then $\map \phi n$ is even {{iff}} $n > 2$.


Cauchy-Goursat Theorem

Let $D$ be a simply connected open subset of the complex plane $\C$. Let $\partial D$ denote the closed contour bounding $D$. Let $f: D \to \C$ be holomorphic everywhere in $D$. Then: :$\displaystyle \oint_{\partial D} \map f z \rd z = 0$


Sigma Function of Integer

Let $n$ be an integer such that $n \ge 2$. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ Then: :$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$


Product of Sigma and Euler Phi Functions

Let $n$ be an integer such that $n \ge 2$. Let the prime decomposition of $n$ be: :$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$ Let $\sigma \left({n}\right)$ be the sigma function of $n$. Let $\phi \left({n}\right)$ be the Euler phi function of $n$. Then: :$\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = n^2 \prod_{1 \mathop \le i \mathop \le r} \left({1 - \frac 1 {p_i^{k_i + 1}}}\right)$


Order Modulo n of Power of Integer

Let $a$ have multiplicative order $c$ modulo $n$. Then for any $k \ge 1$, $a^k$ has multiplicative order $\dfrac c {\gcd \left\{{c, k}\right\}}$ modulo $n$.


Integers with Primitive Roots

Let $n \in \Z: n > 1$. Then $n$ has a primitive root {{iff}} one of the following holds: : $n = 2$ : $n = 4$ : $n = p^k$ : $n = 2 p^k$ where $p > 2$ is prime and $k \ge 1$.


Tau Function of Square-Free Integer is Power of 2

Let $n$ be a square-free integer. Let $\tau: \Z \to \Z$ be the $\tau$ function. Then $\map \tau n = 2^r$ for some $r \ge 1$. The converse is not true in general. That is, if $\map \tau n = 2^r$ for some $r \ge 1$, it is not necessarily the case that $n$ is square-free.


Tau Function Odd Iff Argument is Square

Let $\tau: \Z \to \Z$ be the $\tau$ function. Then $\tau \left({n}\right)$ is odd {{iff}} $n$ is square.


Sigma Function Odd iff Argument is Square or Twice Square

Let $\sigma: \Z \to \Z$ be the sigma function. Then $\sigma \left({n}\right)$ is odd {{iff}} $n$ is either square or twice a square.


Cancellability of Congruences

Let $a, b, c, n \in \Z$ be integers. Then: :$c a \equiv c b \pmod n \iff a \equiv b \pmod {n / d}$ where $d = \gcd \set {c, n}$.


Solution of Linear Diophantine Equation

The linear Diophantine equation: :$a x + b y  = c$ has solutions {{iff}}: :$\gcd \set {a, b} \divides c$ where $\divides$ denotes divisibility. If this condition holds with $\gcd \set {a, b} > 1$ then division by $\gcd \set {a, b}$ reduces the equation to: :$a' x + b' y = c'$ where $\gcd \set {a', b'} = 1$. If $x_0, y_0$ is one solution of the latter equation, then the general solution is: :$\forall k \in \Z: x = x_0 + b' k, y = y_0 - a' k$ or: :$\forall k \in \Z: x = x_0 + \dfrac b d k, y = y_0 - \dfrac a d k$


GCD with Prime

Let $p$ be a prime number. Then: :$\forall n \in \Z: \gcd \set {n, p} = \begin{cases} p & : p \divides n \\ 1 & : p \nmid n \end{cases}$


Prime Divides Power

Let $p$ be a prime number. Let $a, n \in \Z$ be integers. Then $p$ divides $a^n$ {{iff}} $p^n$ divides $a^n$.


Existence of Laurent Series

Let $z_0 \in \C$ be a complex number. Let $R \in \R_{>0}$ be a real number. Let $\map {B'} {z_0, R}$ be the open punctured disk at $z_0$ of radius $R$. Let $f: \map {B'} {z_0, R} \to \C$ be holomorphic. Then there exists a sequence $\sequence {a_n}_{n \mathop \in \Z}$ such that: :$\map f z = \displaystyle \sum_{n = -\infty}^\infty a_n \paren {z - z_0}^n$ for all $z \in B'(z_0, R)$.


Congruence by Divisor of Modulus

Let $z \in \R$ be a real number. Let $a, b \in \R$ such that $a$ is congruent modulo $z$ to $b$, that is: :$a \equiv b \pmod z$ Let $m \in \R$ such that $z$ is an integer multiple of $m$: :$\exists k \in \Z: z = k m$ Then: : $a \equiv b \pmod m$


Congruence of Quotient

Let $a, b \in \Z$ and $n \in \N$. Let $a$ be congruent to $b$ modulo $n$, i.e. $a \equiv b \pmod n$. Let $d \in \Z: d > 0$ such that $d$ is a common divisor of $a, b$ and $n$. Then: : $\displaystyle \frac a d \equiv \frac b d \pmod {n / d}$


Divisibility by 11

Let $N \in \N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ Then $N$ is divisible by $11$ {{iff}} $\displaystyle \sum_{r \mathop = 0}^n \paren {-1}^r a_r$ is  divisible by $11$. That is, a divisibility test for $11$ is achieved by alternately adding and subtracting the digits and taking the result modulo $11$.


Solutions of Polynomial Congruences

Let $P \left({x}\right)$ be an integral polynomial. Let $a \equiv b \pmod n$. Then $P \left({a}\right) \equiv P \left({b}\right) \pmod n$. In particular, $a$ is a solution to the polynomial congruence $P \left({x}\right) \equiv 0 \pmod n$ iff $b$ is also.


Solution of Linear Congruence

Let $a x \equiv b \pmod n$ be a linear congruence. The following results hold:


Chinese Remainder Theorem/General Result

Let $n_1, n_2, \ldots, n_r$ be positive integers such that $n_i \perp n_j$ for all $i \ne j$ (that is, $\gcd \set {n_i, n_j} = 1$). Then the system of linear congruences: :$x \equiv b_1 \pmod {n_1}$ :$x \equiv b_2 \pmod {n_2}$ :$\ldots$ :$x \equiv b_r \pmod {n_r}$ has a simultaneous solution which is unique modulo $\displaystyle \prod_{i \mathop = 1}^r n_i$.


Second Supplement to Law of Quadratic Reciprocity

:$\paren {\dfrac 2 p} = \paren {-1}^{\paren {p^2 - 1} / 8} = \begin{cases} +1 & : p \equiv \pm 1 \pmod 8 \\ -1 & : p \equiv \pm 3 \pmod 8 \end{cases}$


First Supplement to Law of Quadratic Reciprocity

:$\paren {\dfrac {-1} p} = \paren {-1}^{\paren {p - 1} / 2} = \begin{cases} +1 & : p \equiv 1 \pmod 4 \\ -1 & : p \equiv 3 \pmod 4 \end{cases}$


Manipulation of Absolutely Convergent Series

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a real or complex series that is absolutely convergent.


Product of Sums

Let $\displaystyle \sum_{n \mathop \in A} a_n$ and $\displaystyle \sum_{n \mathop \in B} b_n$ be absolutely convergent sequences. Then: :$\displaystyle \left({ \sum_{i \mathop \in A} a_i }\right) \left({ \sum_{j \mathop \in B} b_j }\right) = \sum_{\left({i, j}\right) \mathop \in A \times B} a_i b_j$.


Green's Theorem

Let $\Gamma$ be a positively oriented piecewise smooth simple closed curve in  $\R^2$. Let $U = \map {\operatorname {Int} } \Gamma$, that is, the interior of $\Gamma$. Let $A$ and $B$ be functions of $\tuple {x, y}$ defined on an open region containing $U$ and have continuous partial derivatives in such a set. Then: :$\displaystyle \oint_\Gamma \paren {A \rd x + B \rd y} = \iint_U \paren {\frac {\partial B} {\partial x} - \frac {\partial A} {\partial y} } \rd x \rd y$


Gauss-Ostrogradsky Theorem

Let $U$ be a subset of $\R^3$ which is compact and has a piecewise smooth boundary $\partial U$. Let $\mathbf F: \R^3 \to \R^3$ be a smooth vector function defined on a neighborhood of $U$. Then: :$\displaystyle \iiint \limits_U \paren {\nabla \cdot \mathbf F} \rd V = \iint \limits_{\partial U} \mathbf F \cdot \mathbf n \rd S$ where $\mathbf n$ is the normal to $\partial U$.


Implications of Stokes' Theorem

Stokes' Theorem implies all of the following results: * Classical Stokes' Theorem * Green's Theorem * Gauss's Theorem * Fundamental Theorem of Calculus * Cauchy's Residue Theorem


Divergence Test

Let $\sequence {a_n}$ be a sequence in $\R$. If $\displaystyle \lim_{k \mathop \to \infty} a_k \ne 0$, then $\displaystyle \sum_{i \mathop = 1}^\infty a_n$ diverges.


Eisenstein's Lemma

Let $p$ be an odd prime. Let $a \in \Z$ be an odd integer such that $p \nmid a$. Let $\paren {\dfrac a p}$ be defined as the Legendre symbol. Then: :$\paren {\dfrac a p} = \paren {-1}^{\map \alpha {a, p} }$ where: :$\displaystyle \map \alpha {a, p} = \sum_{k \mathop = 1}^{\frac {p - 1} 2} \floor {\frac {k a} p}$ :$\floor {\dfrac {k a} p}$ is the floor function of $\dfrac {k a} p$.


Law of Quadratic Reciprocity

Let $p$ and $q$ be distinct odd primes. Then: :$\paren {\dfrac p q} \paren {\dfrac q p} = \paren {-1}^{\dfrac {\paren {p - 1} \paren {q - 1} } 4}$ where $\paren {\dfrac p q}$ and $\paren {\dfrac q p}$ are defined as the Legendre symbol. An alternative formulation is: $\paren {\dfrac p q} = \begin{cases} \quad \paren {\dfrac q p} & : p \equiv 1 \lor q \equiv 1 \pmod 4 \\ -\paren {\dfrac q p} & : p \equiv q \equiv 3 \pmod 4 \end{cases}$ The fact that these formulations are equivalent is immediate. This fact is known as the '''Law of Quadratic Reciprocity''', or '''LQR''' for short.


Rational Numbers and SFCFs are Equivalent

Every simple finite continued fraction has a rational value. Conversely, every rational number can be expressed as a simple finite continued fraction.


Properties of Convergents of Continued Fractions

Let $n \in \N \cup \set \infty$ be an extended natural number. Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $\R$ of length $n$. Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators. Let $C_0, C_1, C_2, \ldots$ be the convergents of $\sqbrk {a_0, a_1, a_2, \ldots}$. Then the following results apply:


Manipulation of Exterior Derivative

For the exterior derivative, the following statements are true: :$\map \d {a b} = a \rd b + \paren {\d a} b$ :$\map \d {a \wedge b} = \d a \wedge b - a \wedge \d b$ where $a \wedge b$ is the wedge product.


General Stokes' Theorem

Let $\omega$ be a smooth $\paren {n - 1}$-form with compact support on a smooth $n$-dimensional oriented manifold $X$. Let the boundary of $X$ be $\partial X$. Then: :$\displaystyle \int_{\partial X} \omega = \int_X \rd \omega$ where $\d \omega$ is the exterior derivative of $\omega$.


Kelvin-Stokes Theorem

Let $S$ be some orientable smooth surface with boundary in $\R^3$. Let $\mathbf F:\R^3 \to \R^3$ be a vector-valued function with Euclidean coordinate expression: :$F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$ where $f_i: \R^3 \to \R$. Then: :$\displaystyle \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$ where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.


Even Convergent of Simple Continued Fraction is Strictly Smaller than Odd Convergent

Every even convergent is strictly smaller than every odd convergent.


Denominators of Simple Continued Fraction are Strictly Increasing

Let $N \in \N \cup \set \infty$ be an extended natural number. Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a simple continued fraction in $\R$ of length $N$. Let $q_0, q_1, q_2, \ldots$ be its denominators. Then with the possible exception of $q_0 = q_1$, the sequence $\sequence {q_n}$ is strictly increasing.


Simple Infinite Continued Fraction Converges to Irrational Number

The value of any simple infinite continued fraction in $\R$ is irrational.


Uniqueness of Analytic Continuation

Let $U \subset V \subset \C$ be open subsets of the complex plane. Let $V$ be connected. Suppose: :$(1): \quad F_1, F_2$ are functions defined on $V$ :$(2): \quad f$ is a function defined on $U$. Let $F_1$ and $F_2$ be analytic continuations of $f$ to $V$. Then $F_1 = F_2$.


Zeroes of Analytic Function are Isolated

Let $U \subset \C$ be some open set and let $f$ be an analytic function defined on $U$. Then either $f$ is a constant function, or the set $\set {z \in U: \map f z = 0}$ is totally disconnected.


Irrational Number is Limit of Unique Simple Infinite Continued Fraction

Let $x$ be an irrational number. Then the continued fraction expansion of $x$ is the unique simple infinite continued fraction that converges to $x$.


Analytic Continuation of Riemann Zeta Function

The Riemann zeta function is meromorphic on $\C$.


Accuracy of Convergents of Continued Fraction Expansion of Irrational Number

Let $x$ be an irrational number. Let $(a_0, a_1, \ldots)$ be its continued fraction expansion. Let $\left \langle {C_n}\right \rangle_{n \geq 0}$ be its sequence of convergents. Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators. Then: :$\forall k \ge 1: \left\vert{x - \dfrac {p_{k + 1} } {q_{k + 1} } }\right\vert < \dfrac 1 {q_{k + 1} q_{k + 2} } \le \dfrac 1 {2 q_k q_{k + 1} } < \left\vert{x - \dfrac {p_k} {q_k} }\right\vert$ Thus: :The {{LHS}} of the inequality gives an indication of how close each convergent gets to its true value. :The {{RHS}} gives a bound that limits its accuracy.


Trivial Zeroes of Riemann Zeta Function are Even Negative Integers

Let $\rho = \sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip: :$0 \le \map \Re s \le 1$ Then: :$s \in \set {-2, -4, -6, \ldots}$ These are called the '''trivial zeros''' of $\zeta$.


Poles of Gamma Function

The gamma function $\Gamma: \C \to \C$ is analytic throughout the complex plane except at $\set {0, -1, -2, -3, \dotsc}$ where it has simple poles.


Zeroes of Gamma Function

The Gamma function is never equal to $0$.


Gamma Difference Equation

:$\map \Gamma {z + 1} = z \, \map \Gamma z$


Convergents are Best Approximations

Let $x$ be an irrational number. Let $(p_n)_{n\geq0}$ and $(q_n)_{n\geq0}$ be the numerators and denominators of its continued fraction expansion. Let $\dfrac {p_n} {q_n}$ be the $n$th convergent. Let $\dfrac a b$ be any rational number such that $0 < b < q_{n+1}$. Then: :$\forall n > 1: \left\vert{q_n x - p_n}\right\vert \le \left\vert{b x - a}\right\vert$ The equality holds only if $a = p_n$ and $b = q_n$.


Condition for Rational to be a Convergent

Let $x$ be an irrational number. Let the rational number $\dfrac a b$ satisfy the inequality: :$\left\vert{x - \dfrac a b}\right\vert < \dfrac 1 {2 b^2}$ Then $\dfrac a b$ is a convergent of $x$.


Quadratic Irrational is Root of Quadratic Equation

Let $x$ be a quadratic irrational. Then $x$ is a solution to a quadratic equation with rational coefficients.


Irrational Number has Periodic Continued Fraction iff Quadratic

Let $x$ be an irrational number. Then $x$ is a quadratic irrational {{iff}} its continued fraction expansion is periodic.


Continued Fraction Expansion of Irrational Square Root

Let $n \in \Z$ such that $n$ is not a square. Then the continued fraction expansion of $\sqrt n$ is of the form: :$\sqbrk {a_1 \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1} }$ or :$\sqbrk {a_1 \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1} }$ where $m \in \Z: m \ge 0$. That is, it has the form as follows: :It is periodic :It starts with an integer $a_1$ :Its cycle starts with a palindromic section, either: :::$b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1$ ::or: :::$b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1$ ::which may be of length zero :Its cycle ends with twice the first partial quotient.


Solution of Pell's Equation is a Convergent

Let $x = a, y = b$ be a positive solution to Pell's Equation $x^2 - n y^2 = 1$. Then $\dfrac a b$ is a convergent of $\sqrt n$.


Convergence of P-Series

Let $p \in \C$ be a complex number.


Square Modulo 4

Let $x \in \Z$ be an integer. Then $x$ is: :even {{iff}} $x^2 \equiv 0 \pmod 4$ :odd {{iff}} $x^2 \equiv 1 \pmod 4$


Parity of Smaller Elements of Primitive Pythagorean Triple

Let $\left({x, y, z}\right)$ be a Pythagorean triple, that is, integers such that $x^2 + y^2 = z^2$. Then $x$ and $y$ are of opposite parity.


Dirichlet Series is Analytic

Let $(a_n)$ be sequence of complex numbers. Let :$\displaystyle f \left({z}\right) = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^z}$ be the associated Dirichlet Series, which is defined at the points where the series converges. Then $f$ is analytic in every open set such that the sum converges in the set.


Consecutive Integers are Coprime

$\forall h \in \Z$, $h$ and $h + 1$ have only two common factors: $1$ and $-1$. That is, consecutive integers are always coprime.


Elements of Primitive Pythagorean Triple are Pairwise Coprime

Let $\tuple {x, y, z}$ be a primitive Pythagorean triple. Then: :$x \perp y$ :$y \perp z$ :$x \perp z$ That is, all elements of $\tuple {x, y, z}$ are pairwise coprime.


Odd Square Modulo 8

Let $x \in \Z$ be an odd square. Then $x \equiv 1 \pmod 8$.


Sum of Consecutive Triangular Numbers is Square

The sum of two consecutive triangular numbers is a square number.


If n is Triangular then so is 9n + 1

Let $n$ be a triangular number. Then $9 n + 1$ is also triangular.


Sum of Arithmetic Sequence

Let $\sequence {a_k}$ be an arithmetic sequence defined as: :$a_k = a + k d$ for $n = 0, 1, 2, \ldots, n - 1$ Then its closed-form expression is: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d}       | r = n \paren {a + \frac {n - 1} 2 d}       | c =  }} {{eqn | r = \frac {n \paren {a + l} } 2       | c = where $l$ is the last term of $\sequence {a_k}$ }} {{end-eqn}}


Odd Square is Eight Triangles Plus One

Let $n \in \Z$ be an odd integer. Then $n$ is square {{iff}} $n = 8 m + 1$ where $m$ is triangular.


Triangular Number Modulo 3 and 9

Let $n$ be a triangular number. Then one of the following two conditions applies: : $n \equiv 0 \pmod 3$ : $n \equiv 1 \pmod 9$


Closed Form for Polygonal Numbers

Let $\map P {k, n}$ be the $n$th $k$-gonal number. The closed-form expression for $\map P {k, n}$ is given by: {{begin-eqn}} {{eqn | l = \map P {k, n}       | r = \frac n 2 \paren {\paren {k - 2} n - k + 4} }} {{eqn | r = \frac {k - 2} 2 \paren {n^2 - n} + n }} {{end-eqn}}


Difference Between Adjacent Polygonal Numbers is Triangular Number

Let $\map P {k, n}$ be the $n$th $k$-gonal number. Then: :$\map P {k + 1, n} - \map P {k, n} = T_{n - 1}$ where $T_n$ is the $n$th triangular number.


Method of Infinite Descent

Let $P \left({n_\alpha}\right)$ be a propositional function depending on $n_\alpha \in \N$. Let $P \left({n_\alpha}\right) \implies P \left({n_\beta}\right)$ such that $0 < n_\beta < n_\alpha$. Then we may deduce that $P \left({n}\right)$ is false for all $n \in \N$. That is, suppose that by assuming the truth of $P \left({n_\alpha}\right)$ for ''any'' natural number $n_\alpha$, we may deduce that there always exists ''some'' number $n_\beta$ strictly less than $n_\alpha$ for which $P \left({n_\beta}\right)$ is also true, then $P \left({n_\alpha}\right)$ can not be true after all. This technique is known as the '''method of infinite descent'''. The process of deducing the truth of $P \left({n_\beta}\right)$ from $P \left({n_\alpha}\right)$ such that $0 < n_\beta < n_\alpha$ is known as the '''descent step'''.


Brahmagupta-Fibonacci Identity

Let $a, b, c, d$ be numbers. Then: :$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c + b d}^2 + \paren {a d - b c}^2$


Fermat's Two Squares Theorem

Let $p$ be a prime number. Then $p$ can be expressed as the sum of two squares {{iff}} either: : $p = 2$ or: : $p \equiv 1 \pmod 4$ The expression of a prime of the form $4 k + 1$ as the sum of two squares is unique except for the order of the two summands.


Integer as Sum of Two Squares

Let $n$ be a positive integer. Then: : $n$ can be expressed as the sum of two squares {{iff}}: :each of its prime divisors of the form $4 k + 3$ (if any) occur to an even power.


Square Modulo 8

Let $x \in \Z$ be an integer. : If $x$ is even then: ::: $x^2 \equiv 0 \pmod 8$ or $x^2 \equiv 4 \pmod 8$ : If $x$ is odd then: ::: $x^2 \equiv 1 \pmod 8$


Integer as Sum of Three Squares

Let $r$ be a positive integer. Then $r$ can be expressed as the sum of three squares {{iff}} it is not of the form: :$4^n \paren {8 m + 7}$ for some $m, n \in \Z_{\ge 0}$.


Product of Sums of Four Squares

Let $a, b, c, d, w, x, y, z$ be numbers. Then: {{begin-eqn}} {{eqn | o =        | r = \left({a^2 + b^2 + c^2 + d^2}\right) \left({w^2 + x^2 + y^2 + z^2}\right)       | c =  }} {{eqn | l = =       | o =        | r = \left({a w + b x + c y + d z}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a x - b w + c z - d y}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a y - b z - c w + d x}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a z + b y - c x - d w}\right)^2       | c =  }} {{end-eqn}}


Lagrange's Four Square Theorem

Every positive integer can be expressed as a sum of four squares.


Hilbert-Waring Theorem

For each $k \in \Z: k \ge 2$, there exists a positive integer $\map g k$ such that every positive integer can be expressed as a sum of at most $\map g k$ $k$th powers.


Necessary Condition for Existence of BIBD

Let there exist be a BIBD with parameters $v, b, r, k, \lambda$. Then the following are true: : $(1): \quad b k = r v$ : $(2): \quad \lambda \paren {v - 1} = r \paren {k - 1}$ : $(3): \quad b \dbinom k 2 = \lambda \dbinom v 2$ : $(4): \quad k < v$ : $(5): \quad r > \lambda$ All of $v, b, r, k, \lambda$ are integers. Some sources prefer to report $(3)$ as: :$b = \dfrac {\dbinom v 2} {\dbinom k 2} \lambda$ which is less appealing visually, and typographically horrendous.


Mapping from Singleton is Injection

Let $f: S \to T$ be a mapping. Let $S$ be a singleton. Then $f$ is an injection.


Mapping to Singleton is Surjection

Let $S$ be a non-empty set. Let $f: S \to T$ be a mapping. Let $T$ be a singleton. Then $f$ is a surjection.


Generating Function for Constant Sequence

Let $\sequence {a_n}$ be the sequence defined as: : $\forall n \in \N: a_n = r$ for some $r \in \R$. Then the generating function for $\sequence {a_n}$ is given as: :$\map G z = \dfrac r {1 - z}$ for $\size z < 1$


Area of Circle

The area $A$ of a circle is given by: : $A = \pi r^2$ where $r$ is the radius of the circle.


Distance Formula

The distance $d$ between two points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ on a Cartesian plane is: :$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$


Equation of Circle

The equation of a circle with radius $R$ and center $\tuple {a, b}$ can be expressed in the following forms:


Null URM Program Computes Identity Function

The null URM program computes the '''identity function''' $I_\N: \N \to \N$, defined as: :$\forall n \in \N: \map {I_\N} n = n$


Composition of One-Variable URM Computable Functions

Let $f: \N \to \N$ and $g: \N \to \N$ be URM computable functions of one variable. Let $f \circ g$ be the composition of $f$ and $g$. Then $f \circ g: \N \to \N$ is a URM computable function.


Concatenation of URM Programs is Associative

Let $P, Q, R$ be one-variable URM programs. Then the concatenated URM programs $P * \left({Q * R}\right)$ and $\left({P * Q}\right) * R$ are the same.


Primitive of Cosine Function

:$\ds \int \cos x \rd x = \sin x + C$


Primitive of Sine Function

:$\ds \int \sin x \rd x = -\cos x + C$


Function Obtained by Substitution from URM Computable Functions

Let the functions $f: \N^t \to \N, g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N$ all be URM computable functions. Let $h: \N^k \to \N$ be defined from $f, g_1, g_2, \ldots, g_t$ by substitution. Then $h$ is also URM computable.


Normalized URM Program

Let $P$ be a URM program. Let $l = \map \lambda P$ be the number of basic instructions in $P$. Let $u = \map \rho P$ be the number of registers used by $P$. Then $P$ can be modified as follows: :Every <tt>Jump</tt> of the form $\map J {m, n, q}$ where $q > l$ may be replaced by $\map J {m, n, l + 1}$ :If $u > 0$, a Clear Registers Program $\map Z {2, u}$ can be appended to the end of $P$ at lines $l + 1$ to $l + u - 1$. The new program $P'$ that results from the above modifications produces exactly the same output as $P$ for each input. Note now though that $\map \lambda {P'} = l + u - 1$. Such a program as $P'$ is called a '''normalized URM program'''. The point of doing this is so as to make programs easier to concatenate. Once the above have been done, each program has a well-defined exit line which can be used as the start line of the program that immediately follows it.


Block Copy Program

Let $k, m, n \in \N$ be natural numbers such that: * $k \ge 1$; * $\left|{m - n}\right| \ge k$. The URM program defined as: {| |- ! align="right" | Line !! ! align="left" | Command |- | align="right" | $1$ || | align="left" | $C \left({m, n}\right)$ |- | align="right" | $2$ || | align="left" | $C \left({m+1, n+1}\right)$ |- | align="right" | $\vdots$ || | align="left" | $\vdots$ |- | align="right" | $k$ || | align="left" | $C \left({m+k-1, n+k-1}\right)$ |} is called a '''block copy program'''. It is abbreviated $C \left({m, n, k}\right)$. It has the effect of copying the contents of registers $R_m, R_{m+1}, \ldots, R_{m+k-1}$ into the registers $R_n, R_{n+1}, \ldots, R_{n+k-1}$ respectively. It has length defined as $\lambda \left({C \left({m, n, k}\right)}\right) = k$.


Function Obtained by Primitive Recursion from URM Computable Functions

Let the functions $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be URM computable functions. Let $h: \N^{k+1} \to \N$ be obtained from $f$ and $g$ by primitive recursion. Then $h$ is also URM computable.


Double Angle Formulas

=== Double Angle Formula for Sine === {{:Double Angle Formulas/Sine}} === Double Angle Formula for Cosine === {{:Double Angle Formulas/Cosine}} === Double Angle Formula for Tangent === {{:Double Angle Formulas/Tangent}}


Half Angle Formulas

=== Half Angle Formula for Sine === {{:Half Angle Formulas/Sine}} === Half Angle Formula for Cosine === {{:Half Angle Formulas/Cosine}} === Half Angle Formula for Tangent === {{:Half Angle Formulas/Tangent}} === Half Angle Formula for Tangent: Corollary 1 === {{:Half Angle Formulas/Tangent/Corollary 1}} === Half Angle Formula for Tangent: Corollary 2 === {{:Half Angle Formulas/Tangent/Corollary 2}} === Half Angle Formula for Tangent: Corollary 3 === {{:Half Angle Formulas/Tangent/Corollary 3}} === One Plus Tangent Half Angle over One Minus Tangent Half Angle === {{:One Plus Tangent Half Angle over One Minus Tangent Half Angle}}


Primitive Recursive Function is URM Computable

Every primitive recursive function is URM computable.


Constant Function is Primitive Recursive

The constant function $f_c: \N \to \N$, defined as: :$\map {f_c} n = c$ where $c \in \N$ is primitive recursive.


Heron's Formula

Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively. Then the area $A$ of $\triangle ABC$ is given by: :$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c}}$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.


Addition is Primitive Recursive

The function $\Add: \N^2 \to \N$, defined as: :$\map \Add {n, m} = n + m$ is primitive recursive.


Exact Form of Prime-Counting Function

Let:  :$\displaystyle \map \Pi x = \map \Li x - \sum_\rho \map \Li {x^\rho} - \map \ln 2 + \int_x^\infty \frac {\d t} {t \paren {t^2 - 1} \, \map \ln t}$  where: :$\map \Li x$ is the offset logarithmic integral :the sum $\displaystyle \sum_\rho$ is taken over all $0 < \rho \in \R$ such that the zeta function $\map \zeta {\alpha + i \rho} = 0$ for some $\alpha \in \R$. Then the prime-counting function is precisely: :$\displaystyle \map \pi x = \sum_{n \mathop = 1}^\infty \paren {\frac {\map \mu n} n \map \Pi {x^{1/n} } }$ where $\map \mu n$ denotes the Möbius function.


Multiplication is Primitive Recursive

The function $\operatorname{mult}: \N^2 \to \N$, defined as: :$\map \Mult {n, m} = n \times m$ is primitive recursive.


Exponentiation is Primitive Recursive

The function $\exp: \N^2 \to \N$, defined as: :$\exp \left({n, m}\right) = n^m$ is primitive recursive.


Bézout's Lemma

Let $a, b \in \Z$ such that $a$ and $b$ are not both zero. Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$ That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$. Furthermore, $\gcd \set {a, b}$ is the smallest positive integer combination of $a$ and $b$.


Predecessor Function is Primitive Recursive

The '''predecessor function''' $\operatorname{pred}: \N \to \N$ defined as: :$\operatorname{pred} \left({n}\right) = \begin{cases} 0 & : n = 0 \\ n-1 & : n > 0 \end{cases}$ is primitive recursive.


Cut-Off Subtraction is Primitive Recursive

The '''cut-off subtraction''' function, defined as: :$\forall \tuple {n, m} \in \N^2: n \mathop {\dot -} m = \begin{cases} 0 & : n < m \\ n - m & : n \ge m \end{cases}$ is primitive recursive.


Maximum Function is Primitive Recursive

The maximum function $\max: \N^2 \to \N$, defined as: :$\max \left({n, m}\right) = \begin{cases} m: & n \le m \\ n: & m \le n \end{cases}$ is primitive recursive.


Sum of Maximum and Minimum

For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$: :$a + b = \max \left({a, b}\right) + \min \left({a, b}\right)$


Absolute Difference Function is Primitive Recursive

The '''absolute difference''' function $\operatorname {adf}: \N^2 \to \N$, defined as: :$\map {\operatorname {adf} } {n, m} = \size {n - m}$ where $\size a$ is defined as the absolute value of $a$, is primitive recursive.


Primitive Recursive Set is URM Computable

Every primitive recursive set is URM computable.


Set Containing Only Zero is Primitive Recursive

The subset $\left\{{0}\right\} \subset \N$ is primitive recursive.


Set of Non-Zero Natural Numbers is Primitive Recursive

Let $\N^*$ be defined as $\N^* = \N \setminus \left\{{0}\right\}$. The subset $\N^* \subset \N$ is primitive recursive.


Signum Function is Primitive Recursive

Let $\operatorname{sgn}: \N \to \N$ be defined as the signum function. Then: : $\operatorname{sgn}$ is primitive recursive. : $\overline {\operatorname{sgn}}$ is primitive recursive.