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The integers modulo $n$ form the ring $\mathbb{Z}/n\mathbb{Z}$ of the $n$ residue classes modulo $n$, with the addition and the multiplication of representatives. The definition of the integers modulo n gives the precise form.

The circumcircle of a triangle is the circle that passes through its three vertices. Each triangle has exactly one circumcircle. Its centre is the circumcentre and its radius is the circumradius.

Let $n$ be a natural number. The complete graph $K_n$ is the graph with vertex set $\{1, \dots, n\}$ in which any two distinct vertices are adjacent.

Let $x$ be a real number with $x \neq 1$, and let $n$ be a natural number. Then $$\sum_{k=0}^{n-1} x^k = 1 + x + x^2 + \dots + x^{n-1} = \frac{x^n - 1}{x - 1}.$$

The sample mean of a sample is the arithmetic mean of its values. It is the usual estimator of the mean of the population. The node sample mean gives the precise definition.

Let $D \subseteq \mathbb{C}$ be a domain and let $f : D \to \mathbb{C}$ be holomorphic on $D$. Assume that $f$ is not constant on $D$. Then $f$ is an open map: for each open set $V \subseteq D$, the image $f(V)$ is an open subset of $\mathbb{C}$.

An oriented angle in the plane measures the turn from one direction to another, with a sign. Anticlockwise turns are positive and clockwise turns are negative. It is defined modulo a full turn $2\pi$, and it changes sign when you swap the two directions.

Let $f \colon [0, 1] \to \mathbb{R}$ be the Dirichlet function: $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$. Each subinterval of positive length contains both rational and irrational numbers. So for each partition $P$, $U(f, P) = 1$ and $L(f, P) = 0$. The upper Darboux integral is $1$ and the lower one is $0$. So $f$ is not Riemann integrable.

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, let $x_0 \in U$ and let $t_0 \in \mathbb{R}$. Then there are real numbers $\varepsilon > 0$, $a > 0$ and $L, K \ge 0$ with these properties. The closed ball $B$ of radius $a$ around $x_0$ lies in $U$. The time-independent field $(t, x) \mapsto f(x)$, extended in any way outside $B$, satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ on $[t_0 - \varepsilon, t_0 + \varepsilon]$ with constants $a, L, K$.

Let $n$ be a natural number, and let $F_n$ be the $n$-th Fibonacci number. Let $\varphi = \frac{1 + \sqrt{5}}{2}$ be the golden ratio and let $\psi = \frac{1 - \sqrt{5}}{2}$. Then $$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}.$$

Let $(X, d)$ be a compact metric space and let $(U_i)_{i \in I}$ be an open cover of $X$. Then there is a real $\delta > 0$ such that each subset $A \subseteq X$ with diameter $\operatorname{diam}(A) < \delta$ is contained in $U_i$ for some $i \in I$.

A lattice path is a finite sequence of points of the integer lattice $\mathbb{Z}^2$. Each step between consecutive points is from a given finite set of steps. The most common case uses the steps $E = (1, 0)$ and $N = (0, 1)$. These are North-East lattice paths.

Let $G$ be a cyclic group of finite order $n$, generated by $g$. For each positive divisor $d$ of $n$, $G$ has exactly one subgroup of order $d$, namely $\langle g^{n/d} \rangle$. Each subgroup of $G$ has this form for some positive divisor $d$ of $n$.

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. Then there are unique monic nonconstant polynomials $f_1, \dots, f_r \in F[t]$ with $f_1 \mid f_2 \mid \cdots \mid f_r$ and the following property. The matrix $A$ is similar to the block diagonal matrix $$\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix},$$ where $C(f_i)$ is the companion matrix of $f_i$. This block matrix is the rational canonical form of $A$.

Let $X$ be a compact space, let $Y$ be a Hausdorff space and let $f \colon X \to Y$ be a continuous map that is bijective. Then $f$ is a homeomorphism.

Let $U \subseteq \mathbb{C}$ be an open set and let $f_1, f_2, \dots$ be functions $U \to \mathbb{C}$ that are holomorphic on $U$. Let $f : U \to \mathbb{C}$ be a function such that $f_n \to f$ uniformly on each compact subset of $U$. Then $f$ is holomorphic on $U$. Moreover, $f_n' \to f'$ uniformly on each compact subset of $U$.

Let $p$ be an odd prime number. Then there are integers $x$ and $y$ with $p = x^2 + y^2$ if and only if $p \equiv 1 \pmod 4$.

A multilinear map is a map of several vector variables that is linear in each variable when the other variables are fixed.

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero.

A real random variable has a density when there is a fixed nonnegative function with this property. The probability of each set of values is the integral of the function over that set. The node random variable with a density gives the precise definition.

Let $X$ be a finite set with $|X| = n$. Then the number of derangements of $X$ is the derangement number $D_n$ of the node number of derangements.

A closed curve is a parametrised curve on a closed interval whose end point is equal to its initial point. Its image can be seen as the image of a circle under a continuous map.

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ be events with $A \subseteq B$. Then $P(A) \le P(B)$. More precisely, $P(B \setminus A) = P(B) - P(A)$.

A homogeneous linear system with constant coefficients is a system $x' = A x$ with a constant square matrix $A$. Its solutions are given by the matrix exponential, and the eigenvalues of $A$ control their behaviour. See the definition.

The residue theorem gives the values of many definite integrals of real functions. For an integral $\int_0^{2\pi} R(\cos \theta, \sin \theta) \, d\theta$ with $R$ a rational function of two variables, the substitution $z = e^{i\theta}$ changes it into a contour integral over the unit circle. This substitution uses $\cos \theta = \frac{1}{2}(z + z^{-1})$, $\sin \theta = \frac{1}{2i}(z - z^{-1})$ and $d\theta = \frac{dz}{iz}$. For an integral $\int_{-\infty}^{\infty} f(x) \, dx$, one closes the segment $[-R, R]$ with a semicircle of radius $R$ in the upper half-plane and lets $R \to \infty$. This method gives the integral when the integral over the semicircle tends to $0$, for example by the estimation lemma or by Jordan's lemma.

The variance of a random variable is the expected squared distance between the variable and its mean. It measures how widely the values spread around the mean. The node variance gives the precise definition.

Let $x$ be a real number with $x \ge 0$. Then $x$ has a decimal expansion: there are a natural number $a_0$ and digits $d_k \in \{0, \dots, 9\}$ for $k \ge 1$ with $x = a_0 + \sum_{k=1}^{\infty} d_k 10^{-k}$.

Let $U \subseteq \mathbb{R}^n$ be open. The Jacobian matrix of a map $f \colon U \to \mathbb{R}^m$ at a point $a$ is the $m \times n$ matrix of all first partial derivatives $\partial f_i / \partial x_j$ at $a$. When $f$ is differentiable at $a$, it is the matrix of the total derivative $Df(a)$ in the standard bases.

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an orthonormal basis of $V$. Then, for all $x, y \in V$, $$\langle x, y \rangle = \sum_{i=1}^{n} \langle x, e_i \rangle \overline{\langle y, e_i \rangle} \qquad \text{and} \qquad \norm{x}^2 = \sum_{i=1}^{n} \abs{\langle x, e_i \rangle}^2.$$

The multiplicity of a root $c$ of a nonzero polynomial $f$ is the largest $m$ such that $(x - c)^m$ divides $f$. The definition of the multiplicity gives the details.

Let $D \subseteq \mathbb{C}$ be a domain and let $f : D \to \mathbb{C}$ be holomorphic on $D$. Assume that $|f|$ has a local maximum at a point $z_0 \in D$: there is $r > 0$ such that $|f(z)| \le |f(z_0)|$ for each $z$ with $|z - z_0| < r$. Then $f$ is constant on $D$.

Points are concyclic when they all lie on one circle. A polygon with concyclic vertices is a cyclic polygon, and the circle is its circumcircle.

An open cover of a subset of a topological space is a family of open sets whose union contains the subset. Open covers are used to define compactness.

Let $U \subseteq \mathbb{R}^n$ be open with $n \ge 2$, and let $f, g \colon U \to \mathbb{R}$ be of class $C^1$. Let $c \in \mathbb{R}$, and let $a$ be a point of the level set $\{x \in U : g(x) = c\}$ with $\nabla g(a) \ne 0$. If the restriction of $f$ to this level set has a local extremum at $a$, then there is a unique $\lambda \in \mathbb{R}$ with $$\nabla f(a) = \lambda \, \nabla g(a).$$

Greedy colouring is a simple method that makes a colouring of a finite graph. Put the vertices in an order $v_1, v_2, \dots, v_n$, and use the colours $1, 2, 3, \dots$. For $i = 1, \dots, n$, give $v_i$ the smallest colour that no earlier neighbour of $v_i$ has.

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, 1 - y)$ for all $y \in [0, 1]$. The quotient space $[0, 1]^2 / {\sim}$, with the quotient topology, is the Möbius strip.

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ and $B$ be $n \times n$ matrices over $\mathbb{K}$ with $AB = BA$. Then $$e^{A + B} = e^A e^B,$$ where $e^A$ is the matrix exponential.

A field is a commutative ring in which you can divide by every nonzero element. The rational numbers, the real numbers and the complex numbers are fields. The definition of a field gives the axioms.

For every real $x > 0$, $$\ln x \le x - 1,$$ with equality if and only if $x = 1$.

A real sequence assigns a real number to each natural number. Real sequences are the first setting for limits, convergence and series in real analysis. The node real sequence gives the precise definition.