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Let $K$ be a commutative ring and let $A$ and $B$ be $n \times n$ matrices over $K$. Then $$\det(AB) = \det A \cdot \det B,$$ where $AB$ is the matrix product and $\det$ is the determinant.

Let $A$ be an $n \times n$ complex matrix. Then $A$ is normal if and only if there are a unitary matrix $U$ and a diagonal matrix $D$ with $$A = U D U^*.$$

Let $X$ be an integrable real random variable and let $b \in \mathbb{R}$. Then $$\operatorname{Var}(X + b) = \operatorname{Var}(X).$$

The coordinate vector of a vector in a finite basis is the list of the coefficients of the vector in that basis.

Let $G$ be a finite group and let $H$ be a subgroup of $G$. Then $|G| = [G : H] \cdot |H|$, where $[G : H]$ is the index of $H$ in $G$.

Modular exponentiation computes $b^e \bmod m$ for integers $b$, $e \ge 0$ and $m \ge 1$. The binary method writes $e$ in base $2$. It squares $b$ again and again modulo $m$, and it multiplies the squares $b^{2^i}$ for the bits $i$ of $e$ that are $1$. It needs about $2 \log_2 e$ multiplications modulo $m$, and each intermediate number is smaller than $m^2$.

The natural logarithm is the inverse of the exponential function. It is defined for positive numbers, it turns products into sums, and its derivative is $1/x$. The node natural logarithm gives the precise definition.

The circle group is the set $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$ of the complex numbers of absolute value $1$, with complex multiplication. It is an abelian group with identity element $1$, and the inverse of $z$ is its complex conjugate $\bar{z} = 1/z$. Each element of $\mathbb{T}$ is $e^{i\theta}$ for a real number $\theta$, and $e^{i\theta} e^{i\varphi} = e^{i(\theta + \varphi)}$.

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$. If $b$ and $b'$ are both suprema of $S$, then $b = b'$.

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, with the inner product linear in the first argument. Then, for each linear form $f \colon V \to \mathbb{K}$, there is exactly one vector $y \in V$ with $$f(x) = \langle x, y \rangle \qquad \text{for all } x \in V.$$

The separation axioms are conditions on a topological space that say how well open sets can separate points and closed sets. The main ones, from weak to strong, are T0, T1, Hausdorff, regular and normal.

The divisor function $\sigma_k$ sends a positive integer $n$ to the sum of the $k$-th powers of its positive divisors. The definition of the divisor function gives the precise form.

Let $x$ and $y$ be real numbers with $x < y$. Then there is an irrational number $z$ with $x < z < y$.

Let $\alpha$ be a real number and let $N$ be a positive integer. Then there are integers $p$ and $q$ with $1 \le q \le N$ and $$|q \alpha - p| \le \frac{1}{N + 1}.$$

Let $n$ and $r$ be natural numbers with $r \le n$. Then $$\sum_{i=r}^{n} \binom{i}{r} = \binom{n + 1}{r + 1}.$$

A homogeneous system of linear equations is a system in which every constant term is zero. Such a system always has the zero solution, and its solution set is a vector subspace.

The clique number of a finite graph is the size of its largest clique. It is a lower bound for the chromatic number. The node clique number gives the precise definition.

Let $n$ be a positive integer. Then $n = x^2 + y^2$ for some integers $x$ and $y$ if and only if $\nu_p(n)$ is even for each prime $p \equiv 3 \pmod 4$. Here $\nu_p$ is the p-adic valuation.

A continued fraction is an expression $b_0 + a_1 / (b_1 + a_2 / (b_2 + \cdots))$, a fraction whose denominator contains a fraction, and so on. The definition of a continued fraction gives the precise form.

The geometric distribution describes the waiting time until the first success in a sequence of independent Bernoulli trials. One version counts the failures before the first success, and the other counts the trials up to and including it. The node geometric distribution gives the precise definition.

For every complex number $z$, $$e^{i z} = \cos z + i \sin z.$$ Here $e^{w} = \sum_{n=0}^{\infty} w^n / n!$ is the complex exponential function, and $\cos$ and $\sin$ are given by their power series.

A point of the domain where a real function is not continuous is a discontinuity. Discontinuities are sorted by the behavior of the one-sided limits at the point.

Let $G$ be a cyclic group. Then $G$ is abelian.

Let $X$ be an integrable real random variable on $(\Omega, \mathcal{F}, P)$ and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. Then $$E\left[E[X \mid \mathcal{G}]\right] = E[X].$$ In particular, $E[E[X \mid Y]] = E[X]$ for each random variable $Y$ on the same space.

Assume the axiom of choice. Let $(P, \le)$ be a nonempty partially ordered set in which each chain has an upper bound in $P$. Then $P$ has a maximal element.

A parabola is a plane curve. Each of its points has the same distance from a fixed point, the focus, and from a fixed line, the directrix. It is symmetric about the line through the focus perpendicular to the directrix.

Given the edge of a cube, construct with straightedge and compass alone the edge of a cube with twice the volume.

For each prime power $q = p^n$, the Galois field $\mathrm{GF}(q)$, also written $\mathbb{F}_q$, is the finite field with $q$ elements. It exists, and it is unique up to isomorphism. One model is the splitting field of $x^q - x$ over $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$. Another model is $\mathbb{F}_p[x]/(f)$ for an irreducible polynomial $f \in \mathbb{F}_p[x]$ of degree $n$.

Let $m$ and $n$ be positive integers, and let $a$ and $b$ be integers. Let $g = \gcd(m, n)$ and $M = \operatorname{lcm}(m, n)$. The system $x \equiv a \pmod m$, $x \equiv b \pmod n$ has an integer solution if and only if $a \equiv b \pmod g$. In this case, two solutions are congruent modulo $M$.

Let $p$ and $q$ be propositions. The implication $p \to q$ is logically equivalent to its contrapositive $\neg q \to \neg p$. $$p \to q \equiv \neg q \to \neg p$$

The higher derivatives of a function come from repeated differentiation. The $n$-th derivative $f^{(n)}$ is the derivative of $f^{(n-1)}$. The node higher derivative gives the precise definition.

A definite integral is the integral of a function between two given limits of integration. It is a number, and its sign changes when the two limits are swapped. The node definite integral gives the precise definition.

The continued fraction expansion of a real number is the simple continued fraction found by taking integer parts and reciprocals again and again. The definition of the continued fraction expansion gives the precise form.

A random walk is a stochastic process whose position is a sum of independent random steps. The simple random walk on the integers moves one step up or down at each time. The node random walk gives the precise definition.

Let $k$ be a natural number. The divisor function $\sigma_k$ is a multiplicative function. That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$.

For all real numbers $a$ and $b$, $$\int_a^b \cos x \, dx = \sin b - \sin a.$$

Let $n \ge 1$ and let $v_1, \dots, v_n \in \R^n$ be the columns of a real $n \times n$ matrix $A$. The parallelepiped spanned by them is $P = \{t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1\}$. Its $n$-dimensional volume is $\abs{\det A}$, where $\det$ is the determinant. If $\det A \ne 0$, then its sign is positive when $(v_1, \dots, v_n)$ has the orientation of the standard basis, and negative when it has the opposite orientation.

Let $p$ be a prime number. Then there is a primitive root modulo $p$. That is, the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic of order $p - 1$.

Let $G$ be a connected graph. Then $G$ has a spanning tree.

The elementary row operations are three kinds of operations on the rows of a matrix. They swap two rows, multiply a row by a nonzero scalar, or add a multiple of one row to another row.