Cardinal numbers measure the size of sets, finite or infinite. They generalize the natural numbers as answers to the question "how many elements?".

Twin primes are primes with only one number between them, like $41$ and $43$. The conjecture says that such pairs never stop, even though primes become rarer.

A primitive element of a finite field is a generator of its multiplicative group. The definition of a primitive element states it.

Two matrices are row equivalent when one of them can be changed into the other by a finite sequence of elementary row operations.

Two equal rows mean that the rows span less than the full space. Then the box spanned by the rows is flat, and its volume is zero. So the determinant is zero.

Take a root of a whole number, for example $\sqrt{10}$ or $\sqrt[3]{20}$. Either the root is a whole number, as for $\sqrt{9} = 3$, or it is not a fraction at all. A root of a whole number is never a fraction like $5/2$.

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.

Up to isomorphism, a finite-dimensional vector space is determined by one number: its dimension. All three-dimensional real spaces behave like $\R^3$. Only the names of the vectors differ.

A graph is $k$-colourable when it has a proper colouring with $k$ colours. The least such $k$ is the chromatic number. The node k-colourable graph gives the precise definition.

When you do two permutations one after the other, you join their chains of swaps. The number of swaps adds, so the signs multiply. This is exactly the rule of a homomorphism.

A solvable group is a group with a subnormal series whose factor groups are all abelian. Equivalently, its derived series reaches the trivial subgroup.

Division by a continuous function causes no problem away from its zeros. Near a point where $g$ is not zero, $g$ stays away from zero.

Self-adjoint maps never have complex eigenvalues that are not real. For example, the symmetric matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has the eigenvalues $1$ and $-1$. By contrast, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not symmetric, and its eigenvalues are $\pm i$.

Think of a clock with $n$ hours, where you add hours and wrap around after $n$. This clock arithmetic gives the cyclic group of order $n$. Every cyclic group of order $n$ looks the same as this clock, up to the names of the elements.

The terms shrink very fast, because $n!$ grows fast. Only a few terms give a good value of $e$: the first five terms already give $2.708$.

A sequence of random variables converges almost surely when, for almost every outcome, the sequence of values converges. The node almost sure convergence gives the precise definition.

A group isomorphism is a bijective group homomorphism. Two isomorphic groups have the same structure, and differ only in the names of their elements.

A Pythagorean triple is a triple of positive integers $a$, $b$, $c$ with $a^2 + b^2 = c^2$, the side lengths of a right triangle. The definition of a Pythagorean triple gives the precise form.

A finite set is a set with a natural number of elements.

When the size of a group is a prime, there is no room for proper subgroups other than the trivial one. Any element other than the identity generates a subgroup larger than $1$, so it generates everything. Thus, for example, each group with $7$ elements is a clock with $7$ hours.