Each finite abelian group is built from cyclic groups. You can choose the cyclic pieces to have prime power orders, or to have orders that divide each other. In both forms, the pieces are unique, so this lists all finite abelian groups.

A hard real integral can become easy when you view the real line as part of a closed curve in the complex plane. The residue theorem gives the integral over the closed curve from a few residues. Then you show that the extra part of the curve contributes nothing in the limit.

The edge of the doubled cube is the cube root of $2$. Its minimal polynomial has degree $3$, but constructible numbers have degree a power of $2$.

If continuous graphs approach a limit graph uniformly, the limit graph has no jumps. A jump in the limit would force some $f_n$ to jump as well.

The orthogonal group collects all rotations and reflections of $n$-dimensional space that fix the origin. Composing two of them gives another one, and each one can be undone. In the plane, it consists of the rotations about the origin and the reflections in lines through the origin.

To get a Fibonacci number with a large index, you can combine Fibonacci numbers with smaller indices. For example, with $m = 2$ and $n = 3$: $F_6 = F_2 F_3 + F_3 F_4 = 1 \cdot 2 + 2 \cdot 3 = 8$.

To collect all $n$ pictures of a series, you need about $n \ln n$ purchases, not $n$. The last pictures are the slow ones: when only one type is missing, you wait on average $n$ draws for it. For $n = 50$, $E[T] = 50 H_{50} \approx 225$.

A complicated fraction of polynomials splits into a sum of simple fractions. For example, $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$.

A subbasis of a topology is a family of open sets that generates the topology. The topology is the smallest one in which all members of the family are open. The finite intersections of members of a subbasis form a basis.

Somewhere between $a$ and $b$, the tangent line is parallel to the chord from $(a, f(a))$ to $(b, f(b))$. If a car travels 100 km in one hour, at some moment its speed is exactly 100 km per hour.

The empty set is an empty bag. For example, the set of real numbers $x$ with $x^2 = -1$ is empty. It is useful in the same way that the number $0$ is useful.

A real number is something that rational numbers can approximate as well as you like. The construction takes the approximating sequences themselves as the new numbers. Two sequences that get closer and closer to each other count as the same real number.

The floor function sends a real number $x$ to the greatest integer that is less than or equal to $x$. We write $\lfloor x \rfloor$ for it. For example, $\lfloor 2.7 \rfloor = 2$ and $\lfloor -2.7 \rfloor = -3$.

A gradient field has no swirl. Going around a small loop in a landscape, you return to the same height. So the field $\nabla \phi$ has no circulation, and its curl is zero.

The general linear group collects all the invertible matrices of one size. We can multiply two of them, and we can undo each of them, so they form a group. Geometrically, its elements are all the ways to transform space linearly without flattening it.

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

The mean squared error of an estimator is the expected squared distance between the estimator and the true value. It combines the variance and the bias of the estimator. The node mean squared error gives the precise definition.

A scalar matrix is a diagonal matrix whose diagonal entries are all equal, that is, a scalar multiple of the identity matrix.

A normed vector space is a real or complex vector space together with a norm.

The vector $b \times c$ is perpendicular to the plane of $b$ and $c$. A second cross product with $a$ gives a vector perpendicular to $b \times c$, so the result lies back in the plane of $b$ and $c$. The formula tells you exactly which combination of $b$ and $c$ it is.