The characteristic function of a normal variable is again a bell-shaped function. A narrow normal distribution, with small $\sigma$, has a wide characteristic function, and conversely.

A polynomial splits over a field when it is a product of polynomials of degree one with coefficients in that field. The definition of a split polynomial states it.

The theorem gives a finite list of candidates for the rational roots. You test each candidate by evaluation. For $x^3 - 2$, the candidates are $\pm 1$ and $\pm 2$, and none is a root, so $\sqrt[3]{2}$ is irrational.

A clique in a graph is a set of vertices that are pairwise adjacent. Cliques are the complete subgraphs, and they appear in colouring, in Ramsey theory and in extremal graph theory. The node clique gives the precise definition.

A normal space is a topological space in which any two disjoint closed sets can be put into disjoint open sets. In normal spaces, there are many continuous real functions, by Urysohn's lemma.

The Picard–Lindelöf hypotheses on a vector field are continuity in time, a Lipschitz condition in space and a bound near the initial point. They also ask for a short enough time interval. They are the conditions of the Picard–Lindelöf theorem. See the definition.

A hyperbola is a plane curve with two foci. For each point of the curve, the difference of the distances to the two foci has the same absolute value. It has two branches that are mirror images of each other.

Unless the series stops after finitely many terms, it converges exactly inside $(-1, 1)$, plus possibly some endpoints. This matches the fact that $(1 + x)^{\alpha}$ has trouble at $x = -1$.

The exponential function turns addition into multiplication. This is the rule $e^{x + y} = e^x e^y$ for powers.

The partial sums of $\sum b_n$ do not grow, and the weights $a_n$ shrink steadily to $0$. Together these give enough cancellation for $\sum a_n b_n$ to converge.

A continuous linear map between normed spaces is a linear map that is continuous for the norm topologies. Such maps are the same as the bounded linear maps, which multiply lengths by at most a fixed constant.

A Cauchy–Euler equation is a linear equation $\sum_k a_k x^k y^{(k)} = 0$ with constant $a_k$, in which the $k$-th derivative has the factor $x^k$. It has solutions of the form $x^m$, and the change of variable $x = e^t$ gives constant coefficients. See the definition.

The null space of a matrix is the set of vectors that the matrix sends to zero. It is the solution set of the homogeneous system with this matrix.

Many people feel that $0.999\dots$ is a little less than $1$. But the difference $1 - 0.999\dots$ is smaller than $10^{-n}$ for every $n$. The only nonnegative real number with this property is $0$. Another view: $1/3 = 0.333\dots$, and three times this gives $1 = 0.999\dots$.

A nullcline of a system of autonomous equations $x_j' = f_j(x)$ is a set on which one component $f_j$ of the vector field vanishes. The equilibrium points of the system are the common points of all the nullclines. See the definition.

For a uniformly continuous map, one input tolerance $\delta$ works in the whole space. The map $x \mapsto x^2$ gets steeper and steeper, so near large $x$ you need a smaller and smaller $\delta$, and no single $\delta$ works.

If a quantity grows at most at a rate proportional to itself, it grows at most exponentially. The inequality compares the quantity with the solution of the equation $v' = \beta v$. This is the main tool to compare two solutions of a differential equation.

An increasing function can only move up as $x$ approaches $c$ from the left. Its values are bounded by $f(c)$, so they settle at a limit.

A compact space is a topological space in which each open cover has a finite subcover. Closed bounded intervals of the real line are the basic examples.

In an orthonormal basis, the inner product of two vectors is the standard inner product of their coordinates. In particular, the squared length is the sum of the squared coordinates. This is the Pythagorean theorem in $n$ dimensions.