Sometimes a curve is given by an equation and not by a formula $y = f(x)$. You can still find its slope: treat $y$ as a function of $x$, differentiate the equation term by term, and solve for $y'$.

If a continuous function is negative at one end of an interval and positive at the other, it must cross zero somewhere in between. Check the middle: the sign there tells which half still has a crossing. Keep halving. After $k$ steps the zero is trapped in an interval $2^k$ times shorter than at the start.

Think of the generators as the allowed steps. The length of an element is the fewest steps needed to reach it from the identity. The distance between two elements is the fewest steps from one to the other. In $\Z$ with the steps $\pm 1$, the distance between $m$ and $n$ is $\abs{m - n}$. With the steps $\pm 2$ and $\pm 3$ instead, the number $1$ is at distance $2$ from $0$, since $1 = 3 - 2$ and $1$ itself is not a step.

In a group, two elements multiply to one element. In a hypergroup, two elements multiply to a set of elements. On the real line, let the product of $x$ and $y$ be every number between them; the product of $1$ and $3$ is then $[1, 3]$. This product is associative in a natural sense. Also every number $y$ lies in the product of $x$ and $y$, so the products of a fixed $x$ with all numbers cover the line. These two facts make the line a hypergroup. Hyperstructures study which facts about groups, rings and fields survive in this setting.

A coherent sheaf attaches to each small region of a space a module of functions or sections. Locally everything is described by finitely many generators and finitely many relations. The model is a system of linear equations with holomorphic coefficients: its solutions form a coherent sheaf. For example, on $\C^2$ the germs $(g_1, g_2)$ with $g_1 z_1 + g_2 z_2 = 0$ form a coherent sheaf, generated by the single relation $(z_2, -z_1)$.

When a sequence of non-negative functions converges, mass can escape, but it cannot appear from nowhere. So the integral of the limit is at most the limit of the integrals. A tall thin spike that keeps its area while it shrinks is the typical way mass escapes.

The Borromean rings are three rings that hold together as a group, although no two of them are linked. If any one ring is cut and removed, the other two fall apart. They appear in old coats of arms, among them that of the Borromeo family of Italy, which gives the name.

Knowing all moments of a distribution does not always tell you the distribution. When it does, the moment problem is called determinate. Every distribution on a bounded interval is determined by its moments. So is the normal distribution, because its moments do not grow too fast.

In several complex variables a holomorphic function cannot have a compact hole of singularities. If it is defined everywhere except on a compact set, it automatically extends across that set. This is very different from one variable, where $1/z$ has an isolated singularity at $0$.

In a nuclear space, passing from one seminorm to a stronger one makes things very much smaller, in a summable way. For smooth functions, a bound on many derivatives forces the function to be very close to a finite combination of simple functions. This is why nuclear spaces feel almost finite-dimensional. For example, in the Schwartz space of rapidly decreasing smooth functions, bounded sets are relatively compact, as in $\R^n$. Infinite-dimensional Banach spaces are never nuclear.

You can add, subtract, multiply and divide complex numbers, as with real numbers. The only exception is division by $0$. For example, $\dfrac{1}{i} = -i$, because $i \cdot (-i) = -i^2 = 1$.

A transcendental number is a complex number that is not algebraic, such as $e$ or $\pi$. The definition of a transcendental number gives the precise form.

A nowhere dense subset of a topological space is a set whose closure has empty interior. Such a set is not dense in any nonempty open set.

A field extension is a pair of fields $K \subseteq L$, where $K$ is a subfield of $L$. One writes $L/K$ and says that $L$ is an extension of $K$. The definition of a field extension gives the details.

Take a sequence of nonnegative numbers and replace each term by the average of the terms up to it. Hardy's inequality says the new sequence is not much larger than the old one, when size is measured by the sum of $p$-th powers. The loss is at most the factor $\big(\frac{p}{p-1}\big)^p$, which is $4$ for $p = 2$.

An open book describes a closed three-dimensional space as a family of surfaces, the pages, that all share the same boundary, the binding. They look like the pages of a book turning around its spine. Giroux's theorem says that contact structures on such a space correspond exactly to open books, up to a simple move called positive stabilization. This turns questions about twisting plane fields into questions about surfaces and their diffeomorphisms.

Pick numbers so that no two pairs of them have the same sum. The powers of two $1, 2, 4, 8, 16$ work, because binary expansions show each sum of two of them only once. Such sets cannot be very dense: among the numbers up to $N$, a Sidon set has only on the order of $\sqrt{N}$ elements. The reason is that its roughly $n^2/2$ pairwise sums must all be different and at most $2N$.

Imagine a surface built by gluing pieces of the complex plane, where the gluing maps are holomorphic. Because the gluing respects holomorphy, you can tell whether a function on the surface is holomorphic by looking at it in any piece. The plane itself, the Riemann sphere and the torus $\C/(\Z + i\Z)$ are examples. The graph of $w^2 = z$ is a Riemann surface on which $\sqrt{z}$ becomes a single-valued function, namely $w$.

Flat modules are the modules that behave well under tensor products, and every free module is flat. A general module can be approximated by flat modules in a chain, called a flat resolution. The weak dimension counts how many steps such a chain needs at least. Over the integers, a module like $\Z/2$ needs one step: $0 \to \Z \to \Z \to \Z/2 \to 0$, so its weak dimension is $1$. A ring of small weak dimension has a simple homological theory.

Think of $T(X)$ as the set of formal expressions built from elements of $X$. The unit turns an element into the simplest expression, the element itself. The multiplication flattens an expression made of expressions into a single expression. For example, let $T(X)$ be the set of finite lists of elements of $X$. The unit sends $x$ to the one-element list $(x)$, and the multiplication concatenates a list of lists into one list.