To solve a linear equation with a right side, find one solution by any method. Then add all the solutions of the equation without right side. This gives every solution, and each only one time.

The series for $e$ converges so fast that $e$ cannot be a fraction. If $e = a/b$, a suitable multiple of the tail of the series would be an integer between $0$ and $1$.

A disc of radius $r$ has capacity $r$, the same as its boundary circle, because the charge in equilibrium sits on the edge anyway. A segment of length $L$ has capacity $L/4$. So a thin segment can hold charge, unlike a single point, but it holds less than the disc whose diameter it is. The computation uses the fact that capacity equals the Chebyshev constant, which is easy to find for these two shapes.

In a Diophantine equation, only whole numbers count as solutions. For example, $x^2 + y^2 = 3$ has real solutions, but no integer solutions. The restriction to integers makes such equations much harder than ordinary equations.

Take all fractions of polynomials in $x$, such as $(x^2 + 1)/(x - 3)$, together with the usual rule of differentiation. This is a field where one can add, multiply, divide and also differentiate, and the derivative of every element is again in the field. A differential field is any field with such a differentiation rule. One can then ask whether an equation such as $y' = y$ has a nonzero solution in the field; here it does not, since the nonzero solutions $c e^x$ are not rational functions.

An element is nilpotent when some power of it is zero. A nil ideal is an ideal in which every element has this property, though the power needed may depend on the element. In the ring $\Z/8\Z$, the ideal of even residues is nil: every even residue $a$ satisfies $a^3 = 0$. In $\Z$, the ideal $2\Z$ is not nil, since no power of $2$ is zero.

A universal Turing machine is a single machine that can imitate every other Turing machine. You give it, as input, the rule table of another machine and the input for that machine. It then does whatever that machine would do, step by step. A modern computer running any program you load is the same idea.

A disc is strictly convex: join two of its points and every point strictly between them is well inside the disc. A square is convex but not strictly convex. Two points on the same side give a segment that runs along the edge and never enters the inside. So strict convexity means that the boundary bulges outward everywhere and has no straight parts.

The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both of them. We write it as $\operatorname{lcm}(a, b)$. The definition of the least common multiple gives the precise form and the convention when one of the numbers is $0$.

Imagine that you may only ask yes-or-no questions written as first-order sentences. Two structures are elementarily equivalent when every such question gets the same answer in both. The ordered sets $(\Q, <)$ and $(\R, <)$ are elementarily equivalent, although one is countable and the other is not. The ordered sets $(\Z, <)$ and $(\Q, <)$ are not, because "between any two elements there is a third" is true in $\Q$ and false in $\Z$.

Push the boundary of a region outward a little, in some places more than in others. The area of the region grows, and to first order the growth is the outward push integrated along the boundary. That first-order rate of change is the shape derivative of the area. For other quantities, such as the drag on a wing, the shape derivative tells an engineer where to push the boundary to improve the design.

A Dynkin system is the kind of family one gets when one collects all sets on which two probability measures agree. Such a family is stable under complements and disjoint countable unions, but in general not under intersections. A small example: on $\set{1, 2, 3, 4}$, the subsets with an even number of elements form a Dynkin system. It is not a σ-algebra, since $\set{1, 2} \cap \set{2, 3} = \set{2}$ has one element.

Think of a knotted rope inside space, wrapped in a thin solid tube. Remove the tube and put a solid tube back in, but twisted. Now the disc that used to go around the rope closes off a different loop on the tube's surface. The space you get may be completely new. For the simplest knot, the round circle, surgery with label $0$ gives the product of a sphere and a circle.

The Cantor function climbs from $0$ to $1$ without jumps, yet it is flat almost everywhere. All its growth happens on the Cantor set, a dust of points with total length zero. On the middle third it sits at height $1/2$, on the next two middle thirds at heights $1/4$ and $3/4$, and so on. So knowing that the derivative is zero almost everywhere does not tell you the function is constant.

The functions on a group can be broken into pieces on which the group acts irreducibly, much as a sound breaks into pure tones. For the circle or the real line, all pieces come in continuous families, like frequencies. Some groups, such as $\mathrm{SL}_2(\R)$, also have isolated pieces that appear on their own, with positive "weight". These isolated pieces are the discrete series. For $\mathrm{SL}_2(\R)$, they are realised on spaces of holomorphic functions on the upper half-plane and are closely linked to modular forms.

Write the elements of one set along the rows and the elements of another set along the columns. Put a $1$ in a cell when the row element is related to the column element, and a $0$ otherwise. For the relation "divides" on $\set{1, 2, 3}$, the first row is $1, 1, 1$, because $1$ divides every number. The second row is $0, 1, 0$ and the third row is $0, 0, 1$.

A fair or favourable game whose size stays under control must settle down to a limit. A gambler who keeps playing fair games with a fortune that cannot go negative ends up with a fortune that stops changing. But the limit may lose mass: the expected value can drop in the limit unless the process is uniformly integrable.

Draw a circle in the complex plane that has some roots of a polynomial inside and the rest outside. Integrals around the circle count the roots inside and give the sums of their powers. From these sums one can rebuild the factor of the polynomial that holds exactly the roots inside, and then split again until every root stands alone.

A periodic point comes back to its start after some steps, and then again and again. The minimal period is the length of one trip around the loop. For the rotation of a clock face by $3$ hours, the point at $12$ visits $3$, $6$, $9$ and returns to $12$. So its minimal period is $4$, even though it is also back after $8$ or $12$ steps.

Bayes' theorem turns "the chance of the evidence given the cause" into "the chance of the cause given the evidence". A medical test tells you how often sick people test positive. Bayes' theorem tells you how likely a person with a positive test is to be sick.