Add and subtract the numbers of a row of Pascal's triangle in turn, and you get $0$. For a nonempty set, add or remove one fixed element. This pairs the even subsets with the odd subsets.

The cross product is a binary operation on the vectors of three-dimensional space. It takes two vectors and gives a vector that is perpendicular to both. The length of the result is the area of the parallelogram with the two vectors as sides.

For a Möbius map, a line is only a circle through the point $\infty$. So the theorem says that circles go to circles, if we count lines as circles. For example, $z \mapsto 2z + 3$ sends each circle to a circle with twice the radius.

Add a new element to a partition problem. The new element is alone, or it joins one of the existing $k + 1$ groups, and these two cases give the two terms.

An equation $F(x, y) = 0$ often defines $y$ as a function of $x$, even when you cannot solve it with a formula. The theorem says this works near a solution, provided $F$ really depends on $y$ there, in the sense that $D_y F$ is invertible.

The phase line is a one-dimensional map of where the state can go. Arrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.

In the symmetric group $S_3$, let $H = \{e, (1\,2)\}$. Compose permutations from right to left. Two of the left cosets are $(1\,3)H = \{(1\,3), (1\,2\,3)\}$ and $(2\,3)H = \{(2\,3), (1\,3\,2)\}$. Try to set $(aH)(bH) = (ab)H$. With $a = (1\,3)$ and $b = (2\,3)$, we get $ab = (1\,3\,2)$, which is in $(2\,3)H$. With $a = (1\,2\,3)$ from the same coset, we get $ab = (1\,2)$, which is in $H$. The two results are in different cosets. Thus this product is not defined, and the cosets of $H$ do not form a quotient group.

A bounded sequence is a real sequence whose terms all lie in one fixed interval. Boundedness is a necessary condition for convergence, and it is the hypothesis of the Bolzano–Weierstrass theorem. The node bounded sequence gives the precise definition.

We add two functions by adding their values at each point. We scale a function by scaling its values. With these rules, functions behave like vectors, and the zero function is the zero vector.

Since there is only one finite field of each order, up to a change of names, one gives it a fixed name: the Galois field with $q$ elements. Computations in it are done with polynomials over the integers modulo $p$.

A tangent line to a circle is a line that meets the circle in exactly one point, the point of tangency. It does not enter the inside of the circle, and it is perpendicular to the radius at the point of tangency.

The number $\pi$ is not a root of any nonzero polynomial with rational coefficients. This is much stronger than the irrationality of $\pi$, and the proof is much harder.

Remove the root of a binary tree, and you get a left tree and a right tree. The sizes of the two trees add up to one less than the size of the original tree, and this gives the Catalan recurrence.

Write the pairs $(x, y)$ in an infinite grid. Walk through the grid along the short diagonals, one after the other, and number the pairs as you go. Each pair gets exactly one number, and each number gets exactly one pair.

For square matrices, we only need to check one of $AB = I$ and $BA = I$. The other one then holds automatically. This saves half of the work when we check an inverse.

The curve $\sin(1/x)$ oscillates faster and faster as $x$ goes to $0$. The origin is a limit of the curve, so the space is in one piece. But you cannot walk along the curve to the origin, because the curve never settles down.

The equation splits $f(x + h)$ into three parts: the value at $x$, a linear part in $h$, and an error. The derivative condition says only that the error is tiny compared with the step: halving the step more than halves the error.

Take the triangle with $d(A, B) = 6$ and $d(A, C) = 3$. The bisector of the angle at $A$ cuts $BC$ into two parts. The part next to $B$ is twice as long as the part next to $C$.

A finite separable extension, even one built from many elements, can be generated by one element. So many statements about finite extensions reduce to the case of one algebraic element and its minimal polynomial.

Inside the interval of convergence, a power series converges uniformly once we stay away from the endpoints. Near the endpoints, the convergence can get slower and slower.