A topological space is path-connected when any two of its points can be joined by a continuous path. Path-connected spaces are connected, but the converse is false.

Taking the exterior derivative twice always gives zero, because mixed second partial derivatives are equal while the wedge product changes sign. This one rule contains the facts that gradients have no curl and curls have no divergence.

The greatest common divisor of two Fibonacci numbers is again a Fibonacci number. To find which one, take the greatest common divisor of the two indices.

A full turn around the unit circle has length $2\pi$, and it brings the point back to where it started. So the graphs of sine and cosine repeat after each interval of length $2\pi$.

Take a strip of paper, give one end a half-turn, and glue the two ends together. The result has only one side: an ant can walk to every point of the strip without crossing an edge.

Prey grow when there are few predators, and predators grow when there is much prey. This feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.

A continuous function takes its average value somewhere. The area under the graph equals the area of a rectangle whose height is a value of $f$.

The theorem says that the total circulation of a plane field around a closed curve equals the sum of the small circulations inside. The small circulations of neighbouring squares cancel along shared edges, so only the outer boundary is left.

To test if a linear map is one to one, we only need to look at the vectors that go to zero. If two vectors have the same image, then their difference goes to zero. So if only $0$ goes to $0$, then no two vectors share an image.

The determinant of an endomorphism of a finite-dimensional vector space is the determinant of its matrix in any basis.

"It is not true that $5$ is not prime" says the same as "$5$ is prime". Two negations cancel each other.

When all terms are nonnegative, the running total can only go up. So it either stays below some ceiling and converges, or it grows past every bound.

On a closed bounded interval, a continuous function cannot shoot off to infinity. Its graph stays between two horizontal lines.

The law says that there is no third option: a statement is true, or its negation is true. For example, each natural number $n$ is even or not even.

No two Fermat numbers share a prime factor. For example, $F_2 = 17$ and $F_3 = 257$ are coprime, and $F_5 = 641 \cdot 6700417$ has no factor in common with $3$, $5$, $17$, $257$ or $65537$.

The logarithm goes to infinity, but very slowly. Each positive power of $x$, even $x^{0.001}$, eventually beats it.

A modular multiplicative inverse of an integer $a$ modulo $n$ is an integer $x$ such that $a x$ leaves the remainder $1$ on division by $n$. The definition of the modular multiplicative inverse gives the precise form.

A positive definite matrix stretches each of its eigenvector directions by a positive factor. It never flips or crushes a direction. So its eigenvalues, its determinant and its trace are all positive.

A relation is symmetric when it always holds in both directions.

A harmonic function is a function of two real variables whose two pure second derivatives add up to $0$. Each holomorphic function gives two harmonic functions for free: its real part and its imaginary part.