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In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The ratio of a side of a triangle to the sine of its opposite angle is constant.

Generalization of the Pythagorean theorem to arbitrary triangles.

Expands powers of a binomial into a sum of terms.

Solutions of the quadratic equation ax² + bx + c = 0.

Links five fundamental constants in one equation.

Bridges exponential and trigonometric functions via complex numbers.

Differentiation and integration are inverse operations.

Derivative of a composition of functions.

Derivative of a product of two functions.

Represents a smooth function as an infinite polynomial around a point.

Rules relating negation, conjunction and disjunction.

Updates the probability of a hypothesis given new evidence.

Means of large samples are approximately normally distributed.

For any integer a and prime p.

Bounds the inner product of two vectors.

The length of any side of a triangle never exceeds the sum of the other two.

Area of a triangle from its three sides, with s the semi-perimeter.

For any convex polyhedron, vertices minus edges plus faces equals two.

No positive integers satisfy the equation for any integer exponent greater than two.

There are infinitely many prime numbers.

Every integer greater than one factors uniquely into primes.

Every non-constant polynomial has as many complex roots as its degree.

Evaluates 0/0 or ∞/∞ limits by differentiating numerator and denominator.

A differentiable function attains its average slope at some interior point.

Reverses the product rule to integrate products of functions.

Relates a line integral around a closed curve to a double integral over the enclosed area.

Generalizes Green's theorem to surfaces in three-dimensional space.

Relates the flow of a field out of a volume to the divergence inside.

Represents any reasonable periodic function as a sum of sines and cosines.

Transforms differential equations into algebraic ones.

Values of a holomorphic function inside a contour follow from its values on the contour.

Contour integrals of meromorphic functions equal 2πi times the sum of residues inside.

The divergence of any curl is always zero.

Averages taken inside a convex function are bounded by the function of the average.

Bounds the probability that a random variable lies far from its mean.

Sample averages converge to the true mean as the sample grows.

Straight-line distance between two points in n-dimensional space.

Fundamental measurements of a circle from its radius.

Volume and surface area of a sphere from its radius.

Ratio for which the whole is to the larger part as the larger is to the smaller.

Links the curvature of a surface to its topology through the Euler characteristic.

The standard foundation for modern set theory, with the axiom of choice giving ZFC.