∎Famous Theorems

The Fundamental Theorem of Calculus shows differentiation and integration are inverse operations. It allows computing definite integrals via antiderivatives.

The Fundamental Theorem of Algebra states every polynomial of degree n≥1 has exactly n roots (counting multiplicity) in complex numbers.

For integer n ≥ 3, there are no positive integer solutions x, y, z satisfying xⁿ + yⁿ = zⁿ.

The Prime Number Theorem states that π(x), the count of primes ≤ x, is asymptotic to x/ln(x).

The Mean Value Theorem states for a function continuous on [a,b] and differentiable on (a,b), there exists a point where tangent slope equals secant slope.

Bayes' Theorem is a formula for reversing conditional probabilities. Used to update beliefs given new evidence.

Noether's Theorem states that every continuous symmetry of a physical system corresponds to a conserved quantity.

In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. Mathematics' most famous theorem.

The distribution of sum of many independent random variables approaches normal distribution regardless of original distribution.

If continuous f on [a,b] and k is between f(a) and f(b), there exists c in (a,b) with f(c) = k.

Stokes' Theorem states the surface integral of curl equals the line integral around the boundary. Generalizes the Fundamental Theorem of Calculus.

In any consistent formal system containing arithmetic, there exist true but unprovable statements. Also, the system cannot prove its own consistency.