Fundamentaltheorem

Fundamental theorems are results in mathematics that express essential, broad relationships within a subject. The term denotes results with foundational significance, often providing a bridge between ideas or a unifying framework for a theory.

Fundamental Theorem of Calculus: It links differentiation and integration. If a function is continuous on an interval, the integral of its derivative gives back the original function up to a constant, and the derivative of the accumulated integral equals the integrand.

Fundamental Theorem of Arithmetic: Every integer greater than 1 can be factored uniquely into primes (up to the order of the factors and units). This prime factorization underpins much of number theory and arithmetic.

Fundamental Theorem of Algebra: Every non-constant polynomial with complex coefficients has at least one complex root, so such polynomials factor completely into linear factors over the complex numbers.

Fundamental Theorem of Linear Algebra: The rank of a matrix relates the dimensions of its column space and nullspace, and the rank-nullity theorem ties these dimensions to the number of columns. This theorem summarizes key structural relationships that govern linear systems and transformations.

Other fields also use the label fundamental theorem for central results, and the exact statements vary by context. In each case, the theorem highlights a core, widely applicable principle that supports further theory and applications.