Some Fundamental Theorems of Maths

Every branch of mathematics has key results that are so
important that they are dubbed fundamental theorems.

The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

Pythagoras-Thales

Left: Pythagoras. Right: Thales.

The procedure of definitions and postulates, followed by theorems to be proved, goes back beyond Euclid, to Pythagoras and Thales. The old joke that a mathematician is a machine for changing coffee into theorems is often attributed to Paul Erdös, although it appears to have been said first by Alfréd Rényi, another Hungarian mathematician. The joke captures a critical aspect of the mathematician’s role: proving theorems by using axioms and other theorems already proved.

Fundamental Theorem of Arithmetic

In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers. This identifies the prime numbers as the basic building blocks of all the integers. All other whole numbers are products of primes. The proof of this result goes back to Euclid. The fundamental theorem of arithmetic is essential in the proof of many crucial results in number theory.

This theorem ensures unique factorization of positive integers. It is one of the reasons why 1 is not treated as a prime number. This is largely a matter of convenience.

Fundamental Theorem of Algebra

The fundamental theorem of algebra states that every single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. The theorem has the consequence that every non-zero single-variable polynomial with complex coefficients has precisely as many complex roots as its degree (provided each root is counted up to its multiplicity).

Euler, d’Alembert, Argand, Lagrange, Laplace, Argand and Gauss all worked on proofs. Today, the theorem us usually encountered in a course in complex variable analysis. There is no purely algebraic proof of the theorem; all proofs must use the concept of the completeness of the real numbers, which is an analytical concept. The theorem was given its name when algebra was preoccupied primarily with the theory of polynomial equations. It has been observed that the Fundamental Theorem of Algebra is neither fundamental nor algebraic.

Fundamental Theorem of Calculus

The fundamental theorem of calculus provides a connection between derivatives and integrals. The first part of the theorem shows that an indefinite integration can be reversed by a differentiation:

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int f(x)\,\mathrm{d}x = f(x) \,.

The second part allows us to evaluate the integral of a function {f(x)} by using an `anti-derivative’ {F(x)} of {f(x)}. If {f(x)} is a derivative of {F(x)}, then

\displaystyle  \int f(x) \,\mathrm{d}x = \int F^\prime(x) \,\mathrm{d}x = F(x)  \,.

The theorem was studied by James Gregory and Isaac Barrow. A more comprehensive treatment was provided by Isaac Newton and Gottfried Leibniz.

The theorem can be extended to integrals in higher dimensions and integrals on smooth manifolds. One of the most fruitful generalizations is Stokes’s theorem. This may be written

\displaystyle \int_{A} \mathbf{\nabla\times V\cdot n}\, \mathrm{d} A = \oint_{\partial A} \mathbf{V\cdot\mathrm{d} s}\,.

This result states that the areal integral of vorticity over a surface is equal to the circulation around the boundary. In turn, Stokes’s theorem itself has been generalized to become an important principle in differential geometry: the integral of a differential form over the boundary of an orientable manifold is equal to the integral of its exterior derivative over the manifold; symbolically,

\displaystyle \int_{\partial\Omega} \omega = \int_{\Omega} \mathrm{d} \omega \,.

Fundamental Theorem of Curves

In differential geometry, the fundamental theorem of curves states that any regular curve has its shape and size) completely determined by its curvature {\kappa} and torsion {\tau}. For example, a curve with {\kappa \equiv 1} and torsion {\tau \equiv 0} must be a circle, although further data is required to determine its position and orientation.

Many other Fundamental Theorems

The number of fundamental theorems is large. We may mention

  • Fundamental theorem of surfaces
  • Fundamental theorem of Galois theory
  • Fundamental theorem on homomorphisms
  • Fundamental theorem of linear algebra
  • Fundamental theorem of Riemannian geometry
  • Fundamental theorem of vector analysis
  • Fundamental theorem of linear programming

and many more might be added to this list.

Fundamental Theorem of Geometry (?)

A curious omission from the standard list is geometry. There is no particular theorem named as the fundamental theorem of geometry. Several candidates suggest themselves. One of the leading contenders must be the theorem of Pythagoras for right-angled triangles:

\displaystyle a^2 + b^2 = c^2 \,.

This underlies the structure of Euclidean space. It may be generalized in several directions. For example, the fundamental line-element of Riemannian geometry is

\displaystyle \mathrm{d}s^2 = g_{\mu\nu} \mathrm{d}x^\mu \mathrm{d} x^\nu

from which a wealth of interesting and valuable results follow.


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