Fundamental Theorems

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Fundamental Theorems

A fundamental theorem is a statement or proposition so named because it has consequences for the subject matter that are difficult to overestimate. Put another way, a fundamental theorem lies at the very heart of the subject. A theorem is called fundamental if other theorems can follow from it by accepting it as a foundation to build upon other statements or propositions; that is, by granting its truth as being self-evident. Mathematicians have designated one theorem in each of the main branches of mathematics as fundamental to that branch. Three of the most important ones are: Fundamental theorem of arithmetic, fundamental theorem of algebra, and fundamental theorem of calculus.

Fundamental theorem of arithmetic

The fundamental theorem of arithmetic states that every number can be written as the product of prime numbers in essentially one way. For example, there are no prime factors of 30 other than 2, 3, and 5. One cannot factor 30 so that it contains 2s and a 7 or some other combination.

Fundamental theorem of algebra

The fundamental theorem of algebra asserts that every polynomial equation of degree n ≥ 1, with complex coefficients, has at least one solution among the complete numbers. An important result of this theorem says that the set of complex numbers is algebraically closed; meaning that if the coefficients of every polynomial equation of degree n are contained in a given set, then every solution of every such polynomial equation is also contained in that set. To see that the set of real numbers is not algebraically closed consider the origin of the imaginary number i. Historically, i was invented to provide a solution to the equation x2 + 1 = 0, which is a polynomial equation of degree 2 with real coefficients. Since the solution to this equation is not a real number, the set of real numbers is not algebraically closed. That the complex numbers are algebraically closed, is of basic or fundamental importance to algebra and the solution of polynomial equations. It implies that no polynomial equation exists that would require the invention of yet another set of numbers to solve it.

Fundamental theorem of calculus

The fundamental theorem of calculus asserts that differentiation and integration are inverse operations,

Key Terms

Complex number— The set of numbers formed by adding a real number to an imaginary number. The set of real numbers and the set of imaginary numbers are both subsets of the set of complex numbers.

Composite number— A composite number is a number that is not prime.

Derivative— A derivative expresses the rate of change of a function, and is itself a function.

Integral— The integral of a function is equal to the area under the graph of that function, evaluated between any two points. The integral is itself a function.

Polynomial— An algebraic expression that includes the sums and products of variables and numerical constants called coefficients.

Prime number— Any number that is evenly divisible by itself and 1 and no other number is called a prime number.

a fact that is not at all obvious, and was not immediately apparent to the inventors of calculus either. The derivative of a function is a measure of the rate of change of the function. On the other hand, the integral of a function from a to b is a measure of the area under the graph of that function between the two points a and b. Specifically, the fundamental theorem of calculus states that if f(x) is a function for which f(x) is the derivative, then the integral of f(x) on the interval [a,b] is equal to f(b) - F(a). The reverse is also true, if f(x) is continuous on the interval [a,b], then the derivative of f(x) is equal to f(x), for all values of x in the interval [a,b]. This theorem lies at the very heart of calculus, because it unites the two essential halves: differential calculus and integral calculus. Moreover, while both differentiation and integration involve the evaluation of limits, the limits involved in integration are much more difficult to manage. Thus, the fundamental theorem of calculus provides a means of finding values for integrals that would otherwise be exceedingly difficult if not impossible to determine.

Other fundamental theorems in mathematics include the fundamental theorem of curves, fundamental theorem of linear algebra, fundamental theorem of projective geometry, fundamental theorem of Riemannian geometry, fundamental theorem of surfaces, and fundamental theorem of vector analysis.