![]() ![]() Specifically, if f is any continuous function on the interval, then for every value of x in you can compute the following function: That these should be related is not obvious, but the FTC says that the rate of change for a certain integral is given by the function whose values are being accumulated. Derivatives tell us about the rate at which something changes integrals tell us how to accumulate some quantity. The fundamental theorem of calculus (FTC) connects derivatives and integrals. Like the fundamental theorem of arithmetic, this is an "existence" theorem: it tells you the roots are there, but doesn't help you to find them. The theorem states that every nonconstant polynomial of degree n has exactly n roots in the complex number system. The fundamental theorem of algebra tells us that we need complex numbers to be able to find all roots. The real numbers are not sufficient for solving algebraic equation, a fact known to every child who has pondered the solution to the equation x 2 = –1. The fundamental theorem of algebra connects polynomials with their roots (or zeros). If p is a prime number, sometimes p+2 is also prime (the so-called twin primes), but sometimes there is a huge gap before the next prime. The natural numbers are regularly spaced, but the gap between consecutive prime numbers is extremely variable. It is trivial to enumerate the natural numbers, but each natural number is "built" from prime numbers, which defy enumeration. ![]() ![]() This theorem connects something ordinary and common (the natural numbers) with something rare and unusual (primes). The theorem states that every integer greater than one can be represented uniquely as a product of primes. The fundamental theorem of arithmetic connects the natural numbers with primes. ![]() In this article I briefly and informally discuss some of my favorite fundamental theorems in mathematics and cast my vote for the fundamental theorem of statistics. It is interesting that statistical textbooks do not usually highlight a "fundamental theorem of statistics." " A fundamental theorem is a deep (often surprising) result that connects two or more seemingly unrelated mathematical ideas. In many mathematical fields there is a result that is so profound that it earns the name "The Fundamental Theorem of. Although I currently work as a statistician, my original training was in mathematics. ![]()
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