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Approximation theory is a branch of mathematics, and a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers. Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain.
In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set.
The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p/q is a "good" approximation of a real number α if the absolute value of the difference between p/q and α may not decrease if p/q is replaced by another
Diophantine approximation, the approximation of real numbers by rational numbers Padé approximation , the approximation of functions obtained by set of Padé approximants Any approximation represented in a form of rational function
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely 2 π {\displaystyle {\sqrt {2\pi }}} .
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