Editing Diophantine approximation (section)

{{Short description|Rational-number approximation of a real number}}
{{more citations needed|date=May 2023}}
{{Use American English|date = March 2019}}
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In [[number theory]], the study of '''Diophantine approximation''' deals with the approximation of [[real number]]s by [[rational number]]s.  It is named after [[Diophantus of Alexandria]].

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 rational number with a smaller denominator. This problem was solved during the 18th century by means of [[simple continued fraction]]s.

Knowing the "best" approximations of a given number, the main problem of the field is to find sharp [[upper and lower bounds]] of the above difference, expressed as a function of the [[denominator]]. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for [[algebraic number]]s, which is itself larger than the lower bound for all real numbers. Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a [[transcendental number]]. 

This knowledge enabled [[Joseph Liouville|Liouville]], in 1844, to produce the first explicit transcendental number. Later, the proofs that [[Pi|{{pi}}]] and ''[[e (mathematical constant)|e]]'' are transcendental were obtained by a similar method. 

Diophantine approximations and [[transcendental number theory]] are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of [[Diophantine equation]]s.

The 2022 [[Fields Medal]] was awarded to [[James Maynard (mathematician)|James Maynard]], in part for his work on Diophantine approximation.