Diophantine approximation: Difference between revisions

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The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of [[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]].

ItThis 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]]. Thisknowledge allowedenabled [[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 withby a similar method.

Thus 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.

== Best Diophantine approximations of a real number ==