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Concept
Diophantine approximation
Summary
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. 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 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 fractions.
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 numbers, 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 Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that pi and 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 equations.
The 2022 Fields Medal was awarded to James Maynard for his work on Diophantine approximation.
Continued fraction#Best rational approximations
Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition, the rational number p/q is a best Diophantine approximation of α if
for every rational number math|p/q' different from p/q such that 0 < q′ ≤ q.
Official source
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