Editing Diophantine approximation (section)

== Best Diophantine approximations of a real number ==
{{main|Simple continued fraction#Best rational approximations|Irrationality measure}}

Given a real number {{math|''α''}}, there are two ways to define a best Diophantine approximation of {{math|''α''}}. For the first definition,<ref name="Khinchin 1997 p.21">{{harvnb|Khinchin|1997|p=21}}</ref> the rational number {{math|''p''/''q''}} is a ''best Diophantine approximation'' of {{math|''α''}} if
:<math>\left|\alpha -\frac{p}{q}\right | < \left|\alpha -\frac{p'}{q'}\right |,</math>
for every rational number {{math|''p'''/''q' ''}} different from {{math|''p''/''q''}} such that {{math|0 < ''q''&prime;&nbsp;≤&nbsp;''q''}}.

For the second definition,<ref>{{harvnb|Cassels|1957|p=2}}</ref><ref name=Lang9>{{harvnb|Lang|1995|p=9}}</ref> the above inequality is replaced by
:<math>\left|q\alpha -p\right| < \left|q^\prime\alpha - p^\prime\right|.</math>

A best approximation for the second definition is also a best approximation for the first one, but the converse is not true in general.<ref name=Khinchin24>{{harvnb|Khinchin|1997|p=24}}</ref>

The theory of [[Simple continued fraction|continued fraction]]s allows us to compute the best approximations of a real number: for the second definition, they are the [[Simple continued fraction#Convergents|convergents]] of its expression as a regular continued fraction.<ref name=Lang9/><ref name=Khinchin24/><ref>{{harvnb|Cassels|1957|pp=5–8}}</ref>  For the first definition, one has to consider also the [[Simple continued fraction#Semiconvergents|semiconvergents]].<ref name="Khinchin 1997 p.21"/>

For example, the constant ''e'' = 2.718281828459045235... has the (regular) continued fraction representation

:<math>[2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots\;].</math>

Its best approximations for the second definition are
:<math> 3, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{87}{32}, \ldots\, ,</math>
while, for the first definition, they are
:<math>3, \tfrac{5}{2}, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7},
\tfrac{49}{18}, \tfrac{68}{25}, \tfrac{87}{32}, \tfrac{106}{39}, \ldots\, .</math>