Diophantine approximations height transcendental number theory transcendental number π e Diophantine geometry algebra Diophantine approximation
There is also the closely linked area of Diophantine approximations: given a number 



x


{\displaystyle x}

, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call 



a

/

q


{\displaystyle a/q}

 (with 



gcd
(
a
,
q
)
=
1


{\displaystyle \gcd(a,q)=1}

) a good approximation to 



x


{\displaystyle x}

 if 




|

x
−
a

/

q

|

<


1

q

c






{\displaystyle |x-a/q|<{\frac {1}{q^{c}}}}

, where 



c


{\displaystyle c}

 is large.) This question is of special interest if 



x


{\displaystyle x}

 is an algebraic number. If 



x


{\displaystyle x}

 cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in transcendental number theory: if a number can be better approximated than any algebraic number, then it is a transcendental number. It is by this argument that π and e have been shown to be transcendental.
