Editing Transcendental number theory (section)

==Approaches==

A typical problem in this area of mathematics is to work out whether a given number is transcendental. [[Georg Cantor|Cantor]] used a [[cardinality]] argument to show that there are only [[countable set|countably]] many algebraic numbers, and hence [[almost all]] numbers are transcendental.  Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational).

For this reason transcendence theory often works towards a more quantitative approach.  So given a particular complex number α one can ask how close α is to being an algebraic number.  For example, if one supposes that the number α is algebraic then can one show that it must have very high degree or a minimum polynomial with very large coefficients?  Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental.  Since a number α is transcendental if and only if ''P''(α)&nbsp;≠&nbsp;0 for every non-zero polynomial ''P'' with integer coefficients, this problem can be approached by trying to find lower bounds of the form
:<math> |P(a)| > F(A,d) </math>

where the right hand side is some positive function depending on some measure ''A'' of the size of the [[coefficient]]s of ''P'', and its [[Degree of a polynomial|degree]] ''d'', and such that these lower bounds apply to all ''P'' ≠ 0. Such a bound is called a '''transcendence measure'''.

The case of ''d''&nbsp;=&thinsp;1 is that of "classical" [[diophantine approximation]] asking for lower bounds for
:<math>|ax + b|</math>.

The methods of transcendence theory and diophantine approximation have much in common: they both use the [[auxiliary function]] concept.