Editing Transcendental number theory (section)

===Measure of irrationality of a real number===
{{Main|Irrationality measure}}
One way to define a Liouville number is to consider how small a given [[real number]] '''x''' makes linear polynomials |''qx''&nbsp;−&nbsp;''p''| without making them exactly&nbsp;0.  Here ''p'', ''q'' are integers with |''p''|, |''q''| bounded by a positive integer&nbsp;''H''.

Let <math>m(x, 1, H)</math> be the minimum non-zero absolute value these polynomials take and take:

:<math>\omega(x, 1, H) = -\frac{\log m(x, 1, H)}{\log H}</math>
:<math>\omega(x, 1) = \limsup_{H\to\infty}\, \omega(x,1,H).</math>

ω(''x'',&nbsp;1) is often called the '''measure of irrationality''' of a real number&nbsp;''x''. For rational numbers, ω(''x'',&nbsp;1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. [[Roth's theorem]] says that irrational real algebraic numbers have measure of irrationality 1.