For a rational number "α" the irrationality measure is "μ"("α") = 1. The Thue–Siegel–Roth theorem states that if "α" is an algebraic number, real but not rational, then "μ"("α") = 2. Almost all numbers have an irrationality measure equal to 2. Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number "e" has "μ"("e") = 2. The irrationality measures of , log 2, and log 3 are at most 7.103205334137, 3.57455391, and 5.125, respectively. It has been proven that if the series formula_40 (where "n" is in radians) converges, then formula_41's irrationality measure is at most 2.5. The Liouville numbers are precisely those numbers having infinite irrationality measure. The "irrationality base" is a weaker measure of irrationality introduced by J. Sondow, and is regarded as an irrationality measure for Liouville numbers. It is defined as follows:  Let formula_42 be an irrational number. If there exists a real number formula_43 with the property that for any formula_44, there is a positive integer formula_45 such that  then formula_47 is called the irrationality base of formula_48 and is represented as formula_49  If no such formula_47 exists, then formula_48 is called a "super Liouville number". Example: The series formula_52 is a "super Liouville number", while the series formula_53 is a Liouville number with irrationality base 2. (formula_54 represents tetration.) Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental. However, not every transcendental number is a Liouville number. The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of "e", one can show that "e" is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that is another such example. The proof proceeds by first establishing a property of irrational algebraic numbers. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers, where the condition for "well approximated" becomes more stringent for larger denominators.