of transcendence of x. If the ω(x, n) are bounded, then ω(x) is finite, and x is called an S number. If the ω(x, n) are finite but unbounded, x is called a T number. x is algebraic if and only if ω(x) = 0. Clearly the Liouville numbers are a subset of the U numbers. William LeVeque in 1953 constructed U numbers of any desired degree. The Liouville numbers and hence the U numbers are uncountable sets. They are sets of measure 0. T numbers also comprise a set of measure 0. It took about 35 years to show their existence. Wolfgang M. Schmidt in 1968 showed that examples exist. However, almost all complex numbers are S numbers. Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers: this shows that e is an S number and gives a proof of the transcendence of . This number is known not to be a U number. Many other transcendental numbers remain unclassified. Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in two indeterminates with integer coefficients such that P(x, y) = 0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class. This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or . The symbol S probably stood for the name of Mahler's teacher Carl Ludwig Siegel, and T and U are just the next two letters. Koksma's equivalent classification Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers. Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of this finite set such that |x − α| has the minimum positive value. Define ω*(x, H, n) and ω*(x, n) by: If for a smallest positive integer n, ω*(x, n) is infinite, x is called a U*-number of degree n. If the ω*(x, n) are bounded and do not converge to 0, x is called an S*-number, A number x is called an A*-number if the ω*(x, n) converge to 0. If the ω*(x, n) are all finite but unbounded, x is called a T*-number, Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes. The