A number of the form a to the b, where a is algebraic and not equal to zero or one and b is algebraic and this kind of number, yields a number in this set of numbers according to Gelfond's theorem. This set of numbers include all non-complex transcendental numbers. The proof that the diagonal of a square is incommensurable proved that it was this kind of number. These numbers cannot be expressed in the (*) form a over b, where a and b are integers, and their decimal expansion never repeats or terminates. For 10 points, name this set of numbers 
A number of the form a to the b, where a is algebraic and not equal to zero or one and b is algebraic and this kind of number, yields a number in this set of numbers according to Gelfond's theorem. This set of numbers include all non-complex transcendental numbers. The proof that the diagonal of a square is incommensurable proved that it was this kind of number. These numbers cannot be expressed in the (*) form a over b, where a and b are integers, and their decimal expansion never repeats or terminates. For 10 points, name this set of numbers that includes the golden ratio and the square root of two and that constitutes the real numbers with the rationals.
A number of the form a to the b, where a is algebraic and not equal to zero or one and b is algebraic and this kind of number, yields a number in this set of numbers according to Gelfond's theorem.
{irrational} numbers [accept {transcendental} numbers before it is read]