While all but countably many real numbers are irrational (and even transcendental), it is preciously difficult to decide whether this is the case for concrete real numbers, unless some special reason is apparent. For instance, we all know that e=\sum 1/n! is transcendental, but modify the series slightly and consider the number \sum 1/(n!+1). It is not known whether this number is irrational, let alone transcendental.
In my talk, I will give sufficient conditions for classes of numbers given in terms of series expansions, continued fractions, infinite products and combinations thereof to be irrational and is some cases transcendental. I will also discuss the question of algebraic independence between individual numbers with such representations. The work is joint work with S. B. Andersen, J. Hanˇcl and M. L. Laursen in various combinations.