Pseudo-random number generators (PRNG) based on irrational numbers are proposed elsewhere. They generate random numbers using digits of real numbers which decimal expansions neither terminate nor become periodic and practically their decimal expansion has infinite period. Using that algorithm, we generate sequences of random numbers and then we check their randomness with statistical tests from Diehard battery. Our main idea is to check is there a difference in the randomness of the generated sequences if digits of any irrational non- transcendental number (like √2, √3,√5, … ) are used versus the case when digits of a transcendental number (like π or e) are used. In our experiments we use about 3·107 digits of a given non-periodic irrational or transcendental number. Many experiments were done and all generated sequences by proposed PRNG based on irrational numbers passed the Diehard tests very well. We may conclude that there is not a significant difference in the randomness of the generated sequences in the both cases (irrational nontranscendental versus irrational transcendental number).