Random number generation is crucial in areas such as cryptography, simulations, and gaming. True random number generators (TRNGs) rely on unpredictable physical phenomena (e.g., thermal noise or quantum effects), whereas pseudo-random number generators (PRNGs) use deterministic algorithms seeded with an initial value. The choice of seed can significantly affect the statistical quality and security of PRNG outputs. This paper investigates the use of the irrational number  (approximately 2.2360679…) as a source of randomness. We describe how ’s non-repeating, non-terminating decimal expansion might serve as a high-entropy seed or number stream to enhance unpredictability. The methodology includes theoretical analysis of ’s properties (infinite sequence, normality conjectures) and statistical testing of sequences derived from ’s digits. We present a practical case study—a real-time Monte Carlo simulation using -based random sequences—to demonstrate the feasibility and performance of this approach. Results show that -generated sequences exhibit uniform distribution and pass standard randomness tests similar to conventional PRNGs. These findings imply that certain irrational numbers could be leveraged in hybrid random generation schemes. The paper concludes with implications for using mathematical constants in secure and reproducible simulations and outlines future research directions in irrational number-based PRNG design.