Chance

This book analyzes various aspects of number theory, positional systems, randomness, and information theory. From the introduction, it raises classical paradoxes—such as the one proposed by Zeno of Elea—to question the existence of infinity, or at least to show that it may be a mistaken concept as it is usually interpreted. Based on this, and in order to resolve Zeno’s paradox, the book attempts to demonstrate that there are numbers with decimal digits that, despite having decimal expansions, behave like integers. These numbers are called “high-precision numbers.” They are numbers with decimal digits that nevertheless behave like integers in certain contexts. These numbers make it possible to question the classical idea that every improbable event is possible, showing that there is actually a limit to the probability of occurrence, since certain extremely improbable events become impossible. This is what the book calls “0% probability,” a limit defined by high-precision numbers. The book also analyzes how all of this relates to positional numeral systems, investigating how large such systems would need to be in order to avoid information loss. It shows that there exists a point of “informational saturation,” where very large numbers stop contributing meaningful information and only generate redundancy or inefficiency. This analysis establishes interesting relationships between information saturation and the mathematical constant Euler's number, and even Golden ratio. In relation to randomness, the book also introduces the concept of the “wheel,” which consists of drawing a number as many times as there are possible outcomes. This concept is used to analyze how numbers repeat—or fail to appear—within a finite set of events, such as drawings or combinations, making it possible to distinguish different types of randomness. The wheel also demonstrates the deep relationship between randomness and positional systems, particularly with respect to information saturation. Finally, the book presents a computer program that demonstrates how extremely improbable events, once they exceed certain limits, simply do not occur, thereby confirming their impossibility. Altogether, the work offers a clear and rigorous analysis of randomness, the limits of what is possible, infinity, and information.