Is it Impossible to Generate a Truly Random Real Number in Computing?

The concept of generating a truly random real number might seem straightforward, but in practice, it presents unique challenges due to the nature of real numbers and the finite precision of digital computers. This article delves into the complexities of real number generation and explores the theoretical and practical limitations.

The Nature of Real Numbers

Infinite Precision: Real numbers have the characteristic of infinite precision, meaning there are an infinite number of values between any two distinct numbers. For instance, between 0 and 1, there are infinitely many real numbers. This property is evident when considering the interval [0, 1], where the number of potential values is un countably infinite.

Random Number Generation in Computing

Pseudo-Random Number Generators (PRNGs): The most common method of generating random numbers in computing relies on pseudo-random number generators (PRNGs). These algorithms generate numbers that appear random but are actually deterministic and repeatable. For example, a typical PRNG could produce a random floating-point number between 0 and 1, but the precision is limited by the finite representation of the number.

Uniform Distribution: Even when using a finite representation, you can generate random real numbers uniformly over a specified range. However, this uniformity is constrained by the precision of the representation. For example, generating a random real number in the interval [0, 1] using a PRNG with 32-bit precision will always result in a finite set of possible values.

Theoretical Possibility vs. Practical Constraints

Theoretical Possibility: While theoretically it is possible to generate a random real number in a finite interval [a, b] in a countably infinite number of steps, this process would not be feasible in practice. Consider an alternative universe where a magic genie can roll a 10-sided die in an arbitrarily small amount of time. If the genie were to generate a digit in 1 second, then in 1/2 second, and so on, the time required to generate an infinite number of digits (representing a random real number) would be finite.

Equal Distribution of Integers: Generating a random integer among all integers with an equal probability distribution is not mathematically possible. This is due to the countable and uncountable nature of the sets of integers and real numbers, respectively. Generating a single random integer with equal distribution among all integers would not reflect a uniform probability distribution because the range of integers is infinite.

Implications and Consequences

Random Real Numbers and Intervals: Despite the infinite precision of real numbers, it is possible to generate a random real number within any finite interval [a, b]. However, generating a random rational number over the same interval would not be possible, as rational numbers can be put in a one-to-one correspondence with the set of all integers, thus violating the uniform distribution requirement.

Randomized Distributions and Integers: Generating a random integer with a uniform distribution is not possible, as any integer generated would be infinitely closer to 0 than to positive or negative infinity. This means that the distribution is not truly uniform. For example, generating a million random integers would result in a range of only finite magnitude (e.g., P M - N), which is infinitesimally small compared to the entire set of integers, thereby violating the uniformity of the distribution.

The Uniqueness of Real Numbers: Theoretically, you can generate a random real number with infinite precision within a finite interval, but this process is not practical in computational scenarios. The nature of real numbers ensures that every number generated would represent an irrational, transcendental, and non-computable number, making them quite unique and diverse.