It's impossible to choose a number at random | Veritasium

The Gist

Math cannot truly pick random numbers because formulas always yield deterministic results, forcing mathematicians to rely on specific rules like choosing the smallest option. However, applying this rule to the infinite set of real numbers creates an impossible paradox where no starting point or subsequent number can be identified.

Quick Overview

Computers and mathematical systems lack true random number generators and instead use deterministic algorithms tied to local time to simulate randomness. When attempting to select numbers through strict rules, positive integers and prime numbers are straightforward because they have a clear smallest value. Real numbers break this logic completely by stretching endlessly toward negative infinity without a definable first or next value, sparking a historical mathematical crisis in 1870.

Key Points: Formulas always generate the same result in mathematics, which prevents computers from possessing true random number generators. Computers typically run algorithms on local time to generate numbers that only appear random. Selecting values without true randomness requires following a specific rule, such as always choosing the smallest option available. Positive integers and prime numbers easily yield to this rule because their smallest values are one and two respectively. Real numbers encompass positive, negative, whole, fractional, and irrational values like pi and the square root of two. Applying the rule to real numbers fails because they stretch indefinitely toward negative infinity with no smallest possible value. Attempts to specify a starting point immediately after one fail because numbers like 1.01 and 1.0001 leave an infinite gap of smaller values. The mission to resolve the ordering of real numbers and find a definitive way to select them began with one man in 1870.

Context: Mathematicians and computer scientists have long grappled with the fundamental limits of randomness and deterministic computation. The core challenge of defining order within infinite sets like the real numbers underpins modern mathematical set theory.