---
title: 'It''s Impossible to Choose a Number at Random'
source: 'https://youtube.com/watch?v=5U-V0y-fkSw'
video_id: '5U-V0y-fkSw'
date: 2026-08-27
duration_sec: 100
channel: 'Veritasium'
---

# It's Impossible to Choose a Number at Random

> Source: [It's Impossible to Choose a Number at Random](https://youtube.com/watch?v=5U-V0y-fkSw)

## Summary

This video explores the philosophical and mathematical challenge of selecting a number at random, highlighting why true randomness is impossible for computers and why ordering the real numbers is a fundamental problem. It traces the historical mission beginning in 1870 to resolve this paradox, setting the stage for a deeper discussion.

### Key Points

- **Computers lack true randomness** [00:02] — Computers cannot generate true random numbers because formulas always produce the same result; they rely on algorithms seeded by local time to simulate randomness.
- **Rules for selection** [00:16] — In mathematics, selection requires following a rule, such as choosing the smallest element, which works for sets like positive integers (1) or primes (2).
- **The real numbers problem** [00:31] — The real numbers include all positives, negatives, fractions, and irrationals like pi, making it impossible to define a smallest element because they extend to negative infinity.
- **Specific rules fail** [00:44] — Even a specific rule like 'choose the smallest number after one' fails because there is always a smaller number (1.01, 1.0001, etc.), leading to an infinite regress.
- **The paradox of infinite options** [01:13] — Despite knowing there are infinite real numbers, we cannot specify an order (next, previous, first, last), leaving us stuck on how to pick one.
- **Historical mission begins** [01:27] — In 1870, a man took on the task of ordering the real numbers definitively, risking his life—and nearly dying—in the process.

### Conclusion

The video underscores a deep mathematical paradox: true randomness is unattainable, and even with infinite choices, we cannot define a selection rule for the real numbers. This sets the stage for exploring historical attempts to resolve the ordering problem.

## Transcript

random because formulas always give the same result, which is why computers don't have true random number generators. Instead, they usually run an algorithm on your current local time to generate numbers that appear random. So,
if we can't pick randomly, how do we select anything in math? Well, the only way is to follow a rule of some sort. So, a rule could be always choose the smallest thing. For example, if we're looking at whole positive integers, the
smallest is one. For prime numbers, it would be two. Easy. But, what about the real numbers? That's any number, positive, negative, whole, fraction, even irrational, like pi or the square root of two.
Now, try to choose the smallest one. It's impossible. The real numbers stretch off to negative infinity. Even if we try to fix our rule by making it super specific, like choose the smallest number after one, we still get
stuck. There's 1.01, then 1.0001, then 1.00000001, and so on. and so on. So, really, what number comes after one?
If we can't begin to specify the order of the real numbers, next and previous, first and last, we're stuck. The ridiculous part is we know we have infinite options, but despite that, we can't figure out how to just pick one.
The mission to resolve this began with one man in 1870. He took on the task of putting the real numbers in a definitive order, even if it killed him. And, it nearly did.