Hacker News Reader

Comment by dr_dshiv | Hacker News Reader

You can represent pi on a number line but it is absolutely completely impossible to randomly put a dot down on a number line and have it be pi. You can achieve endless measurable precision with decimal rational numbers. So randomly placing a dot on a line will always be a rational number.

Right???

View on HN

Not necessarily. If you assume an ideal ruler and compass, it‘s pretty easy to construct the square root of any number which means that you can easily put a dot on that number line which is provably not a rational number. There is the question of whether continuity actually exists (are time and space quantized like matter and energy or are they continuous? this is currently unknown) and the fact that your paper and the line on it are composed of discrete molecules. But if the real numbers are, in fact, real, then the probability that your dot is at a rational point is actually 0 since while the number of rationals in [0,1] is infinite, it’s only countably infinite and the number of irrationals in [0,1] is uncountably infinite meaning |ℚ|/|ℝ\ℚ|=0.

Actually, the opposite is true. The rationals have measure zero, so a number selected uniformly at random from 0 to 1 (formalized way of placing a dot on a number line), has a 0% chance of being rational.

You mean in real world or in math?

In real world, correct, assuming "dot" and "number line" are consist of real world materials.

In math, you need to define "randomly placing a dot" first, because it's proven there isn't a uniform distribution over real numbers ("pick randomly" is usually a colloquial way to say "pick from a uniform distribution.")

> it is absolutely completely impossible to randomly put a dot down on a number line and have it be pi

It's not impossible, it just has zero probability of occurring.

Reply on news.ycombinator.com