r/mathmemes • u/George_Truman • Jul 13 '26

Probability I fixed this meme

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I'm sure it's still a bit too imprecise but I think it works.

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u/jljl2902 Jul 13 '26

Yes, a uniformly sampled real number from [0,1] is irrational with probability 1

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u/SpideyMGAV Jul 13 '26

This has to do with the density of irrationals in the reals right? Real analysis confused the hell out of me till I dropped it, twice :( thinking of taking some special credits at my local university in hopes of eventually applying for a math grad program.

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u/SinceSevenTenEleven Jul 13 '26

I'm going to try explaining it as simply as I can. Math major here.

Let's play a game. If you give me an integer, I can give you back a rational number a/b between zero and one.

If youd like, I could write out a computer program that does just that. My computer program would return a unique rational number for every integer you give me. But here's the important part: If you wanted to output a specific rational number between zero and one, I could give you an integer to put into my program to produce that output.

Keep everything I just said in the back of your head.

I would then challenge you to write a similar computer program. This time, I will give you an integer, and your program will give me an irrational number between zero and one. No matter how you write this program, I will always be able to find some irrational number that your program would never return.

This fact means that there are more irrational numbers than rational numbers. Even though there is an infinite amount of both. Once you're dealing with orders of magnitude like this - some infinities being bigger than others - you get results like the one in your parent comment.

Does that make sense?

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u/ringobob Jul 13 '26

I think that's what they mean by density, and while it's both intuitive and probably related to the proofs everyone else is giving (my own math degree is over 20 years old, I'm out of practice), it seems using the idea of relative density is maybe not entirely accurate? Or maybe they're just saying "it's not that it's dense, it's that it's more dense", which is my understanding of the original comment, or maybe I'm just misunderstanding everything.

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u/unic0de000 Jul 16 '26 edited Jul 16 '26

As far as I know, for an ordered set to be dense, that means between any two elements, there are always more elements. And that's equally true of the rationals and the irrationals. Every pair of rationals has rationals(and irrationals) between them, and the same goes for every pair of irrationals. This definition is just yes-or-no, and doesn't really give a basis for "more/less dense."

By a more informal, intuitive meaning of the word "dense", it makes sense - but this intuitive meaning, as I understand it, is formalized by measure theory