[beardyw]

Irrational Numbers

Not sure what I am supposed to get from that. Irrational numbers are just reals that aren't rationals. Why is this enigmatic?

It's marked as Known Unknowns, but what is unknown?

Virtually all numbers are irrational. Are numbers enigmatic?

[MrBlueIncognito]

> It's marked as Known Unknowns, but what is unknown?

The infinite digits of pi that we don't know. But we do know that they exist, so pi, or any other irrational number, will aptly be called a "known unknown."

> Virtually all numbers are irrational.

For every n irrational numbers you name, I can name n+1 rationals. There is no firm basis to the argument that there are somehow more irrational numbers than rational numbers.

[tgv]

Do you reject Cantor's proof then? That takes mathematical bollocks.

[beardyw]

Without wishing to defend it, the gotcha here is "numbers you name". Rational numbers are "named" by putting together a string of digits. That is not possible for the irrationals so the only ones we can name are a handful with alphabetic names. There are more sheep in a field than those.

I think Cantor can rest peacefully.

[ErikCorry]

Extend alphabetical names to alphabetical descriptions and you have a countable number of irrationals.

It's a bit suspicious that we can't name or describe any of the uncountable irrational numbers, isn't it? Not even a single example.

Cantor's proof is not constructive. It doesn't name an example. If you enumerated all irrationals based on the algorithm required to calculate them then the contradiction in Cantor's diagonalization proof just turns into a failure to terminate. But that suggests that there are fewer irrationals than integers, not more.

[tgv]

You're limiting the existence of real numbers to our (human) capability of mentioning them. To me, that seems a rather arbitrary limit. Did I understand you correctly? If so, why do you accept infinity in the first place?

[beardyw]

If you reduce the irrationals to those which can be named you are probably right. I think I said that before.

[ErikCorry]

Named or described.

[johnday]

> For every n irrational numbers you name, I can name n+1 rationals.

I name "x+pi for every rational number x". Good luck!

[jwilk]

https://wikenigma.org.uk/content/mathematics/irrational_numb...

> For many irrational numbers, relatively simple mathematical proofs exist which show that it's impossible to ever arrive at a finite solution.

LOL, what does that even mean?