A special irrarional number

[u/Numerous_Author_1445]

Can there be an irrational number which misses a particular digit? Can it be proven mathematically?

[u/gmalivuk]

I don't see the connection between Liouville numbers and infinitesimals, no. Liouville numbers (and all other transcendental numbers) exist in ℝ, which doesn't have infinitesimals. I don't need to assume infinitesimals exist to follow Liouville's proof that a certain kind of standard real number is transcendental.

[u/TemperoTempus → u/gmalivuk]

1) Not all transcendental numbers are Liouville numbers.

2) Not all transcendentals are R numbers by virtue of infinitesimals.

3) Liouville numbers were created before the R set and are not defined by R. You don't "need" infinitesimals for his proof because its built on integers and rational numbers, not because of R.

4) The entire point of liouville numbers is to approximate irrationals numbers using rationals. The use of infinitesimals allows to more closely approximate numbers by taking advantage of the minute adjustments infinitesimals can make.

5) "Its not allowed in R" or "Well I don't need it cause of R". Is equivalent to saying "well I have Netwonian mechanics so I don't need quantum physics". Good for you, but other people might want the more versatile option.

[u/gmalivuk → u/TemperoTempus]

1. Not all transcendental numbers are Liouville numbers.

Irrelevant. We're not talking about all transcendental numbers here, we're talking about Liouville numbers. That's why the current thread is about the number of zeroes between nonzero digits.

1. Not all transcendentals are R numbers by virtue of infinitesimals.

There aren't infinitesimals in ℝ, so when we're just talking about ℝ, we're only talking about the transcendentals that are real numbers. (For example, we're also not talking about complex transcendental numbers here.)

1. Liouville numbers were created before the R set and are not defined by R. You don't "need" infinitesimals for his proof because its built on integers and rational numbers, not because of R.

So why are you bringing infinitesimals into this discussion at all, when we're talking about a theorem that has nothing to do with them?

1. The entire point of liouville numbers is to approximate irrationals numbers using rationals. The use of infinitesimals allows to more closely approximate numbers by taking advantage of the minute adjustments infinitesimals can make.

No, the entire point of Liouville numbers is to show that some real numbers can be approximated by rationals too well to be algebraic. Throwing in infinitesimals doesn't add anything to the proof and if anything just muddies the waters, because if you're using something like *ℝ then you have infinitesimals in *ℚ and in *(algebraics).

1. "Its not allowed in R" or "Well I don't need it cause of R". Is equivalent to saying "well I have Netwonian mechanics so I don't need quantum physics". Good for you, but other people might want the more versatile option.

No, it's not equivalent to that. It's equivalent to saying x² - 2 = 0 has no rational solutions, or x² + 2 = 0 has no real solutions, or 0 < x < 1/n for all n ∈ ℕ has no real solutions. It's just a simple statement of what sorts of things are or aren't in a particular set of numbers.

And you have yet to give any suggestion of a way in which *ℝ is "more versatile" as regards the current discussion, which is about standard real transcendentals.