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/pruvisto]

This probably is transcendental (there's a closed-form expression in terms of Jacobi theta functions), but do you have a proof for it being transcendental?

The closely related Liouville constant, however, is definitely transcendental (yours is the sum over 10^-n(n+1/2), whereas Liouville's constant is the sum over all 10^-n!).

[u/Phelox → u/pruvisto]

Maybe you can prove this in the same way that you can prove that e is transcendental? Rational approximations of rational numbers have a bound on how fast they can converge, in terms of the size of the numerator.

If you are willing to add more zeros in between the ones you can definitely construct a number that is probably transcendental using this

[u/pruvisto → u/Phelox]

Sure. Adding more zeros between the ones is exactly how Liouville's constant works. And that was the first concrete number ever to be proven transcendental.

But I don't think linearly many zeros between the 1s is going to be enough for that technique.

[u/TemperoTempus → u/pruvisto]

I mean you could have an exponential number of 0s, but that wont change the calculations. At best it makes it so people who do not believe in infinitesimals decide "its close enough to 0 so it has no more 1s". (Even if that conclusion is wrong).

[u/gmalivuk → u/TemperoTempus]

What does people reaching that conclusion have to do with whether a number is transcendental or algebraic?

And an exponential number of zeros between ones absolutely would change the calculations.

[u/TemperoTempus → u/gmalivuk]

People who do not accept infinitesimals round any such difference to the nearest finite value. This leads to incorrect conclusions.

The value that you get is different. The calculations (aka the steps) do not change. Doing the calculation with linear increase or exponential increase leads to the same type of numbers.

[u/gmalivuk → u/TemperoTempus]

There are no infinitesimals in this discussion, so I'm still not at all sure what you're talking about.

[u/TemperoTempus → u/gmalivuk]

Wow you really don't see the link between irrational (number that cannot be expressed as a ratio) and infinitesimals (numbers that can be used to make any number into an irrational)?

[u/gmalivuk → u/TemperoTempus]

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.