Decimal representation of reals

[Bob3141592]

The Wikipedia article on Real Numbers says every "a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way" I don't think this is right. Doesn't there have to be some real numbers that cannot be named, even if we allow that name to be letters from an infinite string of digits?

You can write out a countably infinite list of numbers in [0,1], and each of those numbers can have an infinite number of digits to the right of the decimal point. For any n, you can be certain you've written down the entire list of numbers with n decimal places. According to the Wikipedia article, the infinite list has to contain every real number. But doesn't Cantor's Diagonalization argument say that the list must be incomplete, that there are points which were not on that list?

So either 1) there is no method of knowing you have written all of the decimal representations for any number of digits m > n, at least as n approaches infinity. Not only that, but the decimal representation exists but you must necessarily have missed writing it. Or else 2) there are reals that don't have decimal representation, which I think is to say reals which cannot be named in and of themselves.

Frankly I dislike the first, for this reason. The generating method can include recursively applying the diagonalization procedure and adding any such determined omission onto the list. We can do that countably infinitely often but still Cantor's argument holds, doesn't it?

Am I going wrong, and if so, where?

 

[Hurkyl]

The Wikipedia article on Real Numbers says every "a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way"
...
You can write out a countably infinite list of numbers in [0,1], and each of those numbers can have an infinite number of digits to the right of the decimal point. For any n, you can be certain you've written down the entire list of numbers with n decimal places. According to the Wikipedia article, the infinite list has to contain every real number.

That doesn't follow. Your list only contains real numbers x for which there exists an integer n such that x has a decimal representation of length n -- in other words, it contains only finite decimal strings.

1/9, for example, is not in your list...

 

[mathman]

The Wikipedia article on Real Numbers says every "a Real Number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way" I don't think this is right. Doesn't there have to be some real numbers that cannot be named, even if we allow that name to be letters from an infinite string of digits?

The Wiki statement is correct for any specific real number. However, you cannot make a list of all of them (Cantor's theorem).

 

[Bob3141592]

That doesn't follow. Your list only contains real numbers x for which there exists an integer n such that x has a decimal representation of length n -- in other words, it contains only finite decimal strings.

1/9, for example, is not in your list...

No, the list has infinite entries of infinite decimals. The finite n only comes into prove that subsets of the list contain every possible combination of the first n digits, that no combinations were overlooked. This holds for any arbitrarily large n. I also suggest that any infinitely repeating last digit can be represented with a single special letter, call it "d" for "... dots" (and even another "s" for start repeat sequence, thus giving us a finite representation for any rational). So the only opportunity to miss numbers only comes in at infinite n.

I'm not saying those infinite strings of digits specifying the transcendental numbers can't be on the list. I'm sure any you can name are on the list. But from this I suspect there must exist a class of reals that cannot be named. (My terminology may be off, since "class" has technical meanings I'm not up on).

 

[Bob3141592]

The Wiki statement is correct for any specific real number. However, you cannot make a list of all of them (Cantor's theorem).

I'm not sure if you're agreeing with me or not. Are there necessarily non-specific real numbers? This is the notion I'm trying to get at.

 

[CRGreathouse]

Doesn't there have to be some real numbers that cannot be named, even if we allow that name to be letters from an infinite string of digits?

You're getting into the issue of definable reals here. There are only countably many definable reals, so yes there are some that can't be named. But in some sense even these reals have a decimal expansion -- just not one we can find.

 

[Bob3141592]

You're getting into the issue of definable reals here. There are only countably many definable reals, so yes there are some that can't be named. But in some sense even these reals have a decimal expansion -- just not one we can find.

Thanks, CRGreathouse. That's just what I was thinking, though I didn't know the term definable reals. There's even a Wiki page on it, noted as speculative, but something I can look into.

I think this has relevance to the idea of neighborhoods with or without a distance operator, and then to open balls in topology. At least, that's what I was trying to understand when I got sidetracked by these questions. Not that I understand it now, of course. But maybe I'll be closer tomorrow.

By the way, CRGreathouse sounds so formal. Can I just call you CR, or would you prefer Greathouse?

 

[CRGreathouse]

Thanks, CRGreathouse. That's just what I was thinking, though I didn't know the term definable reals. There's even a Wiki page on it, noted as speculative, but something I can look into.

I hoped it would be useful. Usually just finding enough terminology that you can find an article, paper, or book is all it takes.

By the way, CRGreathouse sounds so formal. Can I just call you CR, or would you prefer Greathouse?

If you'll let me call you Bob, I'll let you call me Charles. Deal?

 

[Hurkyl]

No, the list has infinite entries of infinite decimals. The finite n only comes into prove that subsets of the list contain every possible combination of the first n digits, that no combinations were overlooked.

I took your explanation as a means of constructing this list. (Given that I consider it obvious how to build a list containing all finite decimal strings)

So how exactly were you planning on constructing this list, for which you claim

According to the Wikipedia article, the infinite list has to contain every real number​

 

[Bob3141592]

I took your explanation as a means of constructing this list. (Given that I consider it obvious how to build a list containing all finite decimal strings)

So how exactly were you planning on constructing this list, for which you claim

According to the Wikipedia article, the infinite list has to contain every real number​

You might misunderstand me. I'm saying the list can't be constructed, that there are reals that cannot be explicitly named, and therefore can't have a decimal form. Although they can't be named themselves, they can be referred to as being in the neighborhood of a real that can be named. Ultimately, this seems like the minimal open neighborhood for any point you can name. If I have a distance relation in a topology, I must have a nameable radius r to define what is not in the open ball. Any r. So at some point what is in the open ball has to be unnamably close to the center point.

Within that minimal open neighborhood I can define an equivalence class so that any sequence that converges to any point in that neighborhood is as good as converging into any other point in that neighborhood. And I can do this in a Hausdorff space, where any two points that can be named have disjointed neighborhoods. Personally, I can't seem to understand convergence or continuity otherwise.

I don't know, for me it seems like an important point to get straight. And I tend to talk out loud, trying to figure it out as I go. I'd never realized that rationals could be expressed as finite numbers with the addition of two special digits, and I expect we can even create a finite representation of any transcendentals we can name, using finitely many new coded digits for summation and exponentiation, etc. But either way.

 

[Hurkyl]

I'm saying the list can't be constructed, that there are reals that cannot be explicitly named, and therefore can't have a decimal form.

Depending on what you mean by 'explicitly named', those two clauses are not synonymous. For example:

. If "explicitly named" means "has a decimal form", then every real number can be named. In fact, there is an explicit formula that computes the name of any real number.
. If "explicitly named" means "there exists a Turing machine that outputs its decimal form", then there are real numbers that cannot be named.
. If "explicitly named" means "there exists a set-theoretic formula whose only solution is that real number"... then I strongly suspect the question of whether or not all real numbers have names is independent from the axioms of ZFC.

Although they can't be named themselves, they can be referred to as being in the neighborhood of a real that can be named. Ultimately, this seems like the minimal open neighborhood for any point you can name. If I have a distance relation in a topology, I must have a nameable radius r to define what is not in the open ball. Any r. So at some point what is in the open ball has to be unnamably close to the center point.

Philosophically speaking, this sounds quite similar to an alternate method of mathematical foundations: internal set theory.

 

[Bob3141592]

Thanks, Hurkyl. I'll have to think about the differences in the three interpretations you listed. I've printed out the links you mentioned and will do some reading. Frankly, I'm rather intimidated by the ZF Set stuff, and I'm not at all sure I can make heads or tails of it.

Well, off we go...

 

[CRGreathouse]

Depending on what you mean by 'explicitly named', those two clauses are not synonymous. For example:

. If "explicitly named" means "has a decimal form", then every real number can be named. In fact, there is an explicit formula that computes the name of any real number.

A countable formula, though, not necessarily a finite formula?

 

[Hurkyl]

A countable formula, though, not necessarily a finite formula?

The sequence of digits in a decimal representation $\{ d_n(x) \}$ of a real number x is given by:

$$d_n(x) = \left \lfloor 10 \left( \frac{x}{10^{n+1}} - \left \lfloor \frac{x}{10^{n+1}} \right \rfloor \right) \right \rfloor$$

 

[CRGreathouse]

Specifying that formula requires specifying x, so it is countably infinite. (Obviously, if you can specify an arbitrary real with a finite number of bits the reals are countable, so by Cantor's theorem you can't specify all reals with finite information.)

 

[HallsofIvy]

What definition of real number are you using?

One commonly used definition is this: Take the set of all increasing sequences of rational numbers having an upper bound. Say that two such sequences, {a_n} and {b_n} are equivalent if and only if the sequence {a_n- b_n} converges to 0. The real numbers are defined to be the set of all equivalence classes given by that equivalence relation. Given any "infinite decimal string", a.a₁a₂a₃..., the sequence a, a.a₁, a.a₁a₂, ... is a sequence of rational numbers so is in one of the equivalence classes and so defines that real number. There is no "infinite decimal string" that does not define a real number.

 

[Hurkyl]

Specifying that formula requires specifying x, so it is countably infinite.

Yay semantics! I would have called the formula finite, but the result of plugging in a 'infinite' x might not be.

(Obviously, if you can specify an arbitrary real with a finite number of bits the reals are countable, so by Cantor's theorem you can't specify all reals with finite information.)

On the other hand, you have Skolem's paradox... I haven't completely convinced myself that the ideas there are inapplicable to the situation at some useful level.

 

[Dragonfall]

What is the difference between a definable real and a computable real?

 

[mathman]

What is the difference between a definable real and a computable real?

To some extent it becomes a matter of defining "definable" and "computable". Pi and e are definable, but they can be computed (in a finite amount of time) only to a finite number of decimal places.