An infinite question: Why doesn't .999~ = 1?

Cecil's Columns/Staff Reports
681

Everybody can participate - not only watchwolf49. For those who understand how 1 does represent [0, 1], or how [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1] does represent 1:

You would call me stupid, if I would claim that there are more segments in 1 than the segments [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1], wouldn’t you? And if you ask me what the additional segment is and I don’t answer - wouldn’t it be suspicious?

The segments in 0.999… are [0, 0.9], [0.9, 0.99], [0.99, 0.999], …
So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?

a) There is no such segment
b) The segment is …

682

You are merely restating your flawed argument as one involving segments. It has the same problem it’s always has, it doesn’t actual deal with infinity.

But just for fun, here’s one of the previous counter arguments stated with segments.

Divide the closed interval [0, 1] into intervals [0, 1/n], <1/n, 2/n], <2/n, 3/n] … <n-1/n, 1]

Which interval is 1 in? Which interval is 0.999… in?

Let go towards infinity. Do the answers to the previous two questions ever change?

What’s the length of the segment as we approach infinity? I.e. what is the maximum possible difference between 1 and 0.999…?

What happens to the length of the interval if we, as you’ve said we do when we write 0.999…, “complete” the infinity?

683

1 is in the last interval. 0.999… is not a defined point in any interval.

If we are going to infinity - as you suggest - we are “approaching” uncountably many segments. That’s not what we are dealing with. The length of the segments is 0 at infinity. We can define a first and a last segment then - [0, 0] and [1, 1], but we cannot define a second, or a next to the last segment, or adjacent segments at all.

We are dealing with countably many adjacent segments [0, 0.9], [0.9, 0.99], [0.99, 0.999], …, a first segment but no last segment.

684

That does not make sense in real mathematics, but let us pursue your quirky othermath for a second. Does 0.999… exist at all? If it does, at what step do we jump outside the number line. If it doesn’t, what is the largest number of repeating decimal nines that exist and in what interval does it exist?

685

No, you are dealing with that. The rest of us already know that it’s the same nonsense “non-infinity” argument that you’ve been harping on all along.

686

Yes. It exists. It’s [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…

There is no largest number of repeating decimal nines that exist. There are infinitely many. They are 0.9, 0.99, 0.999, … and all of them exist in the interval [0, 1].

687

Dealing with what?

Don’t tell me that we are dealing with uncountably many 9s. We aren’t. We are dealing with countably infinitely many 9s. None of the segments [0, 0.9], [0.9, 0.99], [0.99, 0.999], … is of zero size.

688

This depends on how we define the ellipsis here:

  1. Generally, it means “and so on without end”. In which case the limit point is an element of the set. The proof of this fact can be found in the first couple of chapters in any decent calculus textbook.

  2. your definition of the ellipsis is “a very very large finite number”. In this case we have a final element of the set and it is not 1.

Your claim I’ve quoted above stands as unsubstantiated, thus your are assuming it. But this is also your conclusion, which makes your entire argument an example of the logical fallacy of “circular reasoning”. It is a fallacy which means you’re wrong …

689

I can’t believe I’ve been reading this thread for this long, but it was almost worth it for this:

Comedy gold.

690

How do you count an infinite number of elements? What is the answer? Be reminded, infinity is not a counting number, that cannot be your answer.

691

Yup, 687 posts … It’s been worth it …

692

So what? So what???

Your entire “point”, that you have celebrated over and over and over with your nursery rhymes, is nothing else than
1 is not a member of the half-open interval [0,1)
Period. Something that follows immediately from definition.

No? You think I’m exaggerating? Then state what your point is in mathematical terms. No “I move my pen …” No gibberish that only looks vaguely like mathematics, e.g.

(Is this your way of writing [0,1) ? )

But something actually mathematically sensical. If you can do that, you will end up with a statement that is either (1) incorrect, or (2), like “1 is not a member of the half-open interval [0,1)” obviously true.

Go ahead: try. Don’t worry; I’m not holding my breath.

693

“Countably infinite” is a standard mathematical term.

694

At what exact point in the accumulation of nines does it cease to be a number?

0.9 is a number.
0.99 is a number.
0.999 is a number.

Where does it stop being a number? Be specific.

695

Can you quote such a proof? Until then the limit of a sequence or set is no element of the sequence or set.

Infinity means there is no final element.

696

We can count an infinite set if there is a counting number n ∈ ℕ for each element of the set.

697

I still don’t see what you mean. It’s not unmathematical to use set theory and intervals. And it’s not unmathematical to ask the question:

The segments in 0.999… are [0, 0.9], [0.9, 0.99], [0.99, 0.999], …
So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?

a) There is no such segment
b) The segment is …

698

If you are talking about the infinite sequence 0.9, 0.99, 0.999, … then every element of the sequence is a number.

699

I still find this absolutely fascinating. Let’s call this pseudo-mathematical theory othermath and examine some of its properties.

In othermath 0.999… exists, but it is not on the number line. I’m not entirely sure if it is then not a number, but I’m sure netzwelter can explain that part …

This property of existence but non-participation is not exclusive to 0.999… any decimal representation that doesn’t terminate has othermathematical existence, but is not present on the number line. 0.333…, 0.444…, decimal representations of most square roots, e, pi., etc.

1/3, 4/9, pi “itself” and so on do exist on the number line though, so it’s basically a question of notation.

Change number system to hexadecimal, or binary or trinary, and there are acceptable decimal notations for a different infinite set of fractions, and a different infinite set of decimal representations that somehow don’t live on the number line.

Now what is so fascinating is that othermath accepts the existence of infinities. Segments of zero length are acceptable. 0.999… exists, even if it isn’t on the number line. Infinities can be “completed”. But instead of coming to the conclusion that maybe the choices of axioms, or application of logic, is flawed somehow, netzwelter just doubles down and seeks out more notation, other approaches, to say what he/she already knows is true. It’s a textbook example of pseudo-science, or pseudo-math if you want. Fascinating mostly because of its resilience to the pointing out of paradoxes it creates.

As long as netzwelter proceeds with the absurd notion that no non-terminating sequence of decimals has a presence on the number line, this will just be an exercise in watching as he/she translates the original flawed argument into a new set of somewhat mathematical terms.

700

Epsilon Delta Proof … short but correct … try reading for comprehension, I gave the citation “The proof of this fact can be found in the first couple of chapters in any decent calculus textbook” … you may find this proof easier to understand in the textbook that spreads it out and explains it in detail over those several chapters …

And how many do you count in the set { 0.9, 0.09, 0.009, … }? All the rest of us have known all this time this is an Aleph-null set, but there’s still nothing that precludes the limit point from existing within this set … and from both citations above I’ve proven it does.