Is 1 = 0.999... ? Really?

Which of the following do you mean:

  1. The infinitesimal portions are connected by the sea.
  2. The sea = All of the infinitesimal portions connected

If you mean 2, what is connecting the infinitesimal portions?

I agree. But as far as I can see, there is only one item that can have that measure.

Does it matter how you write it down? You can do it any way you want. There isn’t much of a difference between your approach and my approach.

Wikipedians do it the same way that I do except they start with (0).

And the reason we do it this way is because it’s more elegant.

Consider this (your approach):

(d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)

And now consider this (Wikipedia approach):

(d_n \dotso d_2d_1d_0 = d_n \times 10^n + \cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0)

Which of the two expressions look more elegent to you?

And this isn’t the only reason I write it that way. There’s another one.

As for why I start with (1) rather than (0), there’s a reason for that too.

Right. If (\infty) stands for “any number greater than every integer” then a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would be an obvious contradiction.

That’s why, as you already know, we have to do one of the following things:

  1. we have to say that, in our case, neither (\infty) nor (\dotso) indicate “any number greater than every integer”, but rather, that both indicate one and the same specific number greater than every integer

  2. we have to come up with a neutral symbol, such as for example (x), which we can say represents some specific number greater than every integer; and we also have to say that the number of digits in (999\dotso) is (x) (to make it clear, we can use something like (999\dotso_x))

Let’s pick the second way because it’s less likely to lead to confusion.

My claim is that (999\dotso_x) stands for (9 \times 10^{x-1} + 9 \times 10^{x-2} + 9 \times 10^{x-3} + \cdots) where (x) stands for some specific number greater than every integer.

I also claim that 1) YOUR expression and MY expression represent two different numbers, 2) my expression represents a larger number, 3) my expression represents (999\dotso_x), and 4) your expression represents (\dotso999_x).

Now, let me take the following statement of mine . . .

. . . and reword it, like so:

The non-zero digits in (999\dotso_x) are digits whose index is less than (x) but greater than every integer.

But I think I made a mistake here.

Let’s see.

The index of the first digit is (1), so the index of the last digit is equal to the number of digits. Since the number of digits is (x), the index of the last digit is (x). Thus, the non-zero digits in (999\dotso_x) are digits whose index is less than (x + 1) but greater than every integer.

Do you agree so far?

(999\dotso) is a decimal representation with the following properties:

  1. it has no fractional part (no decimal point is explicitly stated and the ellipsis does not imply it)

  2. the number of digits is greater than every number (this is implied by the ellipsis)

  3. it has the most significant digit but it has no the least significant digit (this is implied by the fact that the representation does not end with a digit)

  4. every digit is (9)

(999\dotso) is not a number, but rather, a category of numbers. This is because there are two things about it that are not fully specified. These are: the number of digits and the place of the most significant digit.

The number of digits is partially specified because we know it’s greater than every integer. The place of the most significant digit is also partially specified because we know it cannot be a place associated with an exponent that is less than the number of digits minus one.

phyllo appears to be right in that the exponent associated with the first digit in an instance of (999\dotso) is not necessarily the number of digits minus one. It can be the number of digits, or the number of digits plus one, or the number of digits plus two and so on.

Just a general note -

I can understand, although disagree with, wtf’s argument. I understand what he is trying to say and claim - and why.

But when it comes to Certainly real, Magus Anderson, and Ecmandu - you blokes seem to be making some really weird arguments that seem to be just arbitrarily scrambled thinking - like “why not just throw in this confusion - or maybe this ambiguous notion - ah, here’s a conflation that might make things more cloudy…” :confused:

I just have to hope that it isn’t intentional.

What I see is a distinction without a difference.
But actually in A there are no fractions.
And in B there are only fractions (or actually decimals).

    • unless I have them backwards from your intent. :confused:

That part is right (except the use of the word “Infinity” rather than “an infinity”).

A = the big one – 999…
B = the small one – 0.999…

  • that’s what I guess as how you mean them

No, I don’t think so. “0.999…” has nothing at all to do with counting. And it does imply the sum of an infinity of decimals (or fractions). But it is still only one decimal number - that sum, not the list of decimals.

That misunderstanding is one that I understand - I see the confusion (for that one).

B (assumed to be the 0.999…) is a single number. It is NOT a series of actions. The representation of “…” does not mean that anyone is to actually add forever but rather that IF anyone tried to complete the decimal, they would have to add forever (so don’t try).

B is the final sum IF all of the decimals were added regardless of the fact that no one could actually add them. We can know the sum anyway. We can know that something is or isn’t infinite without going and counting it. We can logically deduce a result. We don’t have to always calculate it.

Again, “999…” is a single number. There is no implication of having to add anything unless you just want to try it. And if you add ALL of the infinite parts, the sum of that addition would be the single number “999…”.

And all the above is why this -

  • makes no sense.

I guess you mean this edit:

Is this the edit you meant?

I can agree with it, although there is a a little problem with finding.

Finding is similar to the epistemological subjevtivity/objectivity problem, a duality. But nevertheless I agree to the said sentence “earnestly seek and ye shall find” and the followig sentence.

What you said about truth and reality - “you can hide from one but never escape the other” is even more agreeable, regardless whether I would use the word “facts” instead of the word “reality” as well. Both refer to what we perceive, namely what has become, that is, what is completed through history. So, I would use the word “facts” as well as the word “reality”.

It is right waht you said about truth and reality: “you can hide from one but never escape the other”. If you try to escape from truth, you have to face the reality (facts); if you try to escape from reality (facts), you have to face the truth. You are not as fast as the both are. Lies have short legs; unreality is like a dream in which movement is impossible.

There is no connector. They are merely side by side portions - like the right and left sides of a piece of paper. How one infinitesimal is distinguished from another is merely by which portion you are focused on or talking about. The distinction is an imaginary line drawn between to portions of an otherwise continuous substance - the North portion of the Sea or the South portion of the Sea.

Every portion (an item in our discussing of the whole of the substance) can have that measure (assuming infinitesimal portions). I don’t know where you get that there can be only one. How many infinitesimal portions are there of a pint of beer?


The only reason motion happens in the cosmos is because of infinity: infinity = motion (because it never ends - by definition)

You’re the person here who looks at dictionaries and can’t actually think to this regard, not me.

Just think about it.


Actually I meant the reality of the future consequences due to the reality of the past events. If I say, “facts”, I just throw in more opportunity of get into the semantics of “who’s facts”. :confused:

And even though the following seems off topic here, it is actually related (to why bother pursuing these words and what they mean).

When a society tries to govern by establishing an artificial truth narrative (such as real world Communism and those trying to govern the US right now), they create an artificially induced public bubble of belief and force the issue of having to stop all thinking. Reasoning tends to unveil the reality that it is all artificial - creating doubt in the veracity of the narrative.

I think establishing an artificial reality narrative for a population can be good or bad but is most certainly a very, very dangerous thing (as has been proven over thousands of years of attempts). It stems from the lust to advance by trying to be God - in place of God (in place of Reality). And they actually know that it is extremely dangerous - to everyone else. They fully know that they are going to murder billions of people - they simply don’t care. To them being a god over all life is just too blindingly important. And they know that you can’t catch them. So they proceed.

Those kind of people seem to think that they actually have no choice. They think that the only way to advance is through the means they currently see - their “path to godhood”. And the reason they think that is because of what we are discussing on this very thread -

Getting the real story straight before making presumptions about what is necessarily true - look CLOSER at the details of your own thoughts and language (which largely affects your thoughts). Seek out, not merely A path to victory, but the BEST path at all possible (assuming you are not in a rush).

The problem is that it is an ethical issue. And when people, high or low, simply don’t care - they don’t bother (relating to James’ PHT). I think that on high they are very willing to murder billions of people because they haven’t bothered to seek out a more ethical solution to advancing humanity toward a more harmonious existence on Earth (another seriously big issue with James - “Saint”).

I can see where he is coming from. I started by merely being curious - then impressed - then amazed - then awestruck - then just speechless (as you said - “impressed with him” was only in the beginning). I have been closely examining his proposals and what I call his “final solution” (his SAM Co-op) for the possibility of any tiny little misguiding hidden devil. I haven’t concluded that his final solution is pure and more importantly that it is the BEST solution to human harmony. I think that I have just recently realized who he was talking to all that time (a serious question I have held) - and it was not anything I was guessing. His solution seems to be a calculated far off destiny - and AFTER billions of people have been murdered - exterminated by what he referred to as the source of all of the trouble - the blind lust of the “Godwannabes”.

I have nothing better to do besides my wife, work, and wealth pursuits than to see if his unfortunately distant solution, possibly after the death of literally all human life, is really the BEST possible. And that requires a degree of examination for that elusive devil in the details that a “normal” person ( :blush: ) would never bother to find.

In short - details matter in thinking and trying to solve real world problems involving humanity - not merely maths or scientific war weaponry - “Get the words straight” :smiley:

Even if true, I don’t think that changes what I said. Infinity is not an action or instruction or function - “Get the bloody words straight”.

If there is no connector and there is no separator, then how can the following not be the case: Nothing separates/connects one infinitesimal from/to another.

If I say there is an infinity of infinitesimals in one finite pint of beer, then that either implies:

A) You can have a quantity that is beyond Infinity because other things exist and this would imply Infinity + other things (contradiction)
B) An infinite number of infinitesimals encompass all things. As in every existing thing consists of an infinity of infinitesimals.

A is clearly contradictory. Whether B is contradictory or not is not crystal clear to me.

B suggests that an Infinity of infinitesimals exist. When no existing thing separates/connects one infinitesimal from another, then perhaps B is contradictory because if x is absolutely not separated from x by anything other than x, then can we meaningfully say there is more than one x? Even if we choose to hold on to the belief of there being an infinity of xs/infinitesimals (which is my current belief), I think the following still holds true:

For any given item (finite or otherwise), you either say there are an Infinity of infinitesimals/xs there, or you say the Infinite is there. You cannot say there are 5 xs/infinitesimals here and 6 there. Nor can you say there are 5Inf infinitesimals/xs here and 6Inf infinitesimals/xs there. Can you divide or multiply the quantity of Infinity by anything? I don’t think so. Even if you say you could, it would go something like this. 1/2 of Infinity = Infinity. 2 times Infinity = Infinity. 1/Infinity of Infinity = Infinity. Infinity times Infinity = Infinity. Square root of Infinity = Infinity. Infinity to the power of Infinity = Infinity. Infinity is always the same Infinity.

An infinity of xs can sustain/accommodate any finite number of ys. There is no end to to how big this finite number of ys could be, but this number must be a finite number as it cannot be Infinity.

So, an Infinity of xs cannot accommodate an Infinite number of ys. An Infinite number of xs don’t just sustain/accommodate an Infinite number of xs. They are an infinity of xs. Or as I would say, x IS Infinity. Everything is in Infinity and Infinity is in everything. Only x denotes the measure of Infinite/Infinitesimal. Only x is Infinite in quantity. Nothing else is like this. Given how I have define x and y, to have an endless number of finite numbers (ys), is not the same as being an Infinity (xs). The endless number of xs necessarily = Infinity. The endless number of ys necessarily = not Infinity.

So it really is a new age thing - the over concern for appearing “elegant” - a class struggle issue - everything about appearances and ordained truth narratives - focus on the symbols not the reality so we can keep the peasants confused. Gauging by this board - they have certainly succeeded.

Not exactly.
You say “any number greater than every integer” - and that alludes to the possibility of more than one number greater than every integer. And that’s ok (sort of).

But then you say "a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would require that there be only one “greater than every integer” - else a number could be greater than every integer while another number is greater than that one.

You seem to be talking about greater and lessor infinities (why - I have no idea).

Maybe that is the seed of disagreement. Your (1) doesn’t seem to be true.

Although the symbol (\infty) can possibly indicate ANY infinite “number” (higher cardinality), the “…” ONLY refers to the entire natural number set. I don’t think anyone every uses “…” to refer to a higher cardinality.

And that makes no need for (2).

That doesn’t seem “elegant” to me. It seems a little messy and unnecessarily complicated because of all of the "x - n"s that could have been avoided.

That we disagree on. Again apparently you are focused far too much on the positioning of the symbols (much like Certainly true) - ignoring the relevant meaning of those symbols. the “…” cannot reasonably be placed before the string of digits that define what is to be repeated. And even if it is written that way, it still represents the same value that we both have been discussing. There is no difference in the values.

The “…” isn’t about how to construct the symbol or representation. It is about representing the value.

Two issues -

  1. Why do you keep saying “non-zero digits” when ALL of the digits are clearly non-zero - “9”

  2. If every index is greater than every integer, you certainly have not included either the higher or the lower portion of integers, such as those at “999…” or those down at “…999.0”. It depends on how you set your index.

If we take it that what Certainly real means by “infinity” is “a number greater than every integer” then (0.\dot9), being smaller than some integers, is not a number greater than every integer, and thus, not “infinity”. (Whether we take it that (0.\dot9) is equal to (1) or less than (1) has no impact on that and thus it’s irrelevant.)

But is it a finite number? I can’t tell because I don’t know what that means. What exactly is a finite number?

I can say that it is not an infinitesimal because the word “infinitesimal” is generally taken to mean “a number greater than (0) but smaller than every number of the form (\frac{1}{n}) where (n) is a natural number”. Since (0.\dot9) is greater than (\frac{1}{2}), it’s not an infinitesimal.

I would say that we have no name for this type of number.

My position is that we can. I am, of course, assuming that what you’re asking is “Is it logically possible to count to infinity?” and not “Is it possible to count to infinity in reality?”

I did make a mistake but you did not correct it properly.

Let’s correct it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})

The least significant digit (the one that you should multiply by (10^0)) is the rightmost digit i.e. (d_n).

According to you, (256) is equal to (2 \times 10^0 + 5 \times 10^1 + 6 \times 10^2) which is equal to (2 + 50 + 600) which is (652).

If you make things as simple as possible, you are less likely to make a mistake. That’s why elegance (or simplicity) is important.

Why do you think I made the mistake that you tried to correct but failed? (And why do you think you made the mistake that I corrected, namely, that (999\dotso) is the same as (\sum_{i=0}^{i->\infty} 9 \times 10^i)?)

And that’s also why one should not redefine (0.999\dotso) to mean “the limit of the sequence ((0.9, 0.99, 0.999, \dotso))”. Instead of changing the existing meaning of (0.999\dotso), one should come up with something like “lim(0.999…)” to avoid possible conflation.

That is not what I had represented. That is you trying to conflate a picture with a value (the very source of obfuscating reality and subverting the peasants).

And that is also a part of the game.

Those already experienced with the old simply shortened it because of widespread understanding. But now in the cancel generation, you proclaim an understanding superior to their obviously flawed rhetoric - even though it wasn’t flawed at all. You merely never learned it but then criticize it.

It is like the Americans claiming that the Brits don’t know how to speak English properly. #-o

Okay. I think I see what is going on here - shifting the meanings of words and symbols to make old things “appear” wrong and new to “appear” right. It is all about appearance - focus on the look, not the substance or reality.

No need to go further with ironing out any actual logic.

It is just ordained NEWSPEAK.

You are supposed to explain what I am doing wrong instead of ranting about politics.

That’s because we’re indexing the digits from left to right. The index of the leftmost digit is (1), the index of the one next to it is (2) and so on. That’s what (d_1d_2d_3 \dotso d_n) stands for. In the case of (256), (d_1) is (2), (d_2) is (5) and (d_3) is (6).

That’s what was meant when I introduced this notation in this post of mine:

Here, (d_n) represents the leftmost digit whereas (d_1) represents the rightmost digit.

And this is by no means something new. If you go to Wikipedia, you can see a similar notation. The only difference is they use (a) instead of (d) (which I prefer because it’s short for “digit”) and they start with (0) instead of (1).

So the question is: what exactly is your problem?

Stop ranting about politics and start explaining what’s wrong. Or just shut up.

What “prior”? Who cares about what came before?

The Wikipedia approach is more elegant for very simple reason. The index of every digit is the same as the exponent associated with it. That’s simplicity. It’s easy to memorize.

In your approach, the index of every digit is different from the exponent associated with it. Just look at it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})

The exponent associated with (d_1) is (n - 1). The index of the digit is (1) but the associated exponent is (n - 1). Two different numbers.

That was NOT a statement about how OTHER people define those symbols. That’s why I said “in our case”. That was merely a proposal on how to define those symbols in certain situations so as to increase the chances of other people correctly interpreting what we’re saying.

On the other hand, I do not agree that “…” means “repeat for a number of times equal to the number of naturals”. I would like to know where you got that from.

I think “…” means “repeat endlessly”. The part that you should repeat is the one that precedes the ellipsis. In the case of “999…”, it means “repeat 999 endlessly”. Exactly what is meant by “endlessly” is not clear. What’s clear is that the number of repetitions must be greater than every integer. Whether it refers to any such number or something more specific isn’t clear.

Or it might be the case that you are not paying enough attention to the positioning of the symbols, failing to realize that the meaning of (999\dotso) is decided by its digits and how they are positioned.

Do you agree that if two decimal numbers don’t share the same exact digits, that they are not equal?
(That’s also why (1 \neq 0.\dot9).)

If so, it’s important to compare (999\dotso) to your number to see if they share the same exact digits.

Do they?

Notice that (999\dotso) has the most significant digit. Your number, on the other hand, which is (\sum_{i=0}^{i->\infty} 9\times10^i), does not. It has the least significant digit (evident in the fact that (i) starts at (0)) but it does not have the most significant digit (evident in the fact that (i) tends towards (\infty) without ever reaching it.) That’s precisely the opposite of (999\dotso) which has the most significant digit but no the least significant digit.

Why not? In the case of (\dotso999), it means “repeat the part that comes after the ellipsis, repeat it to the left and repeat it endlessly”.

Well said.

We’re mostly in agreement on what you said in this post.

Well not that, but nevermind.

I perfectly well agree if we focus on the proper subset relation. That’s one way of talking about a set’s size. Cardinality is another. Natural density (which I think I mentioned to you earlier) is yet another. Which is why “size” is ambiguous unless you carefully qualify it.

Hey man thanks, can you please explain that to my friend @obsrvr524? He’s a little confused on this point.

I perfectly well agree that by cardinality, the naturals and the evens are the same size; and by the proper subset relationship, the evens are smaller. That’s not a contradiction, it’s simply a failure to properly disambiguate between size(\text{proper subset}) and size(\text{cardinality}).

Note that the proper subset and cardinality do not play well together. The naturals have the same cardinality as the multiples of 4, via the bijection f(n) = 4n. But the multiples of 4 are a proper subset of the evens. So by proper subsets, mults of 4 < evens < naturals; but the mults of 4 and the naturals are in bijective correspondence. In general, cardinality tends to be the more useful measure of set size, which is why it’s so commonly used. But as long as you are careful to say that you are using the proper subset relation, I have no problem with that. But the thing is, you can’t get too far logically with that idea for the reason I mentioned, that it conflicts with cardinality.

We just need to be precise about which meaning of size is being used at any given time.

The main point on which you and I disagree is that the notation (10^\infty) is not defined and has no meaning; whereas you are casually using it as if it did.