Is 1 = 0.999... ? Really?

The question is what’s the procedure by which we find the result of subtracting one infinite sum from another. There must be such a procedure, right? Otherwise, how can we say that (0.\dot9 - 0.\dot9) is (0)?

Given ((a_1 + a_2 + a_3 + \cdots) - (a_1 + a_2 + a_3 + \cdots)), how do we calculate the result? And you seem to be arguing my case (: Do we pair the terms like this ((a_1 - a_1) + (a_2 - a_2) + (a_3 - a_3) + \cdots) resulting in (0) or do we pair them like this ((a_1 + a_2 - a_1) + (a_3 + a_4 - a_2) + (a_5 + a_6 - a_3) + \cdots) which may be non-zero say if every (a_n) is equal to (1). And to make it worse, the number of ways we can pair the terms seems to be infinite . . . so which way is the right way?

How can you say that (0.\dot9 - 0.\dot9\ = 0) without having an answer to that question? Consider that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0) only and only if we pair the terms like so: ((0.9 - 0.9) + (0.09 - 0.09) + (0.009 - 0.009) + \cdots). But as you say, we can also pair the terms like so ((0.9 + 0.09 - 0.9) + (0.009 + 0.0009 - 0.09) + (0.00009 + 0.000009 - 0.009) + \cdots) in which case the result isn’t zero but approaching zero. And this is a serious problem. It would mean we’re trying to prove that (0.9 + 0.09 + 0.009 + \cdots = 1) by taking for granted that (0.09 - 0.0801 - 0.008901 - \cdots = 0). That would make it . . . not much of a proof. And certainly, noone ever mentioned this anywhere in the proof.

?

But there is no end of the line.

Because (0.\dot9) isn’t an infinite sum, any more than (4) or (\pi) is an infinite sum. (0.\dot9) is a real number, and 0 is the additive identity, so (0.\dot9-0.\dot9=0)

Yes. Also not much of a disproof.

But we aren’t doing something repeatedly, and nothing is changing. I think this is unintuitive but crucial: (0.\dot9) is not changing, it isn’t moving, it isn’t approaching; it is static.

Also, still wondering about your answer to the rational/irrational question.

Opps, was hoping to get back to edit before someone responded.

So the argument was in evaluation of the question 1 = 0.999… ? Really? And I am continuing with decimal notation.

The quoted part is the result of the rephrasing.

So sorry, I didn’t mean to imply any thing other then our minds were changing.

In other words change the question from a math question to a true false question.

Is 0.9 equal to 1. False
Is 0.99 equal to .1 False
Is 0.999 equal to .01 False

This is a pattern that is represented infinitely by the expression 0.9 recurring = 1, and the answer is… it is infinitely false that 1 = 0.9 recurring. From the first decimal position to the infinite, all at once.

What happens when logic and math disagree?

And I respect Magnus for smelling something fishy in the mathematical proofs. This idea of the infinite and how it is applied, is mathematically slippery.

“But there is no end of the line.”

The end of the line rests on the argument that has expanded from meno through the geometers through the conical hyperbole of calculation, which lead to the quantic/uncertainty generated partially reconstructed Fourier series which defined the MC2 transformations.

In fact, it could not have been surmised inductively, anyway.

Meno was right, and Leibnitz through Kant showed metaphysics to have been shut reductively

So how the connect, the touted synthesis?

Maybe Russel’s sense data is right on point, after all, as right as the well repeated proposition:
‘if God does not exist, he had to be created’

The point about transformation is significant because in a sense, something does come from an appearent Nothingness, and that is the point to the underlying functional math analysis , where partial differentiations are subsumed by the calculus of indiscernibles.

So the 'end of time’s is inherent in the declaration that -(.99=1.00) ; because of the same partially derived reconstruction by Fourier transform, validates such a pro-position.

The end is transformed from a logical level, (Russel-Wittgenstein) to one that is inductively not reduceable, is subsumed by the differential levels of quantifiability.

Put it in linear language, upon whose architecture the modality of it has to accorded to: simply: the end(s) justify the means of the reconstruction- that, the original transformation can be signified,

The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, all the way to the missing key of Principia Mathematica:

The ‘sense data’.

This hotly contested concept, overcomes the epoche within which liberal arts still vanquishes, unable to mimic its more quantitatively able cousin , the human brain.

The end of any road is the preintegrated calculus of missing pieces minus the ones left unfilled, but such a presumption fails, unless a transformative filler can replace it.

So, the end of the road could not be in a one dimensional map, but does exist in the calculable certainty in the existence of the perfect monad.

Without that, atomism would/ could not have come about signaling the need: to overcome it.

I shared this with St.James on occasion, and I think there was some concurrent albeit partially limited agreement, there.

The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, > and all the curious minds asking the questions here at ILP, and everywhere < all the way to the missing key of Principia Mathematica: >and beyond <

Come on everyone… in your best Buzz Lightyear impersonation.

Amen.

I have just changed my vote from other to no; pseudo confidently.

Curious if the argument has come up else where in the discussion.

Really? Are you telling me that (0.\dot9 \neq 0.9 + 0.09 + 0.009 + \cdots)?

Note: I am well aware of the fact that (0.\dot9) itself is not an infinite sum of numbers – not even a sum. It’s just a single number. The point is that it’s a number that can be represented as an infinite sum of non-zero terms. That’s the entire point and the only thing that matters.

(4) is not an infinite sum. It’s not a finite sum either. That’s because it’s not a sum. It’s a single number. However, it is equal to a finite sum such as (2 + 2) and an infinite sum such as (2 + 2 + 0 + 0 + 0 + \cdots). This means you can substitute every occurence of (4) with (2 + 2) or (2 + 2 + 0 + 0 + 0 + \cdots).

How do you know that (10 \times 0.\dot9 = 9.\dot9)? What’s the procedure? You avoid this sorts of question like a plague.

By definition, (0.\dot9) is a bit more than you’d like to admit. Yes, it is a real number but . . . it’s also an infinite sum of non-zero terms, specifically, (9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + \cdots). Indeed, every real number can be represented as an infinite sum but not every real number can be represented as an infinite sum of non-zero terms. (1), for example, can be represented as (1 + 0 + 0 + 0 + \cdots). It’s an infinite sum (i.e. there is an infinite number of terms) but the number of non-zero terms is finite.

So you admit that Wikipedia proof is not much of a proof? Good. At least we agree on something.

James S Saint
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Joined: Sun Apr 18, 2010 8:05 pm
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Sun Jun 26, 2016 12:05 am Post
Carleas wrote
In the standard set of real numbers, excluding the hyperreals, do you agree that .999… = 1?

Even more certainly not.

If allowed to proceed to absolute zero (the limit of the hyperreals), it might be arguable that the number string reaches that absolute zero difference between itself and 1.0. But if confined to the standard reals, the string cannot get even down below a single order infinitesimal difference (the rule being that you either cannot divide an infinitesimal or that if you divide an infinitesimal, you get the same infinitesimal remaining). So in the standard reals, there is always a first order infinitesimal difference between the two numbers.

The deeper truth is that even if including the hyperreals, the very definition of “0.999…” still forbids any remaining difference to ever be totally consumed into the accumulated sum. Thus by definition, there must always be a difference between the two numbers.

The “…” simply means, “you can’t get to the limit from here”.

… Sorry to have to be so stubbornly affixed.

{Arguably, the reasoning becomes circular, if no allusion can be made to be a priority.} The circular becomes both: an infinite repetition of what finally is defined as tautological.
The difference between .99999=1 & -(.9999=1) overcomes it
(The tautology) by transforming the function of differentiation. The same way that real numbers can not contain hyperreal numbers, yet hyperreal can contain real numbers.

No presumptions intended here, except philosophical ones, for which I need broadly to review this forum.

Well James S Saint. From this point on, I would like to include myself within the ranks of a “good” philosopher, even if I suspect you beat me to the punch… with your permission. I am a lousy mathematician that believes it accomplishes a great many results. But that mono polar magnetic thing?

For one you gotta respect a question formed through belief? In both directions. Will leave it up to you all whether philosophy is more capable than mathematics.

Different from Uciscore, Urwrongx1000 and a bully mentality, your response offered encouragement and insight to question further. I respect that as grace. Thank you.

So we have two arguments that challenge the truth of the statement.

The convention of decimal notation, the expression itself informs; with a zero to the left of the decimal point, there are zero 1’s. 0 x 1 is not equal to one.

This challenge comes from the perspective that .9 recurring equals 1 is a true statement. The decimal notation “0.9 recurring” declares in it’s own notation, the falseness of the claim.

and

The logic which recognizes the expression 0.9 recurring = 1, represents an infinite falseness in it’s pattern of recurrence as result of offsetting decimals.
0.9 does not equal 1.0, 0.99 does not equal .1, 0.999 does not equal .01 … an infinite string of offsetting decimals that are not equal. From the first to the infinitive.

Interesting that mathematics can “prove it” >within a convention<, while both the expression itself and logic say it just ain’t so. Likely, it has something to do with what is being “accepted” as a mathematical “truth” out of convention.

When an equals sign is used to relate two values to each other the claim is made that it is true there is value equivalency. The two arguments presented establish that it’s truth is falsifiable.

ILP’s very own offset decimal paradox.

(0.999…) represents a number that is larger than every number of the form (0.9_{1}9_{2}9_{3}…9_{n}) (where (n \in N)) but smaller than (1).

Interestingly, whether or not such a number exists is completely irrelevant. That’s because if it does not exist, then it’s not (1), since (1) exists. Remember that the subject of this topic is whether or not (0.999\dotso) is the same number as (1).

Pretty much EVERY single person on this board defending the widely accepted position has been forced to argue that the two numbers, (0.999\dotso) and (1), are MERELY APPROXIMATELY EQUAL. And noone disputes this. But approximately equal is not the same as strictly equal.

0.9 recurring; the largest possible fraction that isn’t a whole. 1 - 0.9 recurring would be the smallest possible fraction greater than zero, leaving all possible fractions existing between None and 1.

An other interesting note: when a whole number is written in the form of a fraction as in 1/1 it doesn’t become a fraction. Fractions are always parts of a whole. And a whole can only be made from all of it’s parts, even if the parts are infinitely small, they remain infinitely small parts.

In the case of the proof that requires us to let X = 0.9 recurring, the requirement treats X and it’s infinite recurrence as a number when Infinity isn’t a number. And now I’m wondering if 0.9 recurring is actually a number, within the same realm that 1 is a number.

0.9 recurring is the fraction 9/10 divided infinitely by 9/10. A fraction divided infinitely by a fraction, is an infinitely divided fraction. The whole number 1 can not be equal to a fraction of itself.

A third potentially valid argument.

But I’m still having trouble with the idea that if I divide 9/10 by 9/10 it does equals one when the infinite division isn’t involved. But then I remember the case of the offset decimal point. Involving infinity offsets the decimal place infinitely. And go figure… 1 divided by 9/10 = 1.111 recurring. It is really hard to do math with recurring numbers. What is the value of any recurring integer?

Does 1 recurring = 9 recurring? Does .1 recurring equal .9 recurring? Fuck it’s difficult to do math at all involving infinity. Infinity ain’t a number, and math requires numbers of some sort.

Ok, so in the same bored way someone asks to be asked any question, I have wasted my time messing around with infinity. Something on which I don’t have an infinite amount of time for.

What math operation can be performed that equals 0.9 recurring as a product of that operation, such that calling it equals makes any sense at all?

A value involving infinity in it’s description isn’t a “number” that can be equal to the number 1.

In other words I can’t get to 1 = 0.9 recurring from here. 1 +1 +1 infinitely = 2 +2 +2 infinitely = 3 +3 +3 infinitely = 4 +4 +4 infinitely = 5 +5 +5 infinitely …= infinity + infinity + infinity… infinitely?

What good does that do? Math was originally conceived for counting and began from the recognition of a unit 1, and out of all the numbers 1 behaves rather strangely when dividing or multiplying by it.

The attempt to distinguish (1) from (0.\dot9) is still going?

To solve a recurring issue that seems to causing doubt in some minds since this was resolved:

(“\infty - \infty”) does not present the exact same issue as e.g. (9.\dot9-0.\dot9)
There is a significant difference resulting from the distance from the radix point (let’s assume a decimal point) and the digits with the highest order of magnitude. To translate:

i) For (0.\dot9) the significant digit with the highest order of magnitude is immediately to the right of the decimal point, and the further you go down the chain of 9s, the smaller any potential discrepancy is going to be between 2 numbers with infinitely recurring digits. The trend is easy to see as converging towards a limit of (0) whether or not you agree that it reaches a literal (0) for (9.\dot9-0.\dot9)

ii) For (\infty) however, any hypothetical “highest order of magnitude” is infinitely far away from the decimal point (the (0.9)), so any potential discrepancy between 2 infinite numbers could be larger and larger - even infinitely large, yet it could also be infinitely small, if existent at all…

So you see why (“\infty - \infty”) is logically and mathematically nonsense, because it could be anything - hence why mathematicians regard such a divergence as undefined.
It’s infinitely unanswerable, with an infinite range of possible answers no more correct than the last: thereby all infinitessimally correct (infinitely incorrect).
By contrast, (9.\dot9-0.\dot9) is far more logically and mathematically viable as the convergence to a single exact limit cannot be anything else other than precisely tending to zero.

It’s been conclusively demonstrated here that non-mathematicians with a chip on their shoulder about this status of theirs cannot cope with all the implications of this concept of “undefined”, and that everyone with a mathematical proclivity has the ability to conceive why (1=0.\dot9) so the vote at the top does more to illustrate the distribution of mathematical backgrounds on this forum than anything else - as mathematicians know, there is nothing democratic about mathematics. It’s an absolute dictatorship.

But as my first contributions to this thread immediately spelled out, advancements in mathematics come about by playing around with what would happen if exactly specific rules of this absolute dictatorship where treated as breakable (e.g. complex numbers). Even though arguments distinguishing (1) from (0.\dot9) are wrong, it’s interesting to see if anything useful happens if we treat e.g. two representations of the same quantity as different. If we were all mathematicians here, we could have moved on long ago to addressing this question - and exploring the utility of hyperreals.

The fact of the matter is that no difference between (1) and (0.\dot9) can be defined - and to this the non-mathematicians cling as evidence that it’s there “because it seems like it ought to be”, but “it’s the fault of the numbers we use” - or something woolly like that.
The point of mathematics and philosophy alike is to be far far more precise and stringent in the definition of what layman intuitions “seem to point towards”. Every single person here has no doubt noticed that something seems odd with the equality between (1) and (0.\dot9) with the only difference seeming to be that mathematicians understand far better than non-mathematicians the issues of holding onto layman intuitions in the face of trying to be exactly definitive about any alleged difference. It’s the most obvious thing in the world to see how there might be issues, but proving it is another matter - and an essential matter at that. A “difference” between (1) and (0.\dot9) evading all definition doesn’t merely mean “approximately equal”. It means any definition between the two representations is impossible to definitely represent, and tends precisely and exactly to no other number than zero. This is entirely mathematically acceptable as equality, so even though the mathematical community is unanimous, the non-mathematicians will apparently never let it go.

I respectfully submit in this:

I think the mistake in this, is zero is only a number through mathematical convention. 1 - 1 = nothing, no quantity to count. Zero is just a symbol used to represent that mathematically. Any number divided by zero is undefined as well. How can you divide a number by a symbol that represents nothing. It is as if nothing at all has taken place. The division can’t occur because there is nothing to divide by.

Additionally, I think there is a sufficiently defined difference between 1.0 and 0.9 recurring. The respective location of the decimal point, and the ontological difference between a whole and a part. Is 0.9 recurring a well defined number, cause 1 seems fairly well defined as numbers go.

I don’t see “why” “∞−∞” is logically and mathematically nonsense. Begin with boundless “what ever that is” and then subtract “what ever that is” and there is nothing left, not even the zero. The zero is sort of stuck as a symbol representing nothing, smack dab in the middle of a set that includes the infinite set of negative numbers on one side and the infinite set of positive numbers on the other.
I can grok why infinity divided by infinity is 1.

So far, infinity is just imaginary boundlessness. I don’t think it’s been proven it actually exists. The difference between a philosopher and a mathematician is a mathematician is willing to forget it was an assumption, treated as if it existed, for the sake of an argument. An axiom *. Philosophy does that as well, but the philosopher, at least a good one, doesn’t forget it.

• en.wikipedia.org/wiki/Axiom

As usual, Silhouette is doing nothing but complaining that the world is not bending to his will the way he’s bending to the will of the mathematicians.