Is 1 = 0.999... ? Really?

James S Saint
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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. :sunglasses:


{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.

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.

Basically, the argument put forward is:

  1. ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) is “undefined” and “mathematical non-sense”.
  2. But ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)) isn’t non-sense and it’s equal to (0).
  3. And this has something to do with the fact that the former series diverges and the latter converges. They have no clue why but it MUST HAVE something to do with it.

In reality, people say that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0) because ((0.9 - 0.9) + (0.09 - 0.09) + (0.009 - 0.009) + \cdots = 0 + 0 + 0 + \cdots = 0). There are many different ways to pair the terms, as I’ve shown before, it’s just that people choose to pair them in this particular way when trying to prove that (0.\dot9 = 1). But when evaluating ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) they choose to pair them in a different way. This lack of consistency is precisely what allows them to errenously conclude that (0.\dot9 = 1).

As I’ve said before:

I don’t think that at all. But I might guess Sil is better inclined as a mathematician than a philosopher, based on his position.

What follows is a bit more “tongue in cheek.”

This idea “zero is a number” likely originated around the time accounting did. When one neighbor had two apples and another neighbor didn’t have any, as a way of determining when the asset column was equal to the debt column, the columns were balanced and there was zero remaining debt.

It was likely not long after that where the idea of interest on a loan originated.
“Well yeah, I know you gave me an apple in exchange for the apple I gave you, but what about the interest?”

And it wasn’t too long after that a capitalist came along and thought, "shit I can get rich quick, if I can figure out a way to profit from a recurring debt.

chuckles, clicks, submit and goes off to do something more important, like flossing teeth or making dinner, thinking we live and breath somewhere between 1 and 64 digits of precision.<
An engineer can figure out how to fly a probe across the solar system with only 16 digits of precision. Fun to think about when you’ve got the time.

“To infinity… and beyond!”, said in his best Buzz Lightyear impersonation.

There are two tells:

  1. frustration caused by other people disagreeing with him
  2. appeals to authority

You don’t do these things UNLESS your goal is to make others agree with you.

You still mad, bro?

No need, I gave up trying to persuade you to further your reasoning so there’s no need to lash out anymore. Chill. Think of anything I’m saying as for the benefit of others who might be able to think past only half the argument. That’s all you were doing wrong - I hoped you took a break from this thread to consider there might be more to your arguments like I was trying to tell you, but you came back as devout as ever, so trust me I’m not going to waste any more time on that :slight_smile:

I wouldn’t assume that just because I show mathematical competence that my other abilities must therefore be lesser in comparison, nor that mathematical and philosophical capability are so very different - they both come back to logic. You only really hear non-mathematicians complaining that mathematicians are bending to the will of mathematics, when all we’re really doing is understanding and appreciating the logic of mathematics more than non-mathematicians. It’s simple cognitive dissonance, like in that story “The Fox and the Grapes”.

It would seem that historically you’re correct that zero was born from accountancy - before then we only had natural numbers. Like I was saying, it was only once the rules of the mathematics of (in this case natural) numbers were considered as breakable that we explored the utility of integers. Then to rational numbers, then to real numbers, then to complex numbers - they’ve all opened up new mathematical utility.

Can you divide by zero? (\frac{x}0=y\to{x}=y\times0) and what values of y can satisfy this equation? Any and all values would work, and x would still be 0. y is undefined, hence why (\frac{x}0) is undefined.
Mathematics is entirely consistent, but if you mess around with what-ifs about infinity, you break that consistency.
That’s how you get these fallacious “proofs” like (“1=2”) etc.
It’s not merely “convention”, there’s very good reason.

See, this is not sufficiently defining any alleged difference between (1) and (0.\dot9)
Speculating on the ontological difference between a whole and a part is hardly a mathematical proof, no?
And what makes you so sure that (0.\dot9) is merely “a part”? Because it “looks like” it’s less than a whole? That’s why its equality with 1 is initially counter-intuitive, because you have to go on to prove what any difference between the numbers actually is, not merely “prove” that it kinda looks like there “should be” a difference between them when looking at it from only one possible angle of many. What is this difference exactly? You can logically prove that there isn’t one as easily as by noting that the endlessness of the 9s leaves no room for any “1” to “top it up” to ever occur ever. No difference ever arrives no matter how far down the chain you look.

You can look at (\infty-\infty) in at least two ways.
i) the second infinity is so large that subtracting it from infinity could leave absolutely nothing left, and therefore result in zero, or maybe even less than zero.
ii) the first infinity is so large that subtracting anything from it couldn’t possibly get you all the way back down to zero.
The crux is that both infinities are undefined, so you can’t define the result of operating on them with respect to either of them, never mind both of them.

That’s why it’s logical and mathematical nonsense.

As I was saying, mathematics has only grown in scope as a result of challenging assumptions - hence how we arrived at treating the square root of minus one as a valid entity with complex numbers etc. etc.
Perhaps you might argue that it was philosophers who advanced the mathematics, but history will show you that the two overlap all-too-often - and no coincidentally.
A mathematician who sticks doggedly a set of accepted rules only is just as bad as a philosopher who thinks speculation without rigor is sufficient, like Magnus. To be good at either is to be good at both.

Silhouette,

1 Is not an algorithm

0.999… is an algorithm

That’s a huge difference between the two.

What you’re stating is that an algorithm equals a non algorithm.

I find that ridiculous and absurd.

Your argument is that there’s no number between them, fine, that does not create an EQUALITY!!!

If you’re just using counting numbers, nothing exists between 1 and 2, that does not instantly make them equalities!

Additionally,

James made this thread. To James INF1 was his way of speaking about one direction in a 3 dimensional universe. INF6 to James was the TOTALITY, there’s only 4 directions in 2 dimensions, 2 more if you add the third dimension … I think James tried to call it InfA^6 … which for some horrible reason, people found profound. James was psychotic. This means nothing. In order to reach all the Cartesian point of infinite space, you need an infinite number of trajectories, not just 6!!

For example, hyper-cube gifs show 4th dimensional space in 2 dimensions.

I could go on for pages here, but I’ll just leave it at this.

I’ve heard this argument enough times already, and it’s still not true.

Anything properly defined is fine, I honestly don’t care and never did whether you use a number for it in particular.
Bear in mind, though, that we’re either talking quantity or exact quality - not anything superficial and non-specific e.g. obviously they qualitatively “look” different, not that anyone is directly being this basic but the arguments of some people are indirectly just as superficial.
All I’m asking is that since some have said “ooo, it looks like there ought to be a difference” - great, we all questioned that far before - now just finish the process by applying sufficient rigour to exactly and validly specifying what it is (on the preliminary assumption that such a thing is possible). Obviously numbers are ideal for this kind of thing, hence why mathematics exists, but in the absence of possibility to use numbers in this case - the same rigour needs to be applied to exactly defining any alleged difference in some other way. Nobody has done that validly - it’s all been superificial - about how it looks. Either that or arguments have been interchangeable with others that contradict it (which tends to be ignored, with people sticking to only one side of the argument because they prefer it, and so they don’t have to change their mind). You see it all over forums like this - when some people are challenged about a flawed position that they’ve put time and effort into, they double down for fear of humiliation, not realising that it deserves respect to change your mind in the face of superior reasoning and the only humiliating thing is to carry on defending a losing position.

You say that (1) is not an algorithm and (0.\dot9) is.
I already explained that both are representations of quantity. Quantity is abstract, numbers are concrete representations of them - numbers give symbolic existence to the mental exercise of dividing up experience. Experience is still continuous but it’s being treated as discrete e.g. two sections of the former whole. You can call one section “1”, and the other “1”, and call their combination “2” - whatever serves your purpose for explanatory power and utility. That’s the only extent to which quantity has existence, and to which numbers have existence.

So as mere representations of quantity, (1) doesn’t necessarily require infinitude to represent a specific quantity, but (0.\dot9) does require infinitude to represent that same specific quantity.
To represent infinitude, algorithms are useful, and you don’t need algorithms to represent finitude. But you could still represent this same quantity algorithmically as (1.\dot0), though this is superfluous to the cause.
The point is, algorithms are representations of quantity just like numbers. Algorithms equalling non-algorithms is just a matter of superficial representation. The quantity that they each represent can still be equal without issue.

And finally, yes, James’ InfA^6 is ridiculous and absurd, but it seems like we long moved on from there thankfully. That’s all I want for this thread - to move on from the superficial: both validly and exhaustively.

Then you are not using the symbol consistently and are representing two different things with the same symbol. That is going to be confusing. I wrote that I did not seen any problem with beginning with infinity “what ever that is” and subtracting infinity “what ever that is” to arrive at nothing. The “same” asset in both columns. Off course not all infinities are equal in all regards but in math with it’s dependence on symbol use and the consistency it requires, it really helps to use the symbol consistently. Imagine what would happen to the results of mathematics if the symbols used didn’t have consistent values.

As long as it is used consistently.

That is funny… good advice, but your examples of i and ii didn’t apply it.

But what of the axiom and it’s use. Do you challenge that the existence of infinity (boundlessness) has been proven? You seem to be saying two different things, first consistency and then mathematicians have to break a few rules. It would help if you stopped contradicting yourself.

You may have your cake and eat it too. I am sensing a lack of rigor in your arguments.