.999 = 1?

Factual Questions
2151

(Bolding mine.) jsum_1 asked you to show a contradiction, not give a vague, hand-waving argument that has nothing to do with field axioms and gets the basic arithmetic of R wrong anyway. The field axioms do not support infinite products. (Certainly R is not closed under this operation; 123*4… does not converge to a real number.)

More importantly, you’re wrong anyway. The number .999… squared is 1 (which isn’t surprising, since .999… = 1). In terms of Cauchy sequences, the product is given by (9/10, 99/100, …) * (9/10, 99/100, …) = (9^2/10^2, 99^2/100^2, …). The latter sequence converges to 1, and so represents 1\in R.

The fact that (1 - 10[SUP]-n[/SUP])[SUP]k[/SUP] -> 0 for fixed n as k -> ∞ is irrelevant, since 0.999… != (1 - 10[SUP]-n[/SUP]) for any n. As I’m sure you know because of your mastery of limits, limits do not necessarily commute. Even though lim[SUB]n -> ∞[/SUB] lim[SUB]k -> ∞[/SUB](1 - 10[SUP]-n[/SUP])[SUP]k[/SUP] = lim[SUB]n -> ∞[/SUB] 0 = 0, it absolutely does not follow that lim[SUB]k -> ∞[/SUB] lim[SUB]n -> ∞[/SUB](1 - 10[SUP]-n[/SUP])[SUP]k[/SUP] = 0.

I’m tired of your inability to provide specific details or precise arguments. This is a General Questions thread, not a Great Debate. jsum_1 asked a totally reasonable question. Please do us all a favor and answer him properly.

2152

You never learn. 0.9999999…is not equal to 1.

The proof was already given to you many times.

You continue to insist that the limit of the series 0.99999…is 0.99999…
but that is wrong because
lim[sub]n -> ∞[/sub] S = a/(1 - q) = 0.9/(1 - 1/10) = 1

The limit of 0.99999…is not 0.99999…
Therefore 1 is not equal to 0.99999…

2153

And so on. Stop spamming the thread with the same nonsense. Go away.

2154

Moderator Instructions

7777777, you can’t simply keep insisting you are right and everyone else is wrong and just continue posting the same things over and over again. If you haven’t convinced anyone with this yet, you’re not going to do so by simply repeating it yet again.

Unless you can come up with a substantially different argument than you have before, I’m instructing you to stop posting in this thread. And you’re not to post further remarks like “you’ll never learn.”

Colibri
General Questions Moderator

2155

nevermind

2156

So you don’t understand my post. Fair enough. I explained how it works. If you don’t get it, don’t say it is handwaving because it most certainly is not.
First you used to the definition to prove equality.
Now use multiplication of the string of terms to ALSO show equality. Then you have a solid proof. That is how proofs work. You need to prove both sides of the equals. Basic stuff here.

2157

nm

2158

Of course **Itself **gets it! Your assertion that the powers of 0.999… converge to 0 is simply false (though appealing – I can see why you imagine we might find a contradiction there, it turns out there isn’t one).

Do you remember Cognitive Tide? He started to complain that he was getting progressively snarky responses to his posts. This was because of his refusal to engage in proper debate. I sense things are going the same way for you.

It seems to me that **Itself **understood your post, but you have failed to understand Itself’s reponse. Maybe I am wrong, but YOU are in the unique position of knowing what aspects **Itself **failed to address.

Acknowledge what **Itself **said about the non-commutativity of limits and add your clearly expressed remaining objections. Certainly don’t patronize us by telling us that we don’t undertstand proof because that is patently false (and also requires that every other mathematician who agrees with us also doesn’t understand proof).

Present your objections cogently using standard notation, and if you wish to use non-standard notation you better start out by defining it.

2159

And just to add, on a personal note, I like debating, I enjoy meeting people with differing points of view or novel ideas, people who don’t necessarily see the world as I do. I am happy to discuss our differing ideas and I flatter myself to believe that I can do so patiently and respectfully. Sometimes I will be persuaded that it is me who should change my view. That preparedness is a sine qua non to proper debate.

You know, I’d LOVE you to persuade me that 0.999… <> 1 – for me that would be an exciting new fact about mathematics, and I’d immediately add it to my repertoire of “things to argue about”.

It is clear that you have discussed your ideas about this fact on other forums, it is obviously a thing for you, but you seem so rarely to acknowledge that those who disagree might have good reasons for their views, that their arguments hold some weight. In fact it seems to me that you do not have the necessary level of empathy required for healthy exchange of ideas. You KNOW you are right, THEREFORE everyone who disagrees must be unable to get you. It REALLY isn’t that, your arguments are just NOT persuasive, plain and simple.

2160

Well, I didn’t understand your post. I don’t know which axiom you were addressing. I couldn’t follow your calc steps because you didn’t show them. And I don’t know why my derivation of the geometric series formula was invalid because it had infinite seps while you do an infinite process an infinite number of times to have all the terms get smaller: you treat them as zero and your process is considered valid. Hand waving seems an apt term.

2161

“And then a miracle happens”?

schooner26, I too didn’t understand how you want us to multiply 0.999… by 0.999… either – it is a bit much to leave such things as an exercise for us, but I did it and I got 0.999…

So what gives?


(0.9 + 0.09 + 0.009 + 0.0009 + ...)(0.9 + 0.09 + 0.009 + 0.0009 + ...) =
0.81 + 0.081 + 0.0081 + 0.00081 + ...
___  + 0.081 + 0.0081 + 0.00081 + ...
____________ + 0.0081 + 0.00081 + ...
_____________________ + 0.00081 + ...
_______________________________ + etc...

The first line of the result is a geometric sum having limit 0.9
The second 0.09
The third 0.009

In fact, the sum of the sums of each line is 0.999…

So there you are, 0.999… x 0.999… = 0.999… (does that look like any other number we can think of?)

2162

[gulp] You noticed me, uh oh …

Ummm … er … I guess I’m wondering that if 0.999… ≠ 1, that means that 0.000… ≠ 0. If that’s correct, then doesn’t that make mischief with what we observe in the universe? If I apply a force over the instantaneous time interval and evaluate it’s effect on an object, I don’t see any change in speed, but that would only be true if 0.000… = 0.

I’d like for Schooner or 7…7 to explain this.

But since I have your attention now, I believe I was wrong earlier when I stated that “real numbers form a vector space”. Unfortunately they are open under scalar multiplication. Is that true, and only the complex numbers form a vector space?

Never am I to dodge pushing an idea too far … how can I, mathematically, convert two linear forces at right angles to each other into a torque?

2163

What bit about limits not necessarily commuting don’t you understand?

Your proof is invalid because you assumed commutativity of limits.

This is pretty basic stuff, and I would hope you were taught this at uni.

2164

Not false. Each term of 0.999… is less than 1, and consecutively smaller as well. When you consecutively multiply each term less-than-1 by other terms less than itself, they all continuously grow smaller.

Actually, you are circular since you assume your conclusion in the proof.
Your counter goes like this “Actually the infinite powers of powers of 0.(9) is 1 because 0.(9) = 1.” When my argument is: “0.(9) not equal 1 because, powers of powers causes each term of the product to converge to 0”
(And when I say 0.(9), I am not talking about the limit, but rather the “number” stated by someone else, which I assume is a series of 9’s).

But that is pretty much how every argument goes. They use the definition on 1 side of the equals sign, and do nothing on the other, and assert conclusion.

As an aside, I have a very busy life and have wasted more time here than I wanted to already. So asking for a detailed proof when I am on my way to bed, and then spouting off when you don’t get it, isn’t very mature.

I have to disagree with you.
I have said perhaps several times “I completely understand what you are saying” to specific people, or something along those lines.

**I went from never saying 0.(9) equals 1, to 0.(9) = 1 because of what I learned in this thread.

We are on a minor detail now where “a bunch of 9’s = 1” where we don’t use a limit. Endless 9’s can never equal 1 unless we use a limit. Using cauchy’s definition, in order to use said definition, people subtract out terms of the series, which is already the limit of another infinite series. I already proved this. The only thing the definition does for an infinite decimal is show that infinity is greater than any finite number (n).
Anyway, an infinite decimal doesn’t show you how much closer you are to another infinite decimal.

Sorry, but I ran out of time for the day last night.
Maybe it was a bit “much” but you all seem to have boat-loads of time here, and love math as much as I do, so why I figured that maybe you would have fun trying something new ?? I actually had lots of fun doing math in my day. If it is a huge chore, then why bother? Anyway, I left it at that because if someone had time, and was interested, they could try it out… like you did !!



(0.9 + 0.09 + 0.009 + 0.0009 + ...)(0.9 + 0.09 + 0.009 + 0.0009 + ...) =
0.81 + 0.081 + 0.0081 + 0.00081 + ...
___  + 0.081 + 0.0081 + 0.00081 + ...
____________ + 0.0081 + 0.00081 + ...
_____________________ + 0.00081 + ...
_______________________________ + etc...

Now, you are on the right track with the above !

But what you are doing part way through is using the definition again (using the limit), which is circular, because you already did that on the left side.
You need to only use multiplication, and powers of powers for the series generated by doing so.
My argument is against the point someone made that “the real number 0.999… corresponds to the sequence {9/10, 99/100, …} whose limit is the real number 1”

But, what I learned in this thread, this means exactly “the real number **limit Σ[sub]n→inf[/sub] 9/10ⁿ **corresponds to the sequence {9/10, 99/100, …} whose limit is the real number 1”

The redundancy here is hard to keep track of. Anyway, everytime anyone references 0.999… they reference the number 1. Someone else referenced 0.999… as a number first, not as a limit, but as an idea in their head as a number. Without any knowledge of the number 1, how does someone conceive the idea of a number as an infinite string of digits ?

This phrase is circular and recursive : "“the real number 0.999… corresponds to the sequence {9/10, 99/100, …} whose limit is the real number 1”
0.999… is already 1 since it is a limit, but then for some reason, “it corresponds to the sequence {9/10, 99/100, …}”
That is circular and requires knowledge of the limit of the sequence FIRST. The correct way would be “the sequence {9/10, 99/100, …} has a limit: the number 1, whose sequence corresponds to {1, 1, 1, …}. Therefore, both sequences have the same limit”. or something like that.

Another example is the series: 1/2 + 1/4 + 1/8 + …
So complete the similar phrase analagous to the 0.999… one above:
“the real number _____… corresponds to the sequence {1/2, 3/4, 7/8, …} whose limit is the real number 1”

Back to the point: what you do now is, instead *only *squaring the original, introduce *more powers of powers *and analyse the results:



[ (0.9 + 0.09 + 0.009 + ...)(0.9 + 0.09 + 0.009 + ...) ][sup]**2**[/sup]

So, [0.81 + 2(0.081) + 3(0.0081) + … ][sup]2[/sup]

Note: A faster way to do see results is to obviously start with much higher powers than say, 2.
Now what do we get ?



(0.81 + 2(0.081) + 3(0.0081) + ...) x (0.81 + 2(0.081) + 3(0.0081) + ...)
0.6561 + 2(0.06561) + ...
------ + 2(0.06561) + ...

And so on, to infinity, and the terms get infinitely smaller. Therefore it doesn’t matter if there is a coefficient or not, it is irrelevant.
Just like {0.1, 0.01, 0.001, …} = 0, and 3 x {0.1, 0.01, 0.001, …} = 0

You can clearly see the first term getting smaller, and with some analysis, we can see each successive term gets smaller than the first, so if the first term goes to 0, then so do the remaining ones.

Another view, (which will of course be rejected immediately,) is the following:
0.9[sup]3[/sup] = 0.729, then cube it again, repeatedly = 0
0.99 cubed repeatedly = 0

The first (n) terms of 0.999… cubed repeatedly is 0.
The likely rejection is “But you only did it for finite 9’s” but…
that is all I *have *to do since by real numbers, I only need to form a sequence of a few terms to show a “patten” which in this case converges to 0. If every term is an infinitesimal, which is 0 by definition, then the condition is satisfied. Is it not?

2165

I’ve been watching this thread with a mixture of amusement and exasperation. I just want to point out a certain muddying going on with the mathematics: The axioms given by j_sum1 in post 2142 don’t define the real numbers, but they do characterize them. Precisely, we call any set with addition and multiplication and an ordering satisfying those axioms a complete ordered field. We can then prove that any two complete ordered fields are isomorphic. So, if there are any complete ordered fields, there is essentially only one.

In order to show that there is a complete ordered field, we need to actually construct one, and we usually do so starting from the rational numbers (which form an ordered field, but not a complete one). Various constructions have been mentioned in this thread, including the one via Cauchy sequences, as sketched most recently I think by Itself in post 2089. To be precise, we can define a real number to be an equivalence class of Cauchy sequences of rational numbers, where we say that one sequence (a[sub]n[/sub]) = (a[sub]1[/sub], a[sub]2[/sub], …) is equivalent to another, (b[sub]1[/sub], b[sub]2[/sub], …), if for every (rational) e > 0, we can find an integer N such that |a[sub]n[/sub] - b[sub]n[/sub]| < e for all n>N. To be clear, write [a[sub]n[/sub]] for the equivalence class of a Cauchy sequence (a[sub]n[/sub]). To be very clear, note that, under this definition, a real number is not a single Cauchy sequence, nor a thing that sequence converges to, but a whole (uncountable) set of equivalent sequences. Addition and multiplication are defined termwise; we say that [a[sub]n[/sub]] ≤ [b[sub]n[/sub]] if either [a[sub]n[/sub]] = [b[sub]n[/sub]] or there exists an N such that a[sub]n[/sub] ≤ b[sub]n[/sub] for all n > N.

Of course, there are things that ought to be proved to make sure that definition works: For example, the operations are defined by choosing representative sequences within each class and operating on them, so we need to show that the choice of representations does not affect the class of the result. It certainly affects the sequence we get, but what we show is that different choices result in equivalent results. And, of course, we need to verify that the real numbers so defined form a complete ordered field, but this is all pretty straightforward stuff.

The advantage of using Cauchy sequences for this discussion was already pointed out by Itself: We can view decimal notation as a way of specifying a particular equivalence class of Cauchy sequences. For example, 0.999… can be viewed as notation for the class [9/10, 99/100, 999/1000, …]. If you don’t like the ellipsis, this is the class of the sequence whose nth term is (10[sup]n[/sup] - 1)/10[sup]n[/sup] = 1 - 1/10[sup]n[/sup]. This is what we mean when we say that 0.999… is notation for a real number.

On the other hand, consider the Cauchy sequence (1, 1, 1, …). This is equivalent to the sequence above: Given e > 0, let N be so large that 1/10[sup]n[/sup] < e for all n > N. Then |1 - (1 - 1/10[sup]n[/sup])| = 1/10[sup]n[/sup] < e for all n > N. Hence, the two sequences are in the same equivalence class, i.e., define the same real number, which is the one we call 1.

(I didn’t set out to prove 0.999… = 1 when I started, but there you are. It just falls out easily from any careful construction.)

No mathematician (pace any constructivists reading) would object to the correctness of the development above, though some might prefer Dedekind cuts or something else on aesthetic grounds.

Rather optimistically, I hope this clarifies some things. I have work to do now, so will go back to lurking.

2166

I like thinking of real world applications of infinity, etc, too.

here is one:

2 objects are 1 meter apart. Object A and B. Say that 2 lines intersect each object (or point) and these lines are parallel (Line A and B). Now move 1 of the objects along its line, say, 1 meter at a time. (Object B moves) Draw a line from Object A to Object B. This is line C. Line O is a line between the 2 points at the start.
Reference to the static object, angle OC is that angle between lines O and C
Angle AC is between Line C and Line A.

Angle OC increases from 0 (the starting angle between both points) as Object B moves away from its origin.
Angle AC = 90 - OC
Angle AC decreases with Object B’s movement.

Link to picture [ HERE]

What happens to angle AC as object B reaches infinite distance from its origin ?

Does it converge to 0 ? If so, then it equals 0 ?
If it equals 0, we know Object B is on Line A. however, this is a contradiction.

2167

Thank you for all that. It is likely the most clear explanation I have heard.

Thanks again.

The issue I was having was someone saying 0.999… and 1 are different real numbers, that are proven equal, or it at least came across that way.
What I mean by that is it came across that “0.999…” was its own real number, and not a “notation for the class [9/10, 99/100, 999/1000, …]”[sub]Topologist[/sub]

2168

This comes down to the notation and what is the correct interpretation.

You are right when people interpret it incorrectly.

The series, a bunch of 9’s to infinity, can’t exist without the limit, because of the involvement of infinity.

So,
0.9(k times) = Σ(n=1 to k) 9/10ⁿ = f(k) = 1 - 1/10[sup]k[/sup]
the idea that you can imagine 0.9(∞ times) without the use of a limit is wrong.
Just as you can’t use the remainder of the above equation with k = ∞ without any limits.

limit[sub]k→∞[/sub] 0.9(k times) = limit[sub]k→∞[/sub] Σ(n=1 to k) 9/10ⁿ = limit[sub]k→∞[/sub] f(k) = limit[sub]k→∞[/sub] 1 - 1/10[sup]k[/sup]

Not saying you can’t imagine a string of digits that is endless, but it just can’t exist as a number because its limit is the number.

I may be way off track here, but thinking about it in steps may interfere with the complete understanding of it.

Someone might think like this:
0.9(n times) [ok, now I have to imagine infinity, so] 0.9(infinite times) [which is the same as] 0.999… [because the … mean endless or to infinity] then [oh, since I have infinity there, I must use the limit] lim 0.999… [which equals] 1.

When the fact is, as soon as you “think” infinity, the limit is there with it.

So I think this is correct: “You have finite digits (finite number)” or you have “the limit of the sequence generated by infinite digits”

2169

I’m sorry, who said that? (7777777 or CognitiveImpairment don’t count.) It doesn’t make the slightest sense:

[QUOTE=no-one nor his dog]
“0.999… and 1 are two different numbers: I shall now prove that they are equal”
[/QUOTE]
That would be nuts!

People have said they are two different representations of the same number, and it’s even possible that someone said something like “Suppose/Assume 0.999… and 1 are distinct numbers” and then went on to provide either a proof by contradiction or a reductio ad adsurdum.

Iff you have been persuaded by Topologist’s argument I shall call it a success. Yay! One less ignorance in the world!

Now, do us all a favour and explain your new found insights to 7777777 will you?

2170

OK, it’s a slow Friday afternoon.

Thanks. As a teacher, that’s one of the nicest things you can hear.

My first reaction was that this is nonsense, but one man’s nonsense is another man’s projective geometry: There is a way to make sense out of this, without contradiction, but not within Euclidean geometry. In Euclidean geometry, there is no sense in which B can reach “infinite distance from its origin.” There simply is no such point in the Euclidean plane.

However, the Euclidean plane embeds in the projective plane, which we get by adding a “line at infinity.” Without getting into the precise definition, you can think of it this way: For each line through the origin, we add a new “point at infinity,” closing that line up into a circle. (This is very similar to the way we can add a point at infinity to the real line, reachable by going “infinitely far” in either direction, turning the line into a circle.) If we start with one line through the origin and sweep it through 180˚, the corresponding points at infinity sweep out the line at infinity.

A line not going through the origin meets the point at infinity corresponding to the parallel line through the origin.

In the projective plane, every pair of lines meets in exactly one point. What we think of as parallel lines in the Euclidean plane meet at one of the new points at infinity. Lines that intersect in the Euclidean plane still intersect in only one (finite) point.

So, to make sense of your scenario above, we can interpret “B reaches infinite distance from its origin” as meaning that B reaches the point at infinity on its line. No contradiction: This point is also on the line through A, being the unique point where those two lines intersect.