Is 0.999... Truly Equal to 1 in the Realm of Infinity?

[russ_watters]

the very definition of convergent means it will never reach its limit.

so there we go.

This is another definition on which you are wrong (or 'don't agree with' - which in this case is pretty much the same thing). "Convergent" means it does reach its limit. "Divergent" means it doesn't.

 

[ram2048]

Convergent: - Mathematics. The property or manner of approaching a limit, such as a point, line, function, or value.

hmm Wolfram Mathworld studiously avoids defining convergent...

possibly another conspiracy... ;D

 

[Hurkyl]

A sequence is convergent iff it has a limit.


BTW, have you considered the sequence: 0, 0, 0, 0, ...?


I think what you meant to say is that, generally, each term of the sequence will be inequal to the limit. (But, of course, that is not always true)

because it makes no logical sense that the sum of a convergent series equals it's limit when the very definition of convergent means it will never reach its limit.

I think what you meant to say is that it violates your "common sense".

Even with your definitions this doesn't follow; even if the partial sums don't "reach" the limit, why should that suggest anything about the infinite sum?

 

[ram2048]

for fun's sake let's go back to Zeno's Paradox

1/2 + 1/4 + 1/8 + 1/16...

every step only completes half of the remaining value converging towards 1.

BUT let us consider the last digit

1/\infty

here's where we get the problem i think

1/\infty would complete only half distance theoretically as well, BUT current math doesn't believe beyond infinity.

AND

they believe that anything divided by infinity is the same number

hence you COULD have 2/\infty and complete the whole remaining distance and it would be the same as adding 1/\infty

pure conjecture of course. i think that in order to continue our understanding of math we have to find ways to define stuff better. having 20 different definitions for infinity is a pain in the patoot

 

[Integral]

having 20 different definitions for infinity is a pain in the patoot

I sure understand what you are saying, that is why I cannot figure out why you insist on having so many definitions of infinity.

If you would simply make an effort to understand the SINGLE definition used by Mathematicians your world would be much simpler.

In the extended real line infinity is simply defined as being larger then all real numbers. A single simple definition. Thus any real number times infinity is still infinity because it is still larger then any real number. Likewise 1 over infinity must be smaller (in absolute value) then all real numbers, the only thing that meets this condition is 0, therefore \frac 1 \infty = 0 simple and consistent. You only add unnecessary complexity with your infinite number of infinites.

You are right that a set of infinitely shrinking intervals does indeed contain something (I think you phrased it that no matter how much you divide it up there is still something there). That something is a single point. Will you agree that the "length" of a single point is 0? That is 1-1=0? So does it not make sense to say that the length of the interval resulting in infinite divisions (which results in a single point) has length zero?

Will wait for your reply to make sure we are on the same page.

 

[matt grime]

Ram seems to be having the standard issue that sequences can tend to limits and not actually reach them after a finite number of steps. That aand the facthe thinks there's a last term in an infinite sequence.

Would you like to list the 20 different definitions of infinity? There's only one meaning for infinite, and no need to use the word infinity if it confuses you.

 

[ram2048]

In the extended real line infinity is simply defined as being larger then all real numbers. A single simple definition. Thus any real number times infinity is still infinity because it is still larger then any real number.

so you consider infinity to be a number governed by its own rules such that infinity is greater than itself (greater than all real numbers)

or is it not a number, such that it doesn't break that definition being greater than itself should infinity + 1 > infinity

and as far as the interval being a single point i don't believe it so. I still believe everything is further infinitely divisible beyond infinity.

 

[arildno]

ram_1024:
(I won't be part of this ego-doubling process from your side; however, I'm not a churl, so I won't start halving it either)

I'd like to point out a few implications of your own ideas, rather than making clear to you why standard maths is consistent, and hence, why your attacks on it is quixotic at best.

(Others on this forum are by far more competent than myself in doing such an explanation, and if you had bothered to read, and tried to digest what they have patiently written to you, you would have stopped these ridiculous attacks long ago)

Implications of your ideas:
a)
Now, you say that a sequence: 0.1, 0.01, 0.001 and so on does not converge to 0,
but to some number 0...1 (with an infinite number of zeroes in between)
Clearly then, a sequence: 0.2, 0.02, 0.002 goes to some number:
0...2 (Right?)

In particular, this is perfectly in accordance that each term in the second sequence is the double value of the same term in the first sequence:
For example, 0.02=2*0.01, so it should make sense that the end value,
0...2 is the double of 0...1 (Correct?)

b) Look now at the sequence:
1, 0.1, 0.01 and so on.
At every single instance, the term in this sequence is 10 times bigger than the same term in the sequence 0.1, 0.01 and so on.

Hence, by your logic, 1, 0.1, 0.01 must go to the number 0...10
(You can't escape this conclusion, sorry about that!)

c)
However, the first sequence is simply a subsequence of the last one; i.e. subsequences in your system doesn't go to/converge to the same values as the sequence itself.
In short, the whole convergence concept is blown to smithereens, and it is meaningless in the first instance to say that something "produces" or goes to
something at all; i.e. every single utterance you have made is mumbo-jumbo and nothing else.

Have a good day!

 

[matt grime]

there is nothing contradictory about this infinity+1 not being greater than infinity, since the definition is that it is greater than all real numbers; infinity is not a real number, so infinity+1 is still infinity, unless you're talking ordinals, when w and w+1 are distinct ordinals, but that is a different system from the extended real line again. amazingly it always seems that these crackpot attempts at showing inconsistency are inconsistent, but also that someone non-crackpot has thought it through and offered something that is consistent and does it properly: surreal numbers, hyperreal numbers, the extended complex plane, ordinals, cardinals...

 

[Hurkyl]

Suppose I do a thorough analysis of red M&Ms. Does that mean there are no green M&Ms?

The destination, indeed, does not appear in Zeno's analysis, but there's no reason in particular to think that Zeno's analysis covers the entirety of the motion in question.

However, there is not a "remaining distance" either. Zeno's analysis covers every position up to (but not including) the destination. If the destination is 1 meter away, then Zeno's analysis covers all positions x where 0 <= x < 1. The proof of this requires the Archmedian property posessed by the real numbers: for every real number r there is an integer n such that n > r.

It goes roughly as follows: let x be any real number in 0 <= x < 1. Let y = 1/(1-x). By the archmedian property, there is an integer n > y. By induction, we can find an integer m with 2^m > n. Thus, there is an integer m with 2^m > y, so 1/(2^m) < 1-x and 1 - 1/(2^m) > x. However, 1 - 1/(2^m) is the "current position" of the runner after the m-th step of Zeno's analysis. Thus, the position x has been considered by Zeno.

1/2 + 1/4 + 1/8 + 1/16...

every step only completes half of the remaining value converging towards 1.

BUT let us consider the last digit

Did you mean last term?

And why would there be a last term? Each term is of the form 1/2^m where m is an integer and there is no last integer...

so you consider infinity to be a number governed by its own rules such that infinity is greater than itself (greater than all real numbers)

The system Integral describes is called the "extended real numbers". In this system, infinity and -infinity are extended real numbers that are not real numbers.

 

[russ_watters]

so you consider infinity to be a number governed by its own rules such that infinity is greater than itself (greater than all real numbers)

Looks like we also need to define "real numbers" for you. The word "real" isn't arbitrary as you are using it, it has a specific definition in math as well. http://en.wikipedia.org/wiki/Real_numbers

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number".

Notice, while the real number line is infinite, "infinity" is not a point on the line, hence "infinity" is not a real number. Your above objection is based on a misunderstanding of the definition of "real numbers." (you also characterized "infinity" as a "digit" a few posts ago - also incorrect).

Seriously, ram, you have a lot to learn about math. What we're talking about here is largely high school stuff. If you would only accept that you have a lot to learn and decide to learn it, you'd be much better off.

 

[ram2048]

i can goto 5 sites and pull a different definition of real numbers, irrational numbers, and infinity.

and you talk to ME about inconsistency. I can't use your words because your definitions are "Mumbo jumbo"

i say something then you correct me with a different definition that means the same thing.

the point of the matter is, you say the number line is infinitely divisible but then you say when you divide it by infinity "it can't be determined" or "is close enough to zero we'll just call it zero"

i'm trying to establish a notational system that allows for infinite division AND allows for division beyond that.

you guys say the current system works, why mess with it, but totally balk at the idea of replacing it with something that works BETTER.

sentimental value i guess... :|

 

[ram2048]

Implications of your ideas:
a)
Now, you say that a sequence: 0.1, 0.01, 0.001 and so on does not converge to 0,
but to some number 0...1 (with an infinite number of zeroes in between)
Clearly then, a sequence: 0.2, 0.02, 0.002 goes to some number:
0...2 (Right?)

these are not even converging sequences under my notation. you would have .00...02, .00...002, .00...0002, . . .

Have a good day!

i didn't! you cursed me didn't you!

 

[ram2048]

Did you mean last term?

And why would there be a last term? Each term is of the form 1/2^m where m is an integer and there is no last integer...

of course i meant last term. Integral has me talking about digits and then you jump on me when i get a word wrong. okay okay spanish inquisition over here..

and yes there is no last integer by MY notation, but there IS by the current one. such that 1/Infinity = 0 = limit.

it's your system though, you KNOW it better than me why do i have to tell you?

 

[Hurkyl]

i can goto 5 sites and pull a different definition of real numbers, irrational numbers, and infinity.

I wouldn't doubt it. There are often many equivalent ways of defining something.

For instance, (IIRC) the first rigorous definition of the real numbers was that a real number is an equivalence class of cauchy sequences of rational numbers. (In laymen's terms, any real number is identified with the sequences of fractions that approach it)

The usual modern definition preferred would be to define the real numbers as a complete ordered field. (In laymen's terms, +, -, *, /, and < all work "properly", and there are no "holes")

And it can be proved that the first definition satisfies the requirements of the second definition. Conversely, anything that satisfies the second definition is isomorphic (in laymen's terms, the same) to the first definition.


(Incidentally, 5? Really? I can only think of 3 definitions of real numbers you could be reasonably expected to find online, and only one definition of irrational. Different definitions of infinity wouldn't surprise me, because there are a lot of different concepts that are (sloppily) called "infinity")

the point of the matter is, you say the number line is infinitely divisible but then you say when you divide it by infinity "it can't be determined" or "is close enough to zero we'll just call it zero"

Actually, we say "Please be more specific".

"Infinitely divisible", taken entirely literally, means that it can be divided into an infinite number of terms. (Notice I did not say "an infinity of terms") In particular, the real line can be divided into c (= |R|) terms.

c is a thing called a cardinal number. It is not a finite cardinal, so it must be an infinite cardinal. (Some call it a transfinite cardinal, simply because so many laymen get confused when things are called infinite)

Division is not well-defined on cardinals, because multiplication is not very nice. For example, 1 * c = 2 * c. If you could divide by c, you would get 1 = 2, which is bad.

i'm trying to establish a notational system that allows for infinite division AND allows for division beyond that.

Learning math might be helpful to see how to do this.

Here's a simple approach to defining such a system:

Consider all (real) rational functions in x. E.G. things like 6, 7 + x^2, and (1 + 3x + 4x^5)/(4x + 3x^7)

+, -, *, and / can all be defined and function "properly" in this field. You can order this field by decreeing that x is bigger than any real number, and then extending the definition of < to accommodate this decree. So, for example, 7 + x^2 is infinite, because it is bigger than any real number. Proof:

Let r be any real number.
r < x
1 < x
r = 1 * r < 1 * x < x * x < x * x + 7
Thus r < x^2 + 7

Similarly, (1 + 3x + 4x^5)/(4x + 3x^7) is an infinitessimal.

If you don't like x, maybe you could use w (omega), a common symbol for transfinite numbers.


There's another number system (whose technical details are very difficult to follow) called the hyperreals which have transfinites and infinitessimals, but, for the most part, behave exactly like the real numbers. You might find information on this by searching for "non-standard analysis". Last time I went looking, there was actually an undergraduate calculus text in PDF format somewhere on the web that develops calculus using nonstandard analysis (i.e. with infinitessimals and transfinite integers, etc) instead of the usual way.

you guys say the current system works, why mess with it, but totally balk at the idea of replacing it with something that works BETTER.

We're "balking" because you are claiming the current system doesn't work.

of course i meant last term. Integral has me talking about digits and then you jump on me when i get a word wrong. okay okay spanish inquisition over here..

Or, maybe I'm just making sure I knew what you meant.

 

[ram2048]

i never claimed it didn't work, i claimed it was inaccurate ;D

i walk on bridges and fly by airplane. i wouldn't do so if i didn't trust our current set of math at all

in any case, there's little point in arguing as it seems apparent that such things are not going to change overnight.

i thank you for the envigorating discussion, and humbly apologize for getting bent out of shape at you in previous threads, Hurk

 

[matt grime]

yes there is no last integer by MY notation, but there IS by the current one. such that 1/Infinity = 0 = limit.

it's your system though, you KNOW it better than me why do i have to tell you?

There is no last integer in 'our' system, that you think we think there is is your error.

 

[ram2048]

ooooooooooooooooookay

define a number less than 1/infinity but greater than 0 in your system

 

[matt grime]

what do you mean by 1/infinity? infinity isn't a real number, so why should i be able to do that? have you not learned anything from this thread? you aren'ty dealing with the real numbers when you write that kind of thing.


in robinsonian analysis i believe the object you are talking about is labelled epsilon.

 

[Zurtex]

Well I'd like to thank all the people who tried to help ram out on this thread, whether or not he still spouting rubbish you have really made me feel a lot better about numbers . I still can't believe your trying to get the point across and have not just banned him from posting on the maths forums and ignored him here, well done your good people.

 

[Hurkyl]

define a number less than 1/infinity but greater than 0 in your system

This question, in general, doesn't make sense.

It cannot make sense in the context of the real numbers.

In the context of the extended real numbers, this cannot be done, since 1/infinity is 0.

In the context of the hyperreal numbers, if we rephrase your challenge (so that it makes sense) as:
"If w is a positive, infinite number, then define a number less than 1/w but greater than 0"
then an answer to your challenge would be 1/(2w).

 

[ram2048]

i somehow wrote that all wrong :O

what i meant to say is the "last" integer in your system would be a function of infinity such that nint(infinity) = that integer (theoretically if you could USE that function)

but that's still getting off the point.

if you COULD then 1/2^nint(∞) would be the closest step for your consideration.

forgot where i was going with this O_O

meh...

 

[Zurtex]

ram2048 what is a "last integer"?

That makes less sense to me than anything else you have written.

 

[ram2048]

hell if i know, Hurkyl was talking about integers and blah blah no last integer..

but there IS a last integer because you have an upwards limit of infinity.

you can't define that number without it being an expression "of infinity" itself, so it kinda defies itself.

but there was a point i was trying to make such that a sum to infinity, even AT infinity does not equal its limit in such a convergent sieries as Zeno's paradox.

as with everything "Infinity related" it's all theory and you have to apply logic. if every "step" in the process or "term" computed is half the remaining, there will never be a process that is "whole of the remaining" because that breaks the rule set forth in the initial exercise. so even at infinity, or beyond infinity in the case of "hyperreal" blah blah "extended irrationals" or whatever you want the sum STILL doesn't equal the limit.

the same is true of .999~ every digit tacked onto the end brings it closer and closer to 1, but no digit is ever a 10 completing a whole "step"

but it really doesn't matter... whatever :D

 

[chroot]

So there's an integer that is larger than every other integer, eh? Could you do us a favor and show us which integer is the biggest?

- Warren

 

[ram2048]

the one closest to infinity? :O

 

[chroot]

That's the stupidest thing I've ever heard.

- Warren

 

[ram2048]

haha my point is proven

not only can you NOT get to infinity, ever. you can NOT get to the largest integer that is not infinity.

so what hurkyl said about every step being considered is kinda out the window eh?

in any case .999 can never be = to 1 because every step is 9/10's the remaining step towards 1 and "the destination is never in consideration"

there is no step covered that is 10/10ths the remaining distance

sums to infinity are a good approximation

 

[Hurkyl]

What is one plus the "largest integer that is not infinity"?
What is two plus the "largest integer that is not infinity"?

(plus, here, means integer addition)

the same is true of .999~ every digit tacked onto the end brings it closer and closer to 1

Why (and how) are you tacking digits onto .999~?

not only can you NOT get to infinity, ever. you can NOT get to the largest integer that is not infinity.

But you forget, you can "get" to every positive integer. It's a simple proof by induction:

I can get to 1.
If I can get to n, then I can get to n+1 by adding 1 to n.
Thus, by induction, I can get to any positive integer.

If the number system about which you are speaking has numbers to which you cannot "get", then you're not speaking about the integers.

in any case .999 can never be = to 1 because every step is 9/10's the remaining step towards 1 and "the destination is never in consideration"

.999 can never be equal to 1 because it is equal to 999/1000. Of what steps are you speaking?

 

[Hurkyl]

sums to infinity are a good approximation

Approximation of what?