# Limits and .9 recurring

1. Mar 29, 2004

### matt grime

In order that that other thread can get locked, here's something for JonF et al (sans organic detritus).

Firstly, 0.999... is the same as 9/10 +9/100 +9/1000+.... that is what the decimal expansion means. This sum is equal to 1.

Now the other quesition involved taking the limit as n tends to infinity of (0.99...)^n now this is ne, but perhaps, JonF, you're switching limits, which you cannot do without proving it is valid. Meaning here, if x_m = 0.99999...9 with m nines that lim_n ( lim_m (x_m)^n) = 1, yet lim_m (lim _n (x_m)^n) =0

this is ok, and not contradictory if you remember that maths is just the manipulation of symbols to follow certain rules. One is that you cannot exchange limits whenever you feel like it. There are times you can times you can't.

Here's a couple of examples I gave in another thread today where it matters when you take limits.

for the interval [0,1] define f_n to be x^n the point wise limit of this os the function f where f(x) = lim_n f_n(x) which is the same as the function that is 0 for all x in[0,1) and 1 at 1. So limits don't commute with continuity.

the second is g_n defined on R as 1+x/n for x in[-n,0] 1-x/n for x in[0,n] 0 elsewhere. the pointwise limit is the zero function and the integral of that over R is zero, but the integral of each g_n is 1, so limits don't commute with integration.

2. Mar 29, 2004

### JonF

Seemed to me that this would mean that .999..... Is infinitely close to 1 but not actually 1.

But now I have figured out the error of my ways, by trying to make an example where: the infinitely small gap is multiplied an infinite many times. But, it turned out my example showed me the exact opposite of what I thought it would.

3. Mar 29, 2004

### matt grime

The idea that it is infinitely close, but not the same, is what Abraham Robinson used in his non-standard analyisis. He defines infinitesimals like this. These ideas are not mainstream (they are intuitive, perhaps, to an untrained mathematician, but do not allow one to define the real numbers in the correct (correct means usual) way, and hence are unintuitive to a trained mathematician). They are of use in parts of economics. The difference is that ordinarily we use this idea of cauchy sequences where a real number is an equivalence class. That's uninterestingly technical, but it just means we say if a sequence of real numbers tends to zero, the limit is zero. In Robinson's version, we say the limit is an infinitesimal, and real, and therefore zero because zero is the only real infinitesimal. It is a subtle distinction that I don't think I've explained properly. But just remember that .99999.... is 1 is a consequence of how we've defined the real numbers.

4. Mar 30, 2004

### Zurtex

Would I be correct in saying that if infinitesimally small numbers existed within a number system then so would infinity. As infinity is not defined within real numbers then neither can infinitesimally small numbers.

5. Mar 30, 2004

### matt grime

The infinitesimal numbers, of Robinsonian analysis, are not part of the real numbers. THe only real infitesimal is 0.

6. Mar 30, 2004

### Hurkyl

Staff Emeritus
No, that would not be correct.

Now,

if the number system was ordered (so you can define infinite and infinitessimal)
if the number system was a field (aka division by nonzero is defined)
and
if there was a nonzero infinitessimal

then you can conclude there is an infinite number. In fact, you can conclude there are lots of infinite numbers; if &epsilon; is a (nonzero) infiniessimal, then, for instance, 1 / &epsilon; and 2 / &epsilon; are distinct infinite numbers.

7. Mar 30, 2004

### Organic

The cardinal of the empty set = 0

The cardinal of a non-empty set is at least 1

So the meaning of small and smaller is meaningless when we compare them to emptiness.

Any transformation from a non-empty set to an empty set cannot be but a phase transition form cardinality 1 to cardinality 0.

Because of this reason we cannot define the smallest number.

A notation that can describe this beautiful situation is this:

[.000..., 1)

The meaning of this notation is this:

There are infinitely many empty levels of scales in the above singleton that cannot change the state of some non-empty set to an empty set.

It means that the cardinality of this half-open interval [.000..., 1) is at least 1

Therefore [.999..., 9) + [.000..., 1) = 1

By the way [.000..., 0) is for nonstandard analysis as [.000..., 1) is for standard analysis.

In both cases the result is a non-empty content of some set ( |{x}|=1 ).

Last edited: Mar 30, 2004
8. Mar 30, 2004

### Zurtex

Oh, O.K

Organic in terms of real numbers would [.999..., 9) + [.000..., 1) not be the same as 1 + 0?

9. Mar 30, 2004

### matt grime

But that isn't the real number system you're talking about. It is not possible to claim there are infinitely many 0s and then a 1 and still be in the real numbers. What you're talking about has been formalized as non-standard analysis. It can easily be shown that it is not the real numbers because then you're implying there is a natural order preserving bijection to an uncountable well ordered set, which does not exist. Now, if you were to care about mathematics you might wish to learn about all the things I've just mentioned. You won't though will you?

Incidentally, why is 0 not the smallest (non-negative) number? You omitted to deal with that. And your analysis doesn't hold up as there's nothing there which doesn't apply to the natural numbers, so you've just "proven" that 1 is not the smallest natural number.

To elaborate on that, at no point do you use what the symbol 0.999..... means ie that it is a decimal expansion of a real number. We could equally be dealing with hexadecimal expansions and all of what you said remains just as "true" and you've just concluded something even less true.

Moreover, the object you write as [.00000000...,1) you call a half open interval. Well do you mean [0,1)? and you're saying that has cardinality is at least 1? Well, it is c, which I suppose is greater than 1. I'm only hesitating because you don't use the correct definition of cardinality.

If you don't mean that half open interval, I'm not sure which one you mean. The last sentence is also equally puzzling as the only interpretation of that is not correct, mathematicall.

Also a supposedly singleton set that is a half open interval of real numbers does not make much sense, as the set [x,x) means all the y greater than or equal to x and srtictly less than x, which is empty - there are no real y satisfying that criterion.

Apart from all those mistakes you've also introduced yet more undefined terms and concepts.

10. Mar 30, 2004

### Organic

Zurtex,

Be aware that what I am doing here is a totally new point of view on Math language.

If you are a student than my ideas can make you a lot of troubles with your teachers, at this stage.

Last edited: Mar 30, 2004
11. Mar 30, 2004

### Zurtex

You can't be serious? I was confusing my teachers with the whole infinitesimally small numbers at GCSE level when I had much less of an understanding of mathematics.

12. Mar 30, 2004

### matt grime

Just because it is a totally new point of view doesn't stop it being totally wrong and inconsistent and not suitable for posting in this thread. Please stick to thoery development as requested.

13. Mar 30, 2004

### JonF

Ok, I get now why .999... = 1

Previously I thought $.\bar{0}1 + .\bar{9} = 1$

Which it does, but that is because: .0000….1 = 0

The infinitely small gap I thought existed is actually 0

14. Mar 30, 2004

### Organic

You did not understand [.000...,1) notation.

Here I am using the idea of the half-open interval on a representation of a single R member, which is a non-standard way of using.

By this notation I mean that there are infinitely many empty levels of scales in the above singleton that cannot change the state of some non-empty set, to a state of an empty set.

You can say: "But number 0 is not an empty set content, but a non-empty set content, for example {0}, so what are you talking about?"

My answer is: "Please don't interfere between the notation and the meaning behind the notation, which are two different things, and I am talking about the meaning behind the notations, which are:

If our singleton's notation is not 0, we are talking about a non-empty set, and if our singleton's notation is 0, we are talking about an empty set.

More than that, in general I am talking about the relative relations between two different representations of to different numbers, where their self unique values are being kept exactly as |{}| and |{0}| cannot have the same cardinality."

I am not talking about 1 notation as some level that we can reach.

1 notation is here to tell us that it does not matter how many levels notated by 0 we have, still it stays a content of a non empty set with cardinality 1 ( |{x}|=1 ).

By the way, by [.000..., 0) notation we get more general representation of the above idea, which says that our singleton cannot get cardinality 0 ( |{}|=0 ) no matter what notation we are using.

therefore the gap between the two numbers remains open(>0), in both standard and nonstandard analysis.

Last edited: Mar 30, 2004
15. Mar 30, 2004

### ahrkron

Staff Emeritus
Organic,

When explaining or elaborating on your own version of math, please do so in the TD forum. It can confuse people that are learning, especially because the definitions you use for many terms are different from the standard ones. It is not enough to state that yours is a "new point of view".

16. Mar 30, 2004

### Organic

Hi ahrkron,

Is General Math is a special forum for students?

17. Mar 30, 2004

### Tom Mattson

Staff Emeritus
No, it's a forum for math!

18. Mar 30, 2004

### Organic

Hi Tom Mattson,

Is there any problem to discuss about non-standard ideas in General Math forum?

19. Mar 30, 2004

### Zurtex

So Organic what exactly do you think you have proved? Because to me it just looks like you are trying to prove something by defining it to be so.

20. Mar 30, 2004

### matt grime

but 0.00000.......1 does not have any meaning as a rea number! Infinitely many zeroes then a 1? Sorry.