Questions about the union of these sets?

[pka]

I don't understand what you are saying

Hello Mates, There is no infinite summation in reply #35.
I think you are thinking about a sum like $\sum\limits_{n-1}^{\infty}[9\cdot10^{-n}]=0.9+0.09+0.009+\cdots=1$ SEE HERE
Now once again that is a limit of sum. It is one number.
$0.9+0.09=0.99$ is one number. $0.9+0.09+0.009+\cdots$ is not one number.
Of course, the representation $0.\overline{\,9\,}=1$ is shorthand and is one number, but not what you asked about.

On the subject of notation, the cardinality of a set is usually denoted by $|A|$
However alternatively maybe denoted by $n(A),~\|A\|,~\text{card}(A),~\overline{\,A\,}$ among others.
Be careful of that last one because almost universally it is used for the negation of $A$.

 

[Agent Smith]

I am not sure if you saw my question (please excuse me if you did). I am interested to know if we can sum to infinity with the method you showed in post #35. Would that make the number 0.999 ...?

Yes, but what would be the larger point to such an exercise? What advantage does this particular approach have in clarifying, decontroversializing, the "problem" of $0.\bar 9 = 1$?

Too, it seems sets are designed for whole numbers and ill-suited for fractions. Can you list the elements of a set A such that $n(A) = 0.9$?

 

[Agent Smith]

@Mates You can look at it this way ... it kinda sorta makes sense to me ...



0.9 wants to become 1, but it's not allowed to add to itself 0.1. That would've been so, so, sooo convenient (0.9 + 0.1 = 1). "However", says God, "you may use the numbers 9 and 10, and the only operation you're sanctioned to perform is division ... and remember 9 has to be the dividend."

So, the maximum our hero can achieve is 0.09 (dividing 9 twice by 10 or 100). The max after that is 0.009. We add = 0.9 + 0.09 + 0.009 = 0.999. What's been accomplished? 1 - 0.9 = 0.1 and 1 - 0.09 = 0.01 and 1 - 0.999 = 0.001. We're getting closer. Keep going and we have 0.9999 The difference gets smaller and smaller: 0.0001, then 0.00001, then 0.000001, so on ... ad infinitum, ad nauseum.

Don't look at the 9 in 0.999..., look at the difference, 0.1 and then 0.01 and then 0.001 and then 0.0001, so on and so forth. The difference tends to 0, so 0.999... tends to 1? Am I correct?

 

[Dr.Peterson]

It's not the segment that I want to focus in on, I am just interested in the number 0.999 ... itself. (This is why I tried to use only 0.999 ... in the OP, but that didn't work)

I don't want to talk about limits because I understand what the limit would say about this. I am interested in exploring what the union produces. Can the union [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... produce the interval [0, 0.999 ...]?

The question about intervals has been fully answered, but the question about the number may need more.

Students often find it difficult to believe that 0.999... = 1, and each of them may need a different explanation to understand it. Here is a collection of answers to the question from my old site, Ask Dr. Math:

Frequently Questioned Answers: 0.999… = 1 – The Math Doctors

www.themathdoctors.org

Perhaps one of these will be of use?

 

[Mates]

Thanks for all of these answers. The more I think about 0.999 ... the more it makes sense that it equals 1.

 

[Mates]

We use to spend a great deal of time on this topic in our foundations course.
For example: $\displaystyle\bigcup\limits_{k = 1}^5 {\left[ {0,1 - \frac{1}{{{{10}^k}}}} \right]}=[0,.9]\cup[0,.99]\cup[0,.999]\cup[0,.9999]\cup[0,.99999]=[0,.99999]$
So that we get $\displaystyle\bigcup\limits_{k = 1}^{\infty} {\left[ {0,1 - \frac{1}{{{{10}^k}}}} \right]}=\left[0,0.\overline{\,9~}\right)$ Note that this is a half-open interval (one not included).

$$$$$$

I was just thinking back to this post. Why isn't 0.999... included? Doesn't 1/10^k equal 0 when k equals infinity?

 

[Dr.Peterson]

I was just thinking back to this post. Why isn't 0.999... included? Doesn't 1/10^k equal 0 when k equals infinity?

These infinite unions do not include k = infinity! The notation means "take the union over all integers greater than 1"; infinity is not an integer. The symbol is just a placeholder, which means "don't stop".

This is why we have the concept of limits. We can't treat infinity as a number; when we talk about it (in this sort of context), it is only as shorthand for a limit.

 

[stapel]

...infinity is not an integer. The symbol is just a placeholder, which means "don't stop".

Nice description! I may steal this....

 

[Mates]

These infinite unions do not include k = infinity! The notation means "take the union over all integers greater than 1"; infinity is not an integer. The symbol is just a placeholder, which means "don't stop".

This is why we have the concept of limits. We can't treat infinity as a number; when we talk about it (in this sort of context), it is only as shorthand for a limit.

Oh I see. But can't we exhaust all k of the naturals?

 

[Dr.Peterson]

Oh I see. But can't we exhaust all k of the naturals?

What is k? There are infinitely many natural numbers; that means they go beyond any (finite) number k. In counting, you never exhaust the natural numbers.

Any k in a summation, product, union, ..., is a finite number. "Finite" means "you can get there by counting". "Infinite" just means "not finite", that is, "unending" or "never finished". It isn't something you can ever get to.

 

[Mates]

What is k? There are infinitely many natural numbers; that means they go beyond any (finite) number k. In counting, you never exhaust the natural numbers.

Any k in a summation, product, union, ..., is a finite number. "Finite" means "you can get there by counting". "Infinite" just means "not finite", that is, "unending" or "never finished". It isn't something you can ever get to.

I thought that they all just exist. I read that we can map all natural numbers into some function. I am quite confused with all of this.

 

[Dr.Peterson]

I thought that they all just exist. I read that we can map all natural numbers into some function. I am quite confused with all of this.

I have no idea what this means. How is it related to what I said?

 

[Mates]

I have no idea what this means. How is it related to what I said?

If we map all infinite natural numbers into the union function, wouldn't we get an infinite number of 9's?

 

[blamocur]

I am quite confused with all of this.

This might be your only sentence in this thread that I understand and agree with

It looks to me that you have your own picture of math in your head, and that picture does not match any actual math theories I am familiar with. Instead of trying to figure out why your picture of math is inconsistent (as demonstrated by this unnecessarily long thread) you might consider studying actual math theories, like Real Analysis and axioms of real numbers, if only to make our conversations more productive.

 

[Dr.Peterson]

If we map all infinite natural numbers into the union function, wouldn't we get an infinite number of 9's?

This is utter nonsense. I give up.

 

[Mates]

This might be your only sentence in this thread that I understand and agree with

It looks to me that you have your own picture of math in your head, and that picture does not match any actual math theories I am familiar with. Instead of trying to figure out why your picture of math is inconsistent (as demonstrated by this unnecessarily long thread) you might consider studying actual math theories, like Real Analysis and axioms of real numbers, if only to make our conversations more productive.

I am not very good at studying on my own. In fact I tried for many years studying on my own and I didn't absorb very much like I did in classes at university. I even asked professors to tell me which textbooks they were using in their courses. I come to websites like this one because I don't know how else to resolve my queries.

I thought that when we sum something to infinity, we end up with an infinite number of summands added together. For example, the infinite sum from 1 to infinity equals infinity doesn't it? Or if we sum 1/2^n from 1 to infinity we get 1.

Aren't we "exhausting" all n of the natural numbers when we have from 1 to infinity?

 

[blamocur]

I am not very good at studying on my own.

I am with you there. I myself spent way too much time stuck on various textbooks without benefit of intelligent help.

I come to websites like this one because I don't know how else to resolve my queries.

This type of query is not really resolvable. I saw questions like this before, a typical one being "Why cannot I do this?". A productive discussion would be "I am trying to understand this theorem, but am stuck here.". As has been mentioned earlier (I believe by @stapel) that this forum is no substitute -- but can be a useful complement -- for a regular class. We simply don't have the bandwidth to teach, only to help with learning.

I thought that when we sum something to infinity, we end up with an infinite number of summands added together.

This is a typical misunderstanding. There are only finite sums, while "infinite sums" are defined through limits.

For example, the infinite sum from 1 to infinity equals infinity doesn't it?

I more correct statement would be that there is no (finite) limit. When we say that something "sums up to infinity" we mean that the some will grow beyond any (finite) number. But before using informal phrases like "sums up to infinity" or "equals infinity" it is important to understand that they actually stand for something defined formally. You are much less likely to come to wrong conclusions if you first reformulate your hypothesis in formal terms, then see if it checks out using formal axioms and theorems. BTW, good textbooks often have counter-examples to hypotheses which seem intuitively right but are actually wrong.

I wish you success in your studies, and we will be happy to help if/when you run into more specific, theory-founded question.

P.S. I don't want to add to your confusion, but felt necessary to mention that unlike "infinite sums" there is a formal definition for infinite unions, just like the one you started this thread with.

 

[Mates]

I am with you there. I myself spent way too much time stuck on various textbooks without benefit of intelligent help.



This type of query is not really resolvable. I saw questions like this before, a typical one being "Why cannot I do this?". A productive discussion would be "I am trying to understand this theorem, but am stuck here.". As has been mentioned earlier (I believe by @stapel) that this forum is no substitute -- but can be a useful complement -- for a regular class. We simply don't have the bandwidth to teach, only to help with learning.


This is a typical misunderstanding. There are only finite sums, while "infinite sums" are defined through limits.


I more correct statement would be that there is no (finite) limit. When we say that something "sums up to infinity" we mean that the some will grow beyond any (finite) number. But before using informal phrases like "sums up to infinity" or "equals infinity" it is important to understand that they actually stand for something defined formally. You are much less likely to come to wrong conclusions if you first reformulate your hypothesis in formal terms, then see if it checks out using formal axioms and theorems. BTW, good textbooks often have counter-examples to hypotheses which seem intuitively right but are actually wrong.

I wish you success in your studies, and we will be happy to help if/when you run into more specific, theory-founded question.

P.S. I don't want to add to your confusion, but felt necessary to mention that unlike "infinite sums" there is a formal definition for infinite unions, just like the one you started this thread with.

Thank you very much

 

[blamocur]

BTW, good textbooks often have counter-examples to hypotheses which seem intuitively right but are actually wrong.

I tried to come up with such counter-example, but managed to remember one only now. Consider these two "infinite sums":
$$S_1 = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ...$$$$S_2= \frac{1}{1} + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6}...$$Both infinite sums have the same (infinite) set of summands (i.e. $\frac{+1}{2k+1}$ and $\frac{-1}{2k+2}$ for $k=0,1,2,3,...$), only their orders are different. For "normal", a.k.a. finite, sums you would expect them to be equal. But $S_1 \approx 0.69315$ and $S_2 \approx 1.03972$.

This example illustrates the danger of approaching "inifinite sums" without formal definitions and axioms. BTW, I believe the proper term for "infinite sums" is "series", probably to underscore that they are not "normal" sums.

 

[Mates]

I tried to come up with such counter-example, but managed to remember one only now. Consider these two "infinite sums":
$$S_1 = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ...$$$$S_2= \frac{1}{1} + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6}...$$Both infinite sums have the same (infinite) set of summands (i.e. $\frac{+1}{2k+1}$ and $\frac{-1}{2k+2}$ for $k=0,1,2,3,...$), only their orders are different. For "normal", a.k.a. finite, sums you would expect them to be equal. But $S_1 \approx 0.69315$ and $S_2 \approx 1.03972$.

This example illustrates the danger of approaching "inifinite sums" without formal definitions and axioms. BTW, I believe the proper term for "infinite sums" is "series", probably to underscore that they are not "normal" sums.

This is very interesting! This is why I am obsessed with infinity. It is so strange.