In regards to 1/3 necessarily having no way of being expressed as a finite sum of rationals, that depends on the base of the number system one is using. Suppose our number system was base 3. That means our number line would look like this:
0 1 2 10 11 12 20 21 22 100 101 …
Here’s what counting from 0 to 1 would look like if went by increments of two decimal places:
0.00 0.01 0.02 0.10 0.11 0.12 0.20 0.21 0.22 1.00
Note the part in bold. This is exactly one third the way to 1.00. Note that it does not require an infinite decimal expansion. It’s just:
0.1
The quantity hasn’t changed. It still represents a third of whatever. The only thing that’s changed is the notation. We use a different notation to represent one third because we are using a different base.
^ This shows, I would hope, that the problem of an infinite decimal expansion, and therefore the problem of an infinite sum of rationals, is a superficial problem having to do only with the notation system in use. Use a different notation system and the problem goes away. It’s not a problem with the quantity itself. That remains the same regardless of the notation system being used.
So a third is not somehow outside the categories of “finite” and “infinite” (what I dubbed “semi-finite” on Magnus’s behalf), and it is not an irrational number, and it is not something that can never be complete, never quite attain it’s limit, it’s just, well, a third.
Now unfortunately, it gets a bit more complicated in the case of (0.\dot9), and I’m not going to tackle that in this post, but I hope the above sheds some light on the difference between the notation and what the notation represents, and how sometimes the problem is only a problem for notation and not for the quantity the notation represents.
Phyllo, did it ever occur to you that people’s intuition that finite numbers are not “infinite digits” (the most blatant contradiction in this whole discussion) means that there’s something wrong with how humans conceive of and understand math?
I mean, if you can really stand there and say “well it makes NO sense but it’s true”
What’s to stop me from saying “well it makes no sense but it’s NOT true”?
I mean, if you’re going with a contradiction, be prepared to have all the nonsense in the world thrown at you as well.
Then find the flaws in the math instead of proposing some bizarre definitions of your own. Start by writing down the definition of “INFINITE number”.
My own solution to the 0.999…=1 question does not depend on infinities at all. I didn’t use an infinite series. I didn’t use infinite sets. I used strictly multiplication and division and alternatively addition and division.
We cannot put in anything for (n). There is an explicit condition: (n) must be greater than (0). It does not have to be something that satisfies Gib’s narrow definition of the word “quantity”, but at the same time, it cannot be something that cannot be said to be greater than (0). If we cannot say that (cow > 0) then we cannot say (\sum_{i=1}^{cow} \frac{9}{10^i} < 1). Fortunately for us, we all agree that (\infty > 0), so we can use it for (n).
Your belief that using the word “infinity” in mathematical equations is the same as using the word “cow” is a mistaken one.
And in order for me to do so, you have to tell me what you think is a necessary and sufficient condition for something to be considered a valid value for (n). And you may also need to explain why.
I think that (n > 0) is a sufficient condition. You obviously don’t. Why? You keep mentioning the word “quantity” without ever bothering to explain what you mean by it and why it’s necessary for (n) to fit your definition of the word “quantity”.
No. I’d ask you what you mean by the word “cow” such that you can say (cow > 0).
It’s not the same onus. We all know that (\infty > 0).
It should tell you that you’re not doing a good job at explaining what needs to be done (and why) in order for you to accept the position that I’m putting forward.
That’s an extremely narrow view. Basically, you’re limiting yourself to what is widely-accepted (or at the very least, to what is familiar to you.)
What makes you think you cannot place (\infty) on the number line? The fact that it is not a widely accepted position?
There is a class of people known as parrots. Parrots are people who adopt other people’s conclusions without knowing how to arrive at them on their own. Since they are not independent thinkers, they can never accept a belief unless their trusted authority approves of it.
How do you convince a parrot? By convincing their trusted authority.
If this is the only way that I can convince you, then we have nothing to discuss.
(\frac{1}{9} + \frac{1}{9} + \frac{1}{9}) is a finite sum of rationals that is equal to (\frac{1}{3}).
And it’s a well-known fact that there is a base-3 representation of (\frac{1}{3}) (which is, as you say, (0.1)). This has been discussed in this very thread (60) pages ago (or more than (3) years ago, if you will.) This is not what’s being disputed.
What’s being disputed is the claim that (\frac{1}{3}), or (0.1) in ternary numeral system, is the same number as (0.\dot3).
You’re not quite getting what (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for n > 0 means. In the case of sums, valid values for n are typically integers greater or equal to whatever initial value is set for i. In other words, n must be an integer, and integers are only found on the number line.
You got this part right:
(\infty) > 0
But you’re making an implicit assumption:
(\infty) has a specific place on the number line.
^ This you got wrong.
(\infty) is not among the values you have at your disposal to choose from for n. IOW, the condition that n > 0 does not just mean “anything greater than 0”, it also means values that fit with sums of the form (\sum_{i=x}^{n} f(i)) (that is, integers, those symbols above the tick marks on the number line).
(\infty) > 0 remains true because > and < don’t signify specific points on the number line. They signify directions on the number line. “Greater than” means: in the positive direction. “Less than” means: in the negative direction. (\infty) need not have a place on the number line for (\infty) > 0 to be true. It just has to be to the right of 0.
Now we’re cooking! You’re finally asking the right questions.
I believe my explanation above should suffice. Valid values for n are any integer found on the number line greater that 0 (at least in the case where i = 1). Why? Because that’s what the condition n > 0 is asking for. (\infty) is not an integer found on the number line. Why? Because it is the endlessness of the number line itself, at least in the positive direction. In other words, (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a special case of (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for any n > 0. It’s an entirely different statement. It says: don’t select a value for n. Just let the sum continue forever.
^ This may not satisfy your request for further clarification on what I need from you, but now your move is to probe deeper into my response. Is there something in the above you need even further clarification on? (wanna get into hyperreals? )
Again, I think the above addresses this. The important point is that (\infty) fails my definition of quantity. A quantity is what points on the number line represent (at least in terms of cardinality). Since (\infty) has no point on the number line, it doesn’t represent a quantity.
You say tomato, I say tomato. (<-- That doesn’t quite come through in text, does it? ).
It doesn’t matter what “we all know”. I’m not a part of “we”. You’re dealing with me in particular. If you want to convince me of your point, the onus falls on you to prove your point to me.
You mean like, oh I don’t know, asking you to prove that what applies to finite sets also applies to infinite sets? How 'bout proving that (\infty) is a valid value for n? Did I forget to ask for that?
It takes two to tango. I very well may not have been detailed enough in explaining to you what I need, but that’s your cue to ask for further clarification. These things may go through a few rounds of back and forth before both parties are clear on what the other needs in order to be convinced. Nothing wrong with that. It’s how it goes sometimes.
You know, this is a desperate cry. You typically see it when your opponent has nothing left. It really means: I’ve got nothing but my going against the grain. I know that I’m taking an unpopular position. I know that all the professionals in the field, all the experts, all the really smart people disagree with me. I know the majority of people who follow this topic are not on my side. I know all my arguments are falling on deaf ears, that my attempts at convincing others seems to be an exercise in futility… but you know what, at least I’m not a sheep, at least I’m not mindlessly conforming to the masses. I’m thinking for myself, I’m exercising my independence of thought. And that’s something to be proud of. I’ve exhausted everything else in the debate, might as well start using this.
For good measure, I will object and say that we can equally well say cow is larger than zero as that we can say infinity is larger than zero; neither are quantities. And yet both clearly more than nothing.
Infinity is not a quantity, because quantities are definite.
You’d have to explain why you’re limiting yourself to integers.
Not necessarily but it can be the case.
Basically, what you’re saying is that (\infty) does not refer to a specific number in the same way that the statement “An integer greater than (10)” does not refer to a specific integer since the quantity that it represents can be (11) or (12) or (100). However, you have yet to explain why it’s invalid for (n) (the upper bound of summation) to be a non-specific number.
My claim is that (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for any (n) that is greater than (0). It does not have to be an integer. The only condition is that it’s greater than (0). Do you think there is (n) greater than (0) such that the sum is not less than (1)? You obviously do (otherwise, you’d not say that (0.\dot9 = 1).)
(\infty) is not an integer. It’s a number greater than every integer.
And (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a completely different statement. The only difference between (\infty) and integers is 1) (\infty) is a bigger number and 2) it’s non-specific. The fact that it’s non-specific shouldn’t be a problem at all. And if you absolutely hate non-specific numbers, you can switch to some specific infinity.
(0.111… )*9=0.999… Equation 2 (ordinary multiplication by 9)
(1/9)*9=1 Equation 3 (ordinary division and multiplication)
Substitute for (1/9) in Equation 3 with the equivalent (1/9) from Equation 1 :
(0.111…)*9=1 Equation 4
The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.
I’m writing this post for Carleas to link to (not because I’m reporting anyone here - that’s absurd!) but because I know he loves numbers and has participated . I love numbers too! Any of you can take a stab at it:
I have two number theory positions:
1.) series don’t converge
2.) there are no orders of infinity
1.) series don’t converge! If a series converges, this means that every counting number is an infinite sequence (proof through contradiction).
2.) there are no orders of infinity!
I use this order symbolically (not using numbers) to disprove this!
A.) rational number
B.) uncounted infinity
C.) different rational number
D.) different uncounted infinity
Etc…
This places all the “lower” infinities in 1:1 correspondence with the “highest order of infinity”, thus there are no orders of infinity.
Abstract concepts in mathematics not using numbers are fair game!
Thus my proof sits.
Magnus, for you personally, if you don’t understand that, I’m not crazy, and it’s not my fault.
I know Carleas doesn’t have the time for this thread, but I know he’s a math pervert (we all should know that) and I don’t think he’s seen these arguments before!
)