1/3=0.333…
3*(1/3)=1
3*(0.333…)=0.999…
Therefore 1=0.999…
If that’s not true, then simple division and multiplication don’t work.
You can even take out the multiplication: 1/3+1/3+1/3= 0.333… + 0.333… + 0.333… = 0.999… = 1
It’s not rocket science.
Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)
Let’s say we hypothetically live in a world where fractions don’t exist… it would be unfathomable that 0.999… = 1.
In a decimal world. 0.111… * 9 equaling 1 is impossible.
The problem is not with my logic, the problem is how operators work with 1 minus base, to make them ‘appear’ equal… but then again, they don’t appear equal at all do they !!!
There seems to be a lot of confusion here about what “infinite” means. “All finite numbers are infinite” doesn’t make any sense as a statement.
That’s simple enough. All you need to do is to restrict yourself to whole numbers, natural numbers or integers.
There is no integer which represents 1/3 or 1/2 or 8/9. When evaluated, those fractions are equal to :
1/3=0 in integer
1/2=0 or 1/2=1 in integer
8/9=0 or 8/9=1 in integer
The two values given for 1/2 and 8/9 depend on whether truncation or rounding(up/down) is the standard procedure when evaluating the results.
A similar thing happens with 0.999… when using real numbers. It “jumps” up to 1.
Yes but in as far as it pertains to quantity, its pertaining makes quantity indefinite.
We agree on that.
Thanks gib.
Yes and I have noticed during the years that the main problem when discussion infinity is its indefinite-ness.
This is the hot core all these debates centre around and it not being made explicit perpetuates the confusion.
How are you not going to be confused if you want definite results out of something that is by definition indefinite?
That seems legit.
Phyllo,
I’ll press you on this for now.
Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”
EXACTLY!!!
That’s my whole point. It makes no sense!
It makes no sense that 0.999… EQUALS 1!!!
In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Who the heck knows what you mean by “INFINITE number”. I certainly don’t.
Infinite digits doesn’t mean infinite number.
Doesn’t matter how you word it… you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! To make this rated g, the pebble never hurts the squirrel
You’re still making the claim that 1 is …
EQUAL!!!
To “infinite digits”
EQUAL!!!
Sure. It looks counter-intuitive but the math equations show that it must be true.
What’s wrong with the argument using 1/3 fractions? Nothing. Unless you want to argue that dividing 1 by 3 doesn’t work.
I am not saying “This is what infinity is”. I am saying “If this is what infinity means, then this is what follows”. My point being that if (\infty) refers to a specific number that is greater than every integer, then (\infty - \infty = 0) is true but (\infty + 1 = \infty) is not; and if it refers to a non-specific number that is greater than every integer then (\infty + 1 = \infty) is true but (\infty - \infty = 0) is not.
Yes, but in such a case, (\infty + 1 = \infty) is not true.
$$ e^{i \pi}=-1$$
Go figure 
Is 0.999…=1 so strange in comparison?
If (\infty) refers to a non-specific number greater than every integer (in the same way that “A number greater than (3)” refers to a non-specific number since it can be (4), (5), (100) or (1000)), then (\infty + 1) equals to (\infty) (because “A number greater than every integer + 1 = a number greater than every integer”) but (\infty - \infty) is indeterminate.
Could you make up your mind what it means instead of using multiple meanings?
Another question:
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.
Not really. The notion of time is completely irrelevant to this thread.
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.
Oh sure… Magnus is a 23105th dimensional being
Magnus, in this thread, I wonder if you even begin to hear yourself talk sometimes.
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.
You can say anything that you like.
That has already happened in this thread.
Phyllo, you’re being disingenuous.
You know that when you divide 1 into decimals by 3 or 9 and then multiply the decimal by 3 or 9 that it NEVER equals 1!!
I know you know this.
No. That’s exactly what I don’t know because I think that I can do it.
Now, explain to me why I am wrong.
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.