There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p
I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“.
If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it.
What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way).
Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery.
[0] http://www.sophiaperennis.com/books/rene-guenon-series/the-m...
Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...
You're demonstrating why so many people get confused by the 0.999... notation: they (and you) are thinking of it as the sequence (0.9, 0.99, 0.999, ...) ; but actually it doesn't represent this sequence, or a process for producing this sequence; rather, it represents the limit of this sequence, which is a different type of object: a real number.
> Mathematicians are using the symbol … ambiguously
Sure, it can mean other things in other contexts (like in my description above of a sequence), but we're talking about what it means in the case 0.j_1 j_2 j_3 ... where the j_k's are base 10 digits. In that case it has a well-defined, non-ambiguous meaning: \lim_{n\to\infty}\sum_{k=1}^n \frac{j_k}{10^k} .
> If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it.
OP is right. I have a degree in math and this is what mathematicians mean by 0.999...
> What I'm saying is that the expression could be taken to be one of any number of things
Yeah we could have had the notation mean something else. But we didn't. We have other notations for the things you're talking about. Who cares?
> That you say the convention is that it most assuredly is 4 is not something I'm buying into.
What would it take to make you believe that this is what mathematicians mean by this notation? I promise with all my heart that this is true.
Don't take my word for it. Here's some random analysis book. Flip to page 31 (51/686 in the numbering of the PDF): http://www.matematica.net/portal/e-books/Apostol%20-%20CALCU...
> (I'd nearly go a step further and assert that it ought not be defined this way).
Plenty of stuff in math has pedagogically dubious notation, it's not really novel or philosophically interesting to claim you dislike some piece of notation or think it could have been better. It's all just convention.
> Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1.
No mathematician is saying that. The sequence (0.9, 0.99, 0.999...) quite clearly never reaches 1, and any mathematician will agree with that. But its limit is 1.
> Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist.
Well, things don't have to "exist" for us to be able to study them in mathematics, so what's the point of claims like this?
This notation is defined as meaning the limit as n approaches infinity of the sum from k=1 to n of 9/10^k . A sequence doesn't have to reach its limit!! For example, the limit of the sequence 1/2, 1/4, 1/8, ... is 0, even though the sequence never reaches 0. The limit of the sequence 0.9, 0.99, 0.999, ... is 1 (and this is BY DEFINITION what 0.999... means).
Or, you can decide to give some arbitrary meaning to all that stuff, and get some incredibly useful results. It's your choice.
Yes, there is. Ask yourself: What's the decimal expansion of 1/3, or pi?
From your examples -- one's a ratio and the other is irrational. Almost as if you suggest "The set of real numbers exists, ergo 0.999999... exists"?
Well, it's technically a claim, but one that is obviously true once we agree on what 0.999... means. Anyone who doesn't agree that 0.999... = 1 has a different definition of 0.999... from the one used in mainstream math notation.
Then, given that 0.99999... exists, what useful properties could it possibly have, if it is to be treated as a number.
0.99999... = 0.9999... * 1 = 0.99999... * 9/9 = 0.99999... * (10 - 1)/9 = (0.99999... * 10 - 0.99999... * 1)/9 = (9.99999... - 0.99999...)/9 = 9.00000.../9 = 9/9 = 1
So if 0.99999... exists and has the same arithmetic properties as decimal expansions of rational numbers, its value must be 1. Now whether you think it should have these properties is an entirely different question, but most mathematicians seem to like it this way.
And the claim is correct. It is the statement that the limit of the partial sums of the inverse powers of 9 is 1, which can be rigorously proven with the epsilon-delta definition of a limit.
For someone who doesn't understand what 0.999... means, why should we expect them to understand what 0.333... means and believe that it exists?
Do you really know "why" 0.99.. = 1 by writing 0.99... = 3 * (0.33...) = 3 * 1/3 = 1 ?
I'd argue that you just kicked the can down the road.
If you happen to somehow built a decimal expansion of 1/3, please share it with us!
This is incorrect. A decimal expansion is just a type of Cauchy sequence, which by definition is infinite (a function from the natural numbers to the reals).
You are confusing the decimal expansion with its truncated approximations.
> If you happen to somehow built a decimal expansion of 1/3, please share it with us!
Sure thing! Here it is:
def oneThird(place): if place >= 0: return 0 else: return 3
Convert to your favorite programming language/Turing machine/other abstract machine of choice.
def oneThird(place):
if place >= 0:
return 0
else:
return 3
def printOneThird():
for x in iter(int, 1):
print(oneThird(-x))
It seems you don't understand what a sequence is. Formally, a sequence is a function whose domain is the natural numbers (or integers). A decimal expansion is simply a sequence whose codomain is the set of digits {0,1,2,3,4,5,6,7,8,9}. I gave you just such a sequence.
Are you really going to continue disputing a basic mathematical fact?