Seems only Fluid Mechanics are in short supply.
Yeah, pretty much.
Sure. You write the decimal point and then 9s until you’re sick of it (you can be more precise if you just hold down the 9 key). Then you look at it and think some and you come to the realization that if that’s not pretty much 1, it might as well be.
Or, having once been a wiseass kid, another effective proof I found is for the teacher to dopeslap me and say, “It just is, so shut up.”
mandrake, wherever you are, I apologize for not posting this reply when you asked. I was new and deferred to my betters, people actually knowledgeable about the topic. Had I provided my succinct and nearly painless proofs then you might still be here. 
You should reinstate this policy.
Seconded. There are already several proper, correct proofs that 0.999… = 1 in this thread. We don’t another incorrect one or some vague statement that 0.999… is “pretty much” 1 or that “if it’s not pretty much 1, it might as well be.” It’s one thing to make an incorrect statement if you don’t know any better; it’s quite another to do so after twenty-nine pages.
In the context of a 14 year old thread that runs to 29 pages, this is hilarious.
Actually, I think this thread is as interesting in the way that it has progressed at least as much as the mathematical content that it has raised. Is there any other zombie thread that has been resurrected as many times as this one?
In fairness, there were only 15 posts when the thread originated in 2000; the 16th post was in 2012, and could just as well have been the start of a new thread rather than a resurrection of the old one. So, while it sounds much less impressive, I think this thread might more accurately be considered two years old (next Sunday) rather than 14 years old (on Wednesday).
I like your succinct and nearly painless proofs. Should we be satisfied that 1=0.999…
because the teacher tells it just is so shut up. Did you ever went to or heard about a shut up and calculate school?
mandrake may not be anymore here, and many others either, but I have demonstrably
proven them wrong who say that 1=0.999999…
My proof is not based on an empirical evidence, I gave up writing a decimal point and then 9s until I was sick of it or holding down the 9 key and then
thinking that it is pretty much 1, or if it isn’t it might as well be, at least it should be because they tell so.
I don’t just say that it is because it is. I don’t prove it true by assuming it is true.
No, I provided a decent proof as mandrake wanted, what I have shown is that
1/3≈0.3333333…
Why is every digit the same as with
1/3=0.3333333…?
Tell me, if every digit is the same what does it mean, should you again continue writing the 3s until you are sick of it, this time you will reach the inaccessible, infinity,
you will manage to write down an infinite amount of numbers. Having achieved what
no-one did before, having solved this old problem, can you give an answer to a new question: what is the smallest amount? That is a sensible question now that you
have reached a huge amount, infinite amount of something, is it perhaps the largest
amount? If you have a largest amount of something, clearly you know the smallest
amount of that something.
Maybe we should have a collation of the correct proofs from this thread repeated in the one post for future reference – beginning with post 36.
As for 7777777, all I can say is <sigh>. I don’t think you’ve got it.
Something being approximately equal to, and equal to, are not exclusive. If you want to argue that that they are, you need to provide a rigorous definition of “approximately equal” and use it. Otherwise it is just a word game, and not mathematics.
Typically, if x ≈ y it means that
|x - y| < epsilon, where epsilon is some usefully defined small value. If epsilon equals zero, then x and y are also equal. ≈ does not require that epsilon is not equal to zero, unless you are using a non-standard definition, and if you do, you will need to show your proof for any non-zero epsilon.
Here, for 1/3 ≈ 0.3333… we will point out that epsilon can be an infinitesimal that has no value in the real numbers other than zero. So the proof is actually that 1/3 = 0.33333… rather than the opposite.
I understand what you are trying to say. If the infinitesimal has no value other than
zero then 1/3=0.33333…
The problem is, there is the word “if”.
According to what I have said, the infinitesimal is never equal to zero, it only
approaches zero, approaching zero but never getting to zero.
Add 1 to …99999 and you will get to zero. Because
…99999 + 1 = 0
Add 0.99999…to …99999 and you never get to zero. Because
0.99999…+…99999 = …99999.99999… ≠ 0
I know , someone insist that …99999.99999…=0 but without providing
the context where these two numbers could be the same although every
digit is different. Just saying that 1=0.99999… and that it what makes
…99999.99999…=0 is not enough now because
we are just proving it either true or false. So saying it must be true because
it is true ignores the fact that it might as well be false.
…How is …999 defined? Is it an infinite string of 9s? The infinite row sum(9, 99, 999, …)? Is it just 9999999999999999999999999999999999999999? Because in none of those cases is what you just said true.
No, nobody has ever insisted that, because the step before that is fundamentally broken.
Here’s another one - can you name me the difference between 0.9999… and 1? Like, if two real numbers a and b are different, then in this equation:
| a - b | > e; e element of Real Numbers
e will, by definition of the real numbers, always be larger than 0. So can you find e? It’s not 0.01. It’s not 0.0001. It’s not 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001.
What is it?
Exactly. Now understand that in the real numbers as normally defined, the if has been proven to be so.
Both of these are vacuous statements, you simply repeat yourself : A because A - they are not even an attempt at a proof.
No, we provide a context where, in the conventional definition of the Real numbers, this number does not exist. It is not a Real number. Be very clear, when we say the Real numbers, we are not talking about some sort of philosophically true numbers, we are talking about a tightly defined set of numbers that adhere to a rigorous set of axioms. Some might argue that there was a level of hubris in calling these numbers the real numbers, as opposed to numbers such as the Integers, Rationals, Imaginary, Complex, or others. But there it is. They are called the Reals, and they are what most people use in day to day work. Engineers and physicists will spend a lot of time with the Complex numbers, and these have some rather more interesting properties, and can have a very different take on the nature of infinity.
But be clear, decimal notation has a very clear and precise definition in the Reals, one that excludes …9999 having any meaning. It defines a divergent series, in a precise and unambiguous manner. To insist it denotes a member of the Reals makes as much sense as demanding that for any x that is a member of the Reals, x/0 is a member of the Reals. Once you allow something like this, you can prove any number of contradictions. This is how you show that propositions are false. If the proposition leads to a contradiction in your chosen system, that proposition is not true.
You can choose modified systems, and Indistinguishable has outlined quite a few, where you can make notations mean other things, or certain assumptions can be made to be true, and within these things like …9999 can be made to have meaning. But …9999 does not have meaning in the Reals. Continued insistence that it does misses the point. The Reals have clear, well defined, and rigorous axioms, and these exclude …9999, and also have a clear result that there are no infinitesimals that have a value other than zero. Whatever system you are trying to work in is either the Reals, or your own different system, and you need to be clear which it is.
worst asciii art ever!
Quibble: I understand what you mean, but I’d say you’ve phrased it inaccurately.
We must continue to distinguish notation from what that notation represents (that being, in some sense, the main idea of this entire thread). What I think you should have said was this: “…9999” does not have meaning in standard decimal notation. Note that that’s different from saying “…9999” does not have meaning in the Reals. “The Reals” comprise a particular mathematical structure we can consider independently of any system for notating its elements.
Supposing you did have a system in mind for interpreting decimal notation which allowed infinite leftwards sequences such as “…9999”, depending on what that system was, it might well give this meaning in the Reals. (For example, this interpretation would). It’s all up to the notation system, which is not determined by fixing the mathematical structure of the Reals themselves.
Anyway, this is all mere phrasing tweaks, but I’m clearly deeply pedantic about it.
Well, the context is really just the real numbers (for the claim that 0.9999~ = 1, that is). You seem to get hung up on the fact that one and the same number can be written in two different ways, but this is not really anything extraordinary.
Consider the number 5—the abstract object, i.e. ‘5 itself’, rather than the numeral. I can write it, in decimal, as ‘5’, in binary, as ‘101’, in ternary, as ‘12’, and so on. These obviously have all different digits, but nevertheless refer to the same abstract object.
Or consider yet other ways of writing 5—3 + 2, for example, or 4 + 1. Now, actually, these two just correspond to the above examples: 3 + 2 = 13[sup]1[/sup] + 23[sup]0[/sup], or 4 + 1 = 12[sup]2[/sup] + 02[sup]1[/sup] + 12[sup]0[/sup]. Thus, the notation ‘12’ in ternary for ‘5’ is just a shorthand for the sum 13[sup]1[/sup] + 23[sup]0[/sup] = 3 + 2 = 5, and the notation ‘101’ in binary is just shorthand for 12[sup]2[/sup] + 02[sup]1[/sup] + 12[sup]0[/sup] = 4 + 1 = 5. The digits you write down are just the coefficients of the powers of the base in which you represent the number. In fact, ‘5’ is just shorthand for 510[sup]0[/sup], since we’re using the decimal system. So in all its glory, we should really say 12[sup]2[/sup] + 02[sup]1[/sup] + 12[sup]0[/sup] = 5*10[sup]0[/sup]. Thus, the two different sets of digits ‘101’ and ‘5’ correspond in a well-defined way to the same number—they’re just different names for the same thing, if you will.
Now, the connection to infinite decimal expansions comes from realizing that you need not limit yourself to finite sums. Certain kinds of infinite sums have, under the standard understanding of real numbers, a well-defined value, so that you can use them just as well to write a given number. The sum 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + …, for instance, is equal to π/4, and thus, can be used to represent that number, even though, if you were to write it as a digit expansion, all the digits would be different from ‘0.78539816…’. So, two numbers can be the same, even though all the digits differ, because digits are really just a naming convention, and different names may refer to the same object—it’s ‘the sun’ in English, ‘le soleil’ in French, ‘die Sonne’ in German, but you wouldn’t argue that ‘the sun’ can’t be ‘die Sonne’, because the letters differ.
So all one has to do is to figure out if one can ‘translate’ between one way of writing a number and another. In this respect, the digit expansion ‘0.9999~’ is, as above, just shorthand for 91/10 + 91/100 + 91/1000 + 91/10000 + … If this infinite sum now converges, then the value it converges to is the value represented by ‘0.9999~’. As I show above, it does converge, and the value it converges to is 1. Thus, ‘0.9999~’ and ‘1’ are just two different names for the same object—that ‘the digits differ’ is as much a red herring for arguing that the numbers represented by those digits differ as saying ‘the letters differ’ is an argument that ‘the sun’ isn’t ‘le soleil’.
No doubt 7777777 would object to the “equal to”. And, that makes me want to ask him/her: what do you think of such infinite series? Like the given expression for π/4, or, for example, e = 1/0! + 1/1! + 1/2! + 1/3! . . . (or, for that matter of e = 2.71828 . . .)? Is this use of the equal signs legit in such expressions?
I’m still hoping for a response to the question I asked 3 pages ago. What’s wrong with this proof?
- x = 0.333…
- 10x = 3.333…
- 3 + x = 3.333…
- 10x = 3 + x
- 9x = 3
- x = 3/9
- x = 1/3
- 0.333… = 1/3
Possibly, but I chose my example with some care: while one might think it reasonable that 0.9 + 0.09 + 0.009 + 0.0009 + … is always an ‘infinitesimal’ bit smaller than 1, or your example of the series expansion of ‘e’ is likewise always a bit smaller, that out isn’t there anymore for the series expansion of π/4 I posted: at different points in the series, it will be both larger and smaller than π/4, so it neither can always remain ‘a little bit below’, nor ‘a little bit above’; but if it can neither be larger nor smaller, then it must be equal (if it is to have a definite value at all, that is—which is of course guaranteed by the fact that it’s a convergent sum).
7777777 obviously does not believe that convergent sums have definite values unless the sum is 9 + 90 + 900 + …
ETA. of course that sum is not convergent so there’s no reason to qualify anything.
Wait, what do you mean by this? What digits would you get instead if you wrote 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + … as a digit expansion?