The Epsilon-Delta proof was the one thing that I learned in Calculus that I immediately thought was useless. Everything else seemed useful or interesting except that. I see now that it’s really necessary for establishing the basis of many things that Calculus shows or solves.
So, thank you for finally resolving this 30+ year conundrum for me! You found a use for Epsilon-Delta – it’s needed for describing the very basics of limits to someone who, for some reason, refuses to accept limits.
Thanks go to Gottfried Wilhelm Leibniz … bless his soul … and this thread is the only time I’ve ever used this proof for any reason … so it solves the same 30+ year conundrum for me too !!!
A proof about linear functions? How is the sequence (0.9, 0.99, 0.999, …) a linear function? The limit is no element of the sequence. Which element of the sequence would it be? The last one?
You can count every number in the set. Which element in the set can not be counted?
If it is not on the number line it is no number. See the fallacy if you are treating it like a number:
The segments in 0.999… are [0, 0.9], [0.9, 0.99], [0.99, 0.999], …
So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?
a) There is no such segment
b) The segment is …
Correct.
Correct.
Correct.
…
b) The segment is [0, 1]
I thought I said this already?
Your whole argument amounts to saying that we cannot express (for example) 1/3 exactly in the decimal system, this is true if we are restricted to decimals with a finite number of non-zero digits. But if we are allowed decimals with (countably) infinite digits we can express it, and that expression is exact.
1/3 = 0.(3) or 0.3 or 0.333… EXACTLY
If you’d like you can call this “definitional” but as it is utterly coherent with standard arithmetic and the theory of limits you can also view it as a logical consequence of the decimal system.
It is true that none of the items in the list {0.3, 0.33, 0.333, …} is equal to 1/3 but that is because each of them has a finite number of decimals. None of them is 0.333… , so we cannot “find” 0.333… in that list – because it has an infinite number of decimals, it therefore cannot physically be written out in its full form.
This doesn’t worry anyone because the same thing can be said for a vast array of rational numbers (*most *of them, in fact) and *all *of the irrationals.
I don’t think I am saying anything that hasn’t been said in this thread before (and repeatedly).
MEANWHILE:
Hmmm. I don’t think 1 is an element of the set {0.9, 0.99, 0.999, … }, nor do I think 1 is an element of the sequence (0.9, 0.99, 0.999, …).
But then 0.999… isn’t an element of that set, nor is it an element of that sequence.
I don’t see why you think that the limit of a sequence has to be an element of the sequence.
Is the square root of -1 a number?
Huh? That’s not my quote … does your browser show the quote this is a response to as coming from me? … what a weird software problem …
I think I got tripped up in all this pseudo-mathematics … where the hell did this set come from? … I’ll withdraw any comment I’ve made about that particular sequence until I understand how it has any equivalence to (0.999…) … and I’ll go ahead and use the lame excuse I thought we were still working with the series (0.9 + 0.09 + 0.009 + …) … of course a sequence and a series are completely different things … whatever is true for a series doesn’t necessarily mean it’s true for a sequence …
HOWEVER:
The series equals 1 … if the sequence doesn’t equal 1, then it doesn’t represent the pseudo-mathematical non-number (0.999…) … and I’m pretty sure this sequence can’t be said to equal any (real) number …
I think there’s a fair chance that Mr. Netz is trying to pledge a frat-house where one of the items on his Scavenger Hunt list is
Find a message board where you can elicit at least 500 responses to a brazenly ignorant piece of pseudo-mathematical tomfoolery.
If so, Congratulations to netzweltler! You’re on the verge of reaching your goal. Please post a Hurrah finale so we can go back to debating whether the recent election results stem from a similar misunderstanding of Monty Hall’s Problem.
But we’re not in the Pit so Board rules require that we proceed on the assumption that your questions are sincere. Please answer the following questions, Mr. Netz, so we can focus in on your particular ignorance. In the sequel an expression like 0.08333 denotes the decimal representation where the final 3 is repeated ad infinitum.
(1) Is it the case that your argument about 0.9999 applies equally to 0.3333 ? This is apparently the case, as you have written
The wording in this seems so similar to the wording of your comments about 0.9999 that it would seem you find the same “paradox” with 0.3333. If so, let me request AGAIN that you discuss 1/3, not 1. That way your instructors need not be confused by wondering if you know the answer to (2):
(2) Do you realize that 0.9999 and 1 are two different names for the same real number?
(3) Is 1/3 a “number on the number line”?
(4) Is 0.3333 a “number on the number line”?
(5) Are 1/3 and 0.3333 two names for the same number?
That’s enough questions for now. We’ll see how you do on these before continuing. If your answers to (3), (4), (5) are Yes, No, Yes — as I assume they will be reviewing your previous posts — then please enclose a brief essay indicating whether you find this inconsistent.
Oh, heck. Here’s two more questions for Mr. Netz:
(6) Can the union of an infinite set of closed intervals, e.g. [0, .3]∪[0, .33]∪[0, .333]∪… be a half-open interval, e.g. [0, 1/3) ?
(7) If you correctly answer Yes to (6), can you point to anyone in the thread saying differently?
~ ~ ~ ~ ~ ~
You are using “segment” to denote a particular kind of set of real numbers. 1 (for which 0.9999 is a synonym) is a real number. You then write
“The segments [sets of real numbers] in 0.9999 [a real number] are …”
Members are in sets. Sets are not “in” their members (nor “in” their non-members). If you can’t remember whether a member is in a set, or a set is in its member, then I think it’s fair to say you are not speaking ordinary mathematics.
We really need series to understand the representation of a real number. So base 10 uses power terms of the form 10^K : where K is zero or a positive integer or a negative integer.
Then we can write 0.3 as 3 ( 10^(-1) ) and
0.33 as 3( 10^(-1) ) + 3*( 10^(-2) ) and so on.
Our partial sum for .3333… out to K places is then :
3* ( 10^(-1) + 10^(-2) + … + 10^(-K) )
These partial sums form a convergent sequence whose limit is a real number. Why ?
First , the sequence of partial sums is a Cauchy sequence - that is, the differences between partial sums become arbitrarily small the farther out we go. Since the real numbers are complete, any Cauchy sequence of real numbers converges to a real number.
So what is this limit of our Cauchy sequence? It must be 1/3.
Your question makes no sense when dealing with an infinity. As we’ve been saying all along, you need to establish some rules in addition to those applying to finites if you’re going to be able to deal with infinities.
You’re trying to have your cake and eat it too when you write the above, and claim it leads to 0.999… not being a number, while at the same time you write:
“Yes. It exists. It’s [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…”
It would be just as meaningless to turn your question around and go 0.9 is a number, 0.99 is a number, 0.999 is a number, what is the action that causes 0.999… to not be a number?
Your above definition [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… doesn’t converge to an interval [0, 0.999…] where 0.999… is not a number. That makes absolutely no sense.
But maybe there’s an opportunity here for insight into your obsession. Would you say that 0.999… represents a sort of non-number end point for an open interval? I.e [0, 1> is sort of like [0, 0.999…], and [0, 1/3> sort of equals [0, 0.333…]?
It was wrong the first time you said this already. At which position can you find a segment containing 1 in [0, 0.9], [0.9, 0.99], [0.99, 0.999], …?
Yes, and I have repeatedly shown that even for an infinite number of 3s its is not true that we reach 1/3.
If it is not on the real number line it is no real number.
Yes.
No.
Yes.
No.
No.
Yes.
Well, it looks to me like The Great Unwashed said it differently if his answer is [0, 1] to the question:
The segments in 0.999… are [0, 0.9], [0.9, 0.99], [0.99, 0.999], …
So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?
0.999… is the union of [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪ …
Better?
The action of completing infinity without reaching a well-defined point on the number line. We have well-defined points for each of 0.9, 0.99, 0.999, …
Doesn’t make sense to me. If it’s a non-number you cannot treat it like an end point.
So… Is it your contention that 1/3, or any real number not of the form, for some integers a, b, c,
a ∙ 2[sup]b[/sup] ∙ 5[sup]c[/sup]
has no representation as a decimal fraction? That an equation like 1/7 = .142857 is an abuse of notation?
Is the following a fair statement of your thesis?:
“While I, netzweltler, do understand that 1/3 and 1/7 are real numbers, they have no legal decimal representation because I, netzweltler, insist that all decimal representations have finite length.”
This seems like a fine (though highly non-standard) position and I won’t argue against it. I do wonder if repeating the nursery rhyme about “moving your pen” was the best way to express this opinion.
And every one of those points is closer to 1 than the previous. No finite number of steps removes us from the number line, it just brings us closer and closer to one. Now if you just refused to accept 0.999… as a meaningful notation that would be one thing, but you insist that 0.999… exists, and is not a number. That is truly baffling.
If it’s a non-number it cannot be the result of mathematical operations. And yet you claim that it’s equal to [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…, but at the same time not a number. Is [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… also equal to [0, 1>?
Not with mathematics you haven’t. Whereas the rest of us have shown that any number of threes that hasn’t brought us to 1/3 is not infinite. Just to repeat that:
If 1/3 - (3/10 + 3/10^2 + 3/10^3 + … 3/10^n)=d then n is a natural number.
But of course you’re not saying 0.333… is different from 1/3 as a number, you are saying it isn’t a number. It’s a non-number result of, according to you, mathematical operations.
Can we do math with these non-numbers though? Or to put it another way, can you show that these operations below are in error somehow?
0.333 = [0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]∪…
30.333 = 3([0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]∪…)=[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…
3*0.333 = 0.999
0.333 / 2 = [0, 0.15]∪[0.15, 0.165]∪[0.165, 0.1665]∪…
0.333/2 = 0.1666
And if you can’t show that these operations are in error (and saying “you can’t do that” isn’t a proof) isn’t it odd that these non-numbers behave exactly like they would do if they were valid representations of fractions and the number 1?
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1]
Is your only lame attempt to disprove this fact is “I said so” … and your only logical step is equivalent to “God can’t make rocks He can’t lift”?
Please, upon which pseudo-theorem do you base the statement “It was wrong the first time you said this” on?
As the number of elements in this union approaches infinity, the length of the line segment approaches 1 … therefore the limit point of the union is 1 … why do you believe then that 1 is not an element of this union?
This question is equivalent to saying, “Which of these finite strings of digits is an infinite string of digits?”
Might just as well say, “Which of these bicycles has 9 wheels?” or “Which of these cinder blocks is an octopus?”