Thanks for the APB !! Already noted! 
100% agree with you.
Sorry, but this is completely wrong, and the PhDs at Dr. Math (themathforum) all say otherwise.
That doesn’t matter. The definition I gave is the definition of the space R of real numbers. It’s the definition I was given in school, as an undergrad, and gave to my own studnts; it’s the definition on wikipedia, in analysis books (baby Rudin, for example), and math dictionaries; and it’s the defintion that professional mathematicians. You don’t get to define what a real number is. If you want to ramble on about schooner26 numbers and set up a different system, go ahead. But the definition I gave is the definition of R, and that is not up for debate.
Congratulations, you’ve shown that a bunch of numbers not equal to 0.999… are also not equal to 1. Huzzah!
This is absolute nonsense and demonstrates that you’re completely out of your depth here. In its basic form, induction means that if a property P holds for 0 and holds for n+1 if it holds for an integer n >= 0, then it holds for all integers n >= 0. (You’re possibly confusing the idea of infinite cardinals with the idea of the point at infinity, but whatever, you’re just wrong.)
For example, take the property P(n) = n is odd or even. Clearly P holds for n = 0. If it holds for n, then it also holds for n + 1: If n is odd, then n+1 is even; and if n even, then n+1 is odd. Hence P holds for all integers n >= 0 by induction. But P(∞) doesn’t even make sense.
For another example, take the property P(n) = there exists a real number x >= 0 with x[SUP]n+1[/SUP] = 1/2." It clearly holds for all integers n. It does not hold for n = ∞, if you interpret P(∞) in the obvious way; lim_{n -> ∞} x[SUP]n+1[/SUP] is either 0, 1, or ∞ for every real x >= 0.
By definition, two real numbers a, b— which, again, are Cauchy sequences a = (a_1, a_2, …) and b = (b_1, b_2, …)— are equal if and only if a_n - b_n is arbitrarily small for large n. That is the definition of equality of real numbers. If you don’t like it, tough shit; make up your own, separate system of numbers.
The rest of your post is just gibberish, and I’m not going to bother responding to it. Please try to understand what I and other posters have been trying at great length to explain to you. It’s clear from your previous posts that you really don’t understand some basic ideas in math like sequences, limits, inductions, and so on. Without that knowledge, there’s really nothing we can do to explain it to you.
You failed to understand the point I was making. The limit of a number c makes no sense; this is clear from the definition of a limit. The limits of the constant function f(x) = c and the constant sequence (c, c, c, c., …) do obviously exist and equal c.
The notation ‘…’ absolutely does not mean the limit of that sequence, and your inability to understand such a simple thing— even after being corrected by multiple people— is making this thread tedious. In a sequence like a = (0.9, 0.99, 0.999, …) the ellipsis means that the sequence continues in the usual way. Thus a_n = 1 - 10[SUP]-n[/SUP]. The sequence a is different from the limit of a. The sequence b = (1, -1, 1, -1,…) is perfectly well-defined: b_n is 1 if n is odd (indexing from 1) and -1 otherwise; it has no limit. The sequence c = (1, 2, 3, …) is perfectly well-defined: c_n = n; it has no limit.
That’s a useless definition, because it requires that real numbers are already defined. The entire point of the definition is that it allows R to be defined entirely in terms of Q.
You failed to understand the point I was making. The limit of a number c makes no sense; this is clear from the definition of a limit. The limits of the constant function f(x) = c and the constant sequence (c, c, c, c., …) do obviously exist and equal c.
The notation ‘…’ absolutely does not mean the limit of that sequence, and your inability to understand such a simple thing— even after being corrected by multiple people— is making this thread tedious. In a sequence like a = (0.9, 0.99, 0.999, …) the ellipsis means that the sequence continues in the usual way. Thus a_n = 1 - 10[1]-n[/SUP]. The sequence a is different from the limit of a. The sequence b = (1, -1, 1, -1,…) is perfectly well-defined: b_n is 1 if n is odd (indexing from 1) and -1 otherwise; it has no limit. The sequence c = (1, 2, 3, …) is perfectly well-defined: c_n = n; it has no limit.
The rewarding you give is utterly useless, because it requires that real numbers are already defined. The entire point of the definition is that it allows R to be defined entirely in terms of Q.
Look, you’re failing to understand a very basic point of mathematical analysis. Go read over the wikipedia page on the set of real numbers, for example, which goes into the matter in far more detail than we can on a message board. You seem to be unclear on what a sequence is, how it’s not a “process”, the difference between a sequence and its limit, the construction of real numbers, mathematical induction, and a host of other mathematical concepts that are necessary to participate usefully in a General Questions thread about math, especially when you’re stubbornly insisting that you’re correct and every other mathematician ever is wrong.
SUP ↩︎
That is the way *you *think. However, I believe everything can be revisited, or explored in more detail, and questioned. Things that were believed “to be true” in the past, have been proven otherwise much later.
So “not up for debate” is certainly close-minded.
Well, going by what someone else on this forum, “infinity is a number” so if :
0.9(n times) < 1 for all n, or for aleph-naught, and is true for n = ∞.
Of course anything that challenges a definition is going to be “nonsense” to you.
Just as
f(n) = 1 - 1/10ⁿ = 0.9(n times)
But f(∞) doesn’t make sense. ie n=∞
But 1 - 1/10[sup]∞[/sup] doesn’t make sense, ie n=∞
However 0.9(∞ times) makes sense, ie, n=∞
You are saying a list of numbers is a real number. (real number a is a sequence)
It *is *tough shit because arbitrarily small is not exactly zero. You are simply declaring it !!
I understand everything you are telling me. I have for decades. The point is that there are some basic contradictions (as the one above).
I did extremely well in those areas. So ,if you would believe, I was offered (and accepted) a full scholarship (with pay) in any degree of my choosing. Of course, you don’t have to believe this, nor do I expect you to be impressed. However, an extremely competitive file is mandatory for even an application to be accepted for consideration, and the faculty in the engineering dept thought otherwise, and maintaining a certain average to keep that scholarship going would indicate more than a “basic idea”. So that is one area of my background at the “basic level” that I have shared with you.
Oh yeah, I was tutoring university students analysis in all those areas you listed, while I was still in high school.
So bring on the next insult if you like. However they are 2-fold: Insults and Lies.
Now we have […] meaning several different things.
[…] means “this” over here, but means something different “over there”, oh, and different again “in this area”.
Thanks for supporting my points that there are contradictions in the current system.
Is that the technical term? How long do I continue “the usual way” ??
Yes of course. Because a sequence is a list of numbers.
The limit is the number that the sequence tends to, (usually by taking the limit of the n’th term function)
,
Yes, your opinions again! … are just the same.
Kinda like PI and sqrt(2). We know the number already and then attempt a sequence to define them.
And like your statement: “the real number 0.999… has a sequence yadda yadda …” You started with your conclusion !
Not at all. Again with the lies. I understand what you wish me to understand.
No thanks. Wikipedia… lol. I won’t go into that one.
.
,
A list of numbers.
I am pointing out inconsistencies that are being glossed over by people. For instance “infinity is a number”.
SUP ↩︎
Congratulations, you finally got it! A real number is a Cauchy sequence of rational numbers modulo equivalence.
Yeah, I’m sure you’re brilliant at math, given your previous discussions about how numbers are only rational in certain bases, and how any statement that holds for each integer n must also hold for n = ∞ by induction. Seems totally legit.
If you want to say that 0.999… = 1 is only true for Itself’s definition of the reals then you must accept that 0.999… = 1 according to that definition (which is the subject of this thread).
But if you believe that the definition is unreasonable you must
b) show us explicitly how assuming it is a good definition leads to a contradiction.
or
c) propose another defintion of the reals for which all the attended properties are true AND 0.999… <> 1
I propose a moratorium on posting to this thread until you are prepared to stump up your end of the argument.
So very seconded. schooner26’s responding to every point with some variant of, ‘Well, I don’t believe that, so I’m going to summarily ignore it,’ is tiresome.
I’m still here, but since I caught a rolled-up newspaper across my snout the last time I posted, I’m not posting. There is a lovely Pit thread already.
What sequence is generated by:
Σ 4.5/10ⁿ ?
s1 = .45
s2 = .495
s3 = .4995
etc
What are the 2 numbers?
Are you asking what sequence corresponds to the real number Σ_{n\geq 1} 4.5/10ⁿ = 1/2? The sequence (1/2, 1/2, 1/2, …), obviously, or s = (s_1, s_2, …) for s_n = 4.5/10 + … + 4.5/10^n = 1/2 - 1/(2(10^n)). Or the sequence t_n = 1/2 + (-1/10)^n, if you prefer; all three sequences are equivalent.
Tee hee hee. This is funny!
What real numbers correspond to the series I posted?
For example, you (or someone) said:
“The real number 0.999… corresponds to the sequence {9/10, 99/100, … } and the limit of the series is 1, therefore equal”
What I am looking for is a direct comparison between that statement and the one for mine:
The real number ______ corresponds to the sequence {0.45, 0.495, 0.4995, … } and the limit of the series is _____
I do believe you were given the number 1/2 (which is equal to 0.5). Three equivalent sequences were given to clarify the point. they were introduced with the question, “Are you asking what sequence corresponds to…”. I am not sure how you could miss it.
1/2 to both, obviously. We’ve gone over this many, many, many times before. There are many different Cauchy sequences that correspond to the same real number. Decimal notation suggests an obvious one, but it isn’t the only one, and (in the case of 0.999… = 1) even a decimal notation isn’t unique.
And the triumph of post-modernism in education comes home to bite.
Sure, there is scope to revisit the manner in which numbers are constructed. And there are mathematicians who spend time doing that still. But you won’t usefully do this without understanding the basics. Nor will you add anything useful by blandly asserting that the mainstream techniques are wrong when you don’t seem to understand what they are. Starting from a base of asserting that the Real numbers are not what everyone else accepts their definition to be is just silly. If you want to explore a different understanding at least start with your own set of axioms and be clear that these differ from those that define the Reals, or minimally, look at the definition of the Reals and suggest what it is that you have a problem in that definition.
Currently you are basically saying that the Real numbers are not what their definition says they are. Which is just plain impossible to discuss sensibly. Note that “Real” is a proper noun. That is the name given to the numbers that are defined by the accepted axioms. Just as the Natural, Integer, and Rational numbers are names given to sets of numbers defined by their respective axioms. So to are the transfinite numbers defined. If you want to address the construction of the Reals, don’t start by telling everyone that the definition of them is not what it is. Start by providing a your definition of numbers, one that addresses the questions that the definition of the Reals does. Show that it meets the same objectives as the Reals, and that your version of 0.999… fits into this scheme.
Right now you are simply asserting that the Real numbers are not what their definition says they are. And that is futile.
To that end, it might be appropriate to have the axioms of the Reals stated.
So, if you want to assert your claim on the Reals, you need to show which of these axioms leads to a contradiction. Then you need to provide an alternative that does not lead to a contradiction and show that it is indeed consistent.
Good Luck.
Multiplication.
Multiply the “decimal” version of 1, namely, the series 0.999… by itself.
Then multiply that product by itself. (Powers of powers). Now do it infinite times.
You will see some magic happen as the terms themselves start converging on 0.
You can do it for a few iterations using distributive property.
Basically, you end up a bunch of finite terms less than 1, that are raised to powers, repeatedly. I am sure you are aware what happens to a number between 0 and 1 when this happens.