I have to follow along, I just can’t stop myself, I’ve been having nightmares about this. If 0.999… ≠ 1, then zero can’t have a direction. That would be very bad for me …
I have an issue with this definition. It is clearly based on finite (n), therefore, mathematical induction can also be used for the same (n).
0 < 1 - a_1
0 < 1 - a_2 < a_1
0 < 1 - a_n+1 < 1 - a_n
Clearly, for any (n), 1 - a_n > 0
Then the crying starts that “but that doesn’t work for infinity”. If infinity is all the natural numbers, then actually, yes it does.
This is simply a declaration, since it was a declaration that said if the difference between the sequences “tends to” 0, then it *is *0 . So what you did was assert that “tends to zero” *equals *0.
I proved that 1 - 0.999… is greater than 0 for allof your (n) , but you use a made up rule to say “successive subtractions get smaller and smaller, we will consider that 0”.
You are subtracting sequences and not the numbers.
I’d like to know how {9/10, 99/100, 999/1000, …} tends to the number “0.999…” ?? By putting the […] you have just said “limit” and limit is a concept, not a number.
I have already shown this, too, more than once:
{9/10, 99/100, 999/1000, … , 1 - 1/10[sup]n[/sup] , …}
The limit of the partial sums is clearly 1.
If you mean “a string of endless 9’s” then nothing can “tend to infinite 9’s” because the result of of a limit cannot “tend to infinity”
In essense, this is the idea you are portraying:
{9/10, 99/100, 999/1000, … , 1 - 1/10[sup]n[/sup] , …}
f(n) = 1 - 1/10[sup]n[/sup] = 0.9(n times)
Then you go ahead and say “I will just carry this to infinity and conceive infinite 9’s in my head as a number”
On what basis ? We can’t say “infinite 9’s” is a number because it automatically induces a limit as soon as you attempt to conceive the thought of “going to infinity” !!
You can’t have infinite anything. Someone recently quoted “infinite 9’s are all there!” Oh really ? Never ending 9’s are “all present” If they are all there, then how many are there ? This is nonsense speak.
It is the same as trying to conceive f(∞) = 1 - 1/10[sup]∞[/sup]
We can not put ∞ there because the minute we try, BAM!, the limit is already done for you.
You have 2 choices:
Finite amount of 9’s after the decimal; or
the limit of infinite 9’s.
Infinity is tied to limit. Just like we *cannot *write: 1/∞ = 0 ! We can write f(n) = 1/n; then write:
limit f(n) = limit (n→∞) 1/n = 0
n→∞
This means “The Limit as n tends to infinity of 1/n = 0”
Not: 1/n = 0 as n→∞
Not: 1/∞ = 0
Yes, and for the umpteenth time:
THE STANDARD MATHEMATICAL INTERPRETATION OF THE NOTATION […] IMPLIES A LIMIT.
You may have your own personal understanding of what the notation […] means, but nobody uses it except you, and so like a language with only one speaker it’s not much use.
what do you think I just finished saying ?
Others are saying 999… and 1 are different numbers, that are equal to each other.
Saying things like "0.999… is the limit of {9/10, 99/100, …} " is the same as saying "the limit of 1 is the limit of {9/10, 99/100, …}
Except that infinity is not “all the natural numbers”. This is quite explicit and with reason. Infinity is the cardinality of the set of natural numbers. These two things are not the same. Not even vaguely. The cardinality of a set is a single number. It is not a selection mechanism from a set. Sets that contain things other than natural numbers can have a cardinality.
|{cat, dog, fox}| = 3
|{{},{{}},{{{}}}}| = 3 as well.
|{ ℕ }| = ℵ[sub]0[/sub] - aka ∞
How many natural numbers are there?
How about how many rational numbers?
Which pretty much sums up the problem. You don’t get infinity. Somehow your mind recoils from the concept. Which isn’t surprising. It isn’t easy. Discussion doesn’t come into most university level mathematics courses unless you are studying pure mathematics. Which is a problem. Standard calculus courses do tend to gloss over things, and leave students with a less than total understanding of what is happening. It is subtle, and if you are studying applied mathematics, you have equations to solve and bridges to build, rather than worry about the deep underpinnings. But that doesn’t mean they are not there, and have not been worked over in excruciating detail.
Since you used […] in your sequence, then yes, you ARE talking about limits by that notation, just as Buck has just finished yelling.
Don’t forget that the […] means the limit of that sequence.
Perhaps worded like this: “0.abc… corresponds to the sequence (a/10, a/10 +b/100, a/10 +b/100 + c/1000, …) of which the limit is a real number”
It seems you want to use just the string of digits as a number *and *the limit as the same number, but you can’t have infinite digits without limit.
So if there are 12 eggs in a carton, you say the cardinality is 12, but it is wrong to say “all the eggs in the carton” or “for each egg…” ?
And you are saying that number is *infinity *?
The number infinity numbers by what you just said.
Sure I do. I don’t just regurgitate what you want me to say about it.
IF I don’t say exactly how you were told to understand it, then “I” don’t understand it.
Fabricated
Insult
I’ll take your word for it.
What are you on about?
Yes, it is wrong. There are two things here, and you don’t get to interchange them. Infinity is the cardinality of the natural numbers. It is not the elements of the natural numbers. Otherwise I could say.
limit (for n = each egg in the dozen) ∑ 10[sup]-n[/sup]
It doesn’t make any sense to say 10[sup]egg[/sup], but it does make sense to say
limit (for n = |{dozen eggs}|) ∑ 10[sup]-n[/sup] because |{dozen eggs}| = 12
There is some scope for confusion between the numbers in the set and the size of the set, since both are numbers. But the cardinality of the set is simply not a way of providing an enumeration of the elements of the set. As it is, you can easily prove that the rationals has the same cardinality. So perhaps you could consider your view by substituting “the size of the set of all rational numbers” for infinity.
The cardinality of the set of natural numbers is infinity. Obviously sets with fewer (finite) elements have smaller cardinality, and sets with more numbers have greater cardinality.
Wrong.
Limit[sub]x→a[/sub] C = C
Very basic stuff.
The real number is one because limit is automatically implied.
So “is” should be “equal to a” or “outputs”
It is? Or is the list something we can generate by analysis ??
Because this does not look like any *list *to me: 0.999…
When you use cauchy definition, you are subtracting using an implied limit already if you do something like this: 0.999… = 0.9 + 0.0999…
because 0.(9) - 0.0(9) = 0.8999…1 or more “correct notation” : Σ 81/100ⁿ
Therefore subtracting off finite terms from an infinite series exploits the concept of infinity.
So it is a little circular to say the infinite decimal number equals the limit of the infinite decimal number, which is another decimal number.
Because “0.9999…” means "limit {9/10, 99/100, …} " which equals 1.
If you don’t take 0.999… to mean limit, but rather a string if digits which is the real number (not 1), that is what he is understanding that you are trying to assert.
This is Redundant: lim 0.999… = 0.999…
If 0.999… = limit of {9/10, 99/100, …} = 1
then the redundant statement lim 0.999… = 0.999…
is the same as : lim 1 = 1
So what is the point of writing this exactly ?? Embedding limits serves no purpose here.
You said the cardinality of a set is a single number.
If you say the cardinality of N is ∞, then you are saying ∞ is a number.
They are two different representations for the same number.
Of course it is a number. It has a very clear definition, and follows some clear rules. They are not the same rules as finite numbers, but there is a consistent logic. These are generally known as the transfinite numbers, and guess what? - There are an infinite number of them, all quite distinct (and that infinite number is bigger infinity than the cardinality of the natural numbers.)
As a first step, whilst there are the same number of natural numbers, integers, and rational numbers, there are more irrational numbers than this, and the number of them (*) is the next transfinite number. ℵ[sub]1[/sub]
There are lots of different sorts of numbers. Quaterions are a personal favourite. There is a certain satisfaction I find in them.
(*) with continuum hypothesis
The word “number” by itself doesn’t have a precise mathematical definition; so the argument over whether or not something is a number is not a well-defined question. Rather, mathematicians have precise definitions for specific types or sets of numbers. So we can talk about whether or not something is a member of the set of ____ numbers (real numbers, complex numbers, transfinite numbers, surreal numbers, etc.).
To wade (yet again) back into this, this is pretty much it (except for the pages and pages and pages of butting-heads-into-walls over the difference between representation/notation and the platonic notion of “number”).
In number theory, we generally mean (with some exceptions) positive integers and occasionally then number zero.
We can divide “numbers” into primes, composites, and the number 1. But it doesn’t make sense to ask if pi is prime or composite. In context, we realize a discussion about primes/composites is limited to positive integers. Just as in context, it makes no sense to ask about “infinity” in a discussion about real numbers.
Those with a mathematical background do generally understand the context of questions, but often students and/or those who don’t have that background may not realize or accept mathematicians use different sets of numbers in different contexts. Worse, they may begin mixing sets of numbers interchangeably without clearly understanding those sets or how to do so without errors (as seen repeated ad nauseum in this thread).
Does lim ad nauseum = ad infinitum?
We’ll hit the heat death of the universe before then.
I hope.
And ad absurdum.
The numbers 1 and 0.999999…are not equal to each other. Because the limit of 0.99999…is not equal to 0.99999…
Proof:
lim[sub]n -> ∞[/sub] S = a/(1 - q) = 0.9/(1 - 1/10) = 1
The limit of 0.99999…is equal to 1
Therefore 1 (= the limit of 0.99999…) is not equal to 0.99999…
Right.