azzaman333
meh
In the realm of real numbers, 0.999...8 does not exist as there are infinitely many 9s before it.
1=.999999...?
- Thread starter Mouthwash
- Start date
In the realm of real numbers, 0.999...8 does not exist as there are infinitely many 9s before it.
That number makes no sense.
Can you write it using a different formula, much like you can write '4' as '2+2'?
Can you write it using a different formula, much like you can write '4' as '2+2'?
Hygro
soundcloud.com/hygro/
There are also an infinite number of 9s before the infinite number of 9s.
I don't see the problem with the rule of the number is that there are infinite 9s before it never-but-would end on an 8.
hoplitejoe
Top fun-poster
- Joined
- Mar 22, 2010
- Messages
- 5,470
As soon as you put an 8 on the end you no longer have an infinite string of 9's, as there is an end at the 8, so it is not equal to 1. An 8 anywhere in the number would make it less than 1. As soon as the 8 happens the value is no longer as close to a value of 1 as it can be for that number of digits.
azzaman333
meh
Because it's no longer infinite 9s if there's an 8 at the end.
That's the conceptual. You also can't make a function that generates that entity.
It's not true that these are the same. Here is one important difference:
In fact, in 0.99999..., before any given "9", there are only FINITELY many "9"'s before and thus every "9" has an actual, FINITE position.
In your proposed example, the "8" does not have a defined finite position, so this is not a well-defined real number.
onejayhawk
Afflicted with reason
Not exactly. Such a number does exist. After all, you just constructed it. Is it distinguishable from 0.999...? Not easily. We can say that one is less than the other, but not much else. Both fit the definition of lim .9+.09+.009+...=1. Of course, there are an uncountable infinity of points that do.
J
azzaman333
meh
Not when you're talking about real numbers it doesn't.
onejayhawk
Afflicted with reason
Onejayhawk, I really believe that you on the other hand want to understand maths, but it seems like you haven't learnt it the rigorous way (the definition of the open interval you quoted above is an example of that).
The halo you described above, that's the infinitesimals. People used them around 17th century, but they were highly controversial and were banned when the rigorous concept of the limit was introduced in mid 19th century, although they're still popular today in the maths taught to the appliers.
However, in the 1960s it was shown that infinitesimals can be rigorously constructed. The maths involving them is called nonstandard analysis. This is a thing I am very hesitant to tell, since IMO people should learn the standard maths properly first before getting into nonstandard analysis (=NSA). It's very easy to form all kinds of misunderstandings of it and for many people the license to use infinitesimals is a license to do bad maths. For example the infinitesimals you encounter in the maths for appliers courses are rarely the consistent infinitesimals.
Now, one thing to notice about these infinitesimals is that they are not real numbers. They are an addition to the real numbers, just like complex numbers are. They are something stuffed in between the reals. So, in real analysis there is no way any sum, limit or series can be an infinitesimal (or to differ by an infinitesimal from 1, as it would be in the case of this thread).
Another good thing to know is that they don't change anything in the real analysis: everything you can prove of the real numbers with them you can prove without, and vice versa. So, it's a matter of taste whether you use them or the standard analysis. The NSA people are very few, and to much of them it's only a curiosity. Some on the other hand remind more religious fanatics.
The infinitesimals carries a baggage though: If you accept the construction of them, you must accept also the axiom of choice, and if you accept that, you have to accept the Banach-Tarski paradox. Most mathematicians don't have anything against that axiom, but I thought you might. At least the ultrafinitists in this thread would.
Notice also that this doesn't change pretty much anything said in this thread: your intuition isn't necessarily contradictory, but the way you tried to force it in the normal maths is. Your proofs were still faulty, 0.999... still is equals to one, it still isn't in ]0,1[. A limit still can be exactly 1 although the elements of the sequence are strictly less. There still is just one limit of a real number sequence (with the usual topology). A series is still a number and not a sequence, and the sum of it is still by definition the limit of the partial sums.
I will admit I have not studied infinitesimals. As you point out, I have never heard much good about them. So I will take much of this on faith.
That said, I still insist that 0.999...<1, which is a sufficient condition to the main point of contention, ie 0.999...=/= 1.
If you choose to define a series as the limit of an infinite sum, what do you call the sum to distinguish it from the limit? How do you defend the statement that an open set has no end point.
J
Thank you, i will have a look
So this is also about the need to have multiplicative inverses for any Real number? (even if not so, i thank you for your reply cause you did give insight as to where to look for why 0.9999... is currently set as 1, which is what a lot of people here had been asking. Unlike you some others were merely -at best- alluding to some reason).Of course infinity is one of the most fundamental concepts in math. How to define it in the context of other math notions used does tie to internally practical reasons in a set axiomatic system anyway. When i have the time i will read more of this
@Atticus: Hey! :\ We all know that in art anyone can be less checked in what he thinks he is
At any rate i would rather we put the unpleasantness behind us, and why not you even wish me good luck with my book! 
I do regret that you (apparently) did not note that much that i specifically said many times here that i was just looking for a specific place to start examining why in current math the limit -as it seems- included stuff like 0.999...; in my view this is not a position to be attacked, given math is a logic system and as such it has a vast number of definitions of such things subject to change, even if changes like the one discussed may indeed mean massive alteration to analysis as a field. That is not my aspiration, i just wish to be known as a major writer
Not the limit of the infinite sum, but the limit of the finite sums. There's no difference between the limit of the finite partial sums and "the infinite sum". The latter is defined to be the former.
Why it's so? Because originally you have just a sum of two objects. You can extend it to more recursively, for example you can define a+b+c = (a+b) + c. That is, first sum a and b and then sum the result with c. This way you can define sum for all finite number of summands, but not for infinite. That's why the infinite sum requires a different definition. Notice also that it's different from a finite sum in many other ways too: finite sums are always defined, infinite not necessarily. Finite sums can be summed in any order, infinite sums not.
You can even go further and define sum for an uncountable amount of numbers, but need a different definition for that too.
EDIT: Forgot the second question. I don't understand what you're exactly getting at. It's easy to show that there's no biggest number in ]0,1[ for example: By definition ]0,1[ = { real x: 0<x<1}. If there would be the biggest number in ]0,1[, call it a, these would be true:
a < 1
For every x in ]0,1[ x < a.
Now, choose b = (a+1)/2, i.e. the average of a and 1. It's easy to see that a<b<1, so b is in ]0,1[ and on the other hand b>a. That is a contradiction with a being the biggest element of ]0,1[.
Besides, if an open interval included it's endpoints, how would it differ from the closed interval? In no way. It would be exactly the same thing. The difference between them is however huge, as you can see already in the elementary maths, for example a continuous function is bounded and reaches it's maximum and minimum on a closed interval, but not necessarily on an open one.
@Atticus: Hey! :\ We all know that in art anyone can be less checked in what he thinks he is
Suits me. Good luck with your book, sir.
Hygro
soundcloud.com/hygro/
This is reasonable but seems arbitrary. I'll take your word for it.
Akka
Moody old mage.
No you didn't. The example you gave precisely show you don't understand what "infinity" is.
You use "they will never manage to write the number" as an argument against it, when it's THE WHOLE POINT of infinity : it NEVER ends.
You basically attempt to prove you understand infinity by using an example based on NOT understanding infinity. Hardly convincing...
If there is a "final 8", then the number of 9 is finite, then it's not 0,999... anymore. I honestly don't know how you can imagine it's a legitimate comparison, when the whole concept of 0,999... is that it's an INFINITE amount of 9.
Again, people have no idea what infinity IS.
Akka
Moody old mage.
It is VERY easily distinguishable from 0,999... : one is finite, the other is not. How does it makes it "not easily distinguishable" ?
Also, one includes an 8 while the other don't. That's pretty easily distinguishable to me. Or are you going to say that 0,98 is hardly distinguishable from 0,9999999999999999 ?
Despite being proven wrong constantly, you stubbornly claims you're right and all the mathematicians are wrong, but you admit here you can't even read a number and distinguish it from another ? Seriously ?
Terxpahseyton
Nobody
- Joined
- Sep 9, 2006
- Messages
- 10,759
Infinite continuation of 0.999... is exactly what 0.999... is. Or isn't it? And that is what I am talking about. You get too hung up on those ancestors trying to reach 1. It is just an illustration of what 0.999... supposedly does. But as you note yourself, 0.999.. can't reach an end. Which is just another way of saying that it can never be one.
A list of written numbers is a physical representation, ultimately. Or at least can be, in principle.
An infinite process is just that. A process, no goal line. That does not at all mean that it does not exist. That would be like saying that space can not be infinite because we could never reach the end... The opposite is true, of course. Never reaching the end means space's infinity is real (supposing one continuously try's to reach the end).
So my illustration is fully fine with 0.999... existing.
I am not disputing that. That is rather my point. If 0.999... never ends - it will also never reach one, even if it goes on forever, which is all 0.999.. means and I just illustrated how that works. Hence it cannot be one. Otherwise the ancestors would reach it, since what they are carrying out for eternity is exactly what 0.999... represent: an infinite process. It can not be a simple number exactly because it is infinite, but allegedly, the opposite was true. Which makes no sense what-so-ever.
Borachio
Way past lunacy
- Joined
- Jan 31, 2012
- Messages
- 26,698
Well, I don't know.
It makes perfect sense to me: that an infinite number, in this case 0.99.. ,would exactly equal 1. I can't see how it wouldn't. If it didn't, it wouldn't be infinite.
I can see how it might look as if it doesn't. But it really does.
It makes perfect sense to me: that an infinite number, in this case 0.99.. ,would exactly equal 1. I can't see how it wouldn't. If it didn't, it wouldn't be infinite.
I can see how it might look as if it doesn't. But it really does.
Terxpahseyton
Nobody
- Joined
- Sep 9, 2006
- Messages
- 10,759
:sigh: I don't suppose that you can articulate this "perfect sense", while accounting for how I articulated it made no sense, can you?
Borachio
Way past lunacy
- Joined
- Jan 31, 2012
- Messages
- 26,698
:sigh: No, it seems I can't. Maths is sometimes like that for me. Something goes "ping" in my brain, and I "get" it. Very often after a period (more or less prolonged) of not "getting" it.
Sometimes it's not a matter of "understanding" an idea, but just of getting used to it.
But can't you appreciate that I can't see how it wouldn't equal 1? My previous post expresses my position very well and succinctly, I feel.
No, it seems I can't. Maths is sometimes like that for me. Something goes "ping" in my brain, and I "get" it. Very often after a period (more or less prolonged) of not "getting" it.Sometimes it's not a matter of "understanding" an idea, but just of getting used to it.
But can't you appreciate that I can't see how it wouldn't equal 1? My previous post expresses my position very well and succinctly, I feel.
Similar threads
