Is 0.999... Truly Equal to 1 in the Realm of Infinity?

[ram2048]

that's what .999~ means. infinite digits. meaning if i were to COUNT each digit as f(n) i would count to infinity (magnitude).

it's your number and your definition. is the number of digits NOT equal to infinity were they to be matched on a 1 to 1 basis?

if no then you would have to define exactly what ~ or ... or _ means in your statement.

and even supposing you DO define it, come back and explain to me how in a system where everything to the left of a number is greater than and everything to the right is less than, you have come up with a number to the right of another that is exactly equal to a number to the left.

 

[Integral]

Please repeat that post in meaningful terms. What are you saying? It does not make any sense to me.

Once again ALL real numbers have an infinite number of digits, NO real number is infinite in magnitude. Now what is your point.

 

[ram2048]

you know perfectly well what I'm talking about.

feigning ignorance doesn't help your case one bit.

 

[Integral]

Am I supposed to read you mind? Your post does not make any sense. It does not appear to address any of issues being discussed. How does it relate to your previous nonsense statement that 1-1=1 ?

For the third time .
All real numbers have an infinite number of digits NO real number is infinite in magnitude.

So yes, .999... being a real number has an infinite number of digits. What is the problem?

 

[ram2048]

i'm just going to assume then that you've not read any of my posts concerning this matter, Integral, as i have explained several times in exacting detail WHAT I'm talking about.

how many digits of with value 9 does .999~ have?

i'm also assuming you KNOW what a digit is since you claim all "real numbers" have infinite of them

each digit position has a value from 0 to 9. simple question to you, how many digits have value of 9?

 

[hello3719]

that's what .999~ means. infinite digits. meaning if i were to COUNT each digit as f(n) i would count to infinity (magnitude).

it's your number and your definition. is the number of digits NOT equal to infinity were they to be matched on a 1 to 1 basis?

if no then you would have to define exactly what ~ or ... or _ means in your statement.

and even supposing you DO define it, come back and explain to me how in a system where everything to the left of a number is greater than and everything to the right is less than, you have come up with a number to the right of another that is exactly equal to a number to the left.

I don't see a problem. When I write ~ or ... or _ we mean that the pattern repeats itself never-endingly. We can then say that .999... = .999... (obvious)
do you agree that .999... is between 0 and 2. If yes, knowing that 0 and 2 are real numbers then .999... must be a real number. let's call x this real number. Then the way real numbers are defined x = x implying x - x = 0
x=.999...
10x = 9.999... = 9 + .999... = 9 + x
10x - x = 9
9x = 9
x=1

even if .999... has an infinity of digits we can still do the 1 to 1 basis since .999... = .999... and it is a real number. there is then no contradiction
with the fact that infinity- infinity being undeterminate since the number .999... itself isn't infinity.
And even if there is an infinity of the digit "9" it doesn't mean that infinity - infinity is ALWAYS equal to 0, in this case it is in some other cases it isn't our system is totally consistent for this matter.In your system you say that infinity - infinity is AlWAYS = 0 , the possibility of being 0 in our system is existent but not exclusive to it. Try to understand the uses of limits with infinity and everything will be clear.

 

[ram2048]

you don't understand my system AT ALL.

my system allows for infinities to be equal, greater than, or less than each other. complete with a way to differentiate between such expressions of infinity so that there is no confusion.

with your system, maybe it's 0 maybe it's undeterminable or undefined or maybe it's equal. you never know because you use the term "infinity" as an all-inclusive term for so many things.

 

[hello3719]

didn't your system say that infinity(d) - infinity(d) = 0 ?
it is always equal to 0 in your system isn't that right ?
then logically your other "infinities"(ridiculous) are built on infinity(d).

and about our system i said that infinity - infinity is AN UNDETERMINATE FORM not INFINITY. As i said a thousand times try to understand the avantages of undeterminate forms. Tell me, do you know how to use limits, find derivatives etc...
If you do you wouldn't see the undeterminate forms as an inconvenience but as a logical consequence and totally treatable form with regards to the concept.

 

[ram2048]

hello3719 - and about our system i said that infinity - infinity is AN UNDETERMINATE FORM not INFINITY.

Well first of all our infinite isn't a real number if you didn't understand that yet. Saying "exactly infinite" isn't proper concerning our infinite.
Tell me, if i have a never-ending supply of apples and I eat one will i still have a never-ending supply of apples?

i made the comparison that .999~ was infinite un ending number of 9's just like in the case with your apples. you never replied back when i said that if they're infinite un-ending i should be able to eat an infinite amount of them without affecting that property. such that:

.999~ - .999~ = .999~ (unending) (minus) (infinite) (equals) (unending)

please verify.

didn't your system say that infinity(d) - infinity(d) = 0 ?
it is always equal to 0 in your system isn't that right ?
then logically your other "infinities"(ridiculous) are built on infinity(d).

i said that's how my system is built AT THE VERY BEGINNING. OMG you finally understand :P

 

[Integral]

Ram,
Let's get one thing straight. You do NOT have a system. You have nothing of any use to anybody. Since you do not care to learn what generations of mathematicians have developed why should anybody care what nonsense you have cooked up in the last couple of hours?

Of course I do not understand your system, you have northing but misconceptions, why should I waste my time making any effort to understand nonsense?

You are the loser here, you have an opportunity to learn something useful but instead continue to insist that everyone but you is wrong. What a waste of time.

 

[hello3719]

i made the comparison that .999~ was infinite un ending number of 9's just like in the case with your apples. you never replied back when i said that if they're infinite un-ending i should be able to eat an infinite amount of them without affecting that property. such that:

.999~ - .999~ = .999~ (unending) (minus) (infinite) (equals) (unending)

please verify.

Did you read my last post well? It seems you didn't, i explained you that .999... is a REAL NUMBER so .999... - .999... = 0
Re-read well my last post since you did not understand it in depth.

.999~ - .999~ = .999~ (unending) (minus) (infinite) (equals) (unending)

.999... is infinite?
i explained the number of digits thing in the last post

eating apples non-endingly from a non-ending supply of apples
can only be represented as infinite - infinite which is undeterminate
you cannot say that it is equivalent to .999... - .999...


and when i talked about the apples i meant to tell you that you can take a finite number of apples of a non-ending supply but there will always be a non-ending supply of apples.
This was to show you the true nature of infinite.
I think you should have understood it by now.

 

[Hurkyl]

i would count to infinity (magnitude).

But never arrive at infinity.

 

[ram2048]

hello3719

I don't see a problem. When I write ~ or ... or _ we mean that the pattern repeats itself never-endingly

i explained you that .999... is a REAL NUMBER so .999... - .999... = 0
Re-read well my last post since you did not understand it in depth.

eating apples non-endingly from a non-ending supply of apples
can only be represented as infinite - infinite which is undeterminate

.999~ - .999~ = indeterminate then?

oh apples can't be 9's? why can't they be. numbers are just tools for describing reality. if i substitute an apple for every 9 there's no failure in logic, just the contradiction you happily supplied

 

[ram2048]

Integral: you don't want to read about it then you're free to leave and never look back.

fact of the matter is you're so scared i might be right that you're unwilling to apply any effort at all to understand me.

my "system" irons out a lot of kinks in the current system and replaces a lot of false notions and assumptions with perfect logical ones.

you wouldn't lash out like this if you weren't feeling threatened by me.

 

[ram2048]

But never arrive at infinity.

true enough

 

[matt grime]

ram, when we say there are an infinite number of nines in the expansion 0.999... we mean exactly what the word literally means: that the number of them is not finite. that is all. we also can go further and say they are in bijection with N, tha natural numbers, by place.

we are not using infinity as a number in the same sense as a real number. that is why we have cardinals, and ordinals, which are dsitinct, and have different arithmetics. there are also infinitesimals. we also have analysis.

so you have developed a symbol, call ik K, that indicates the 'number' of 9s in the expression 0.99999...

how many elements are there in N or Z? how many finite groups are there?

seems like you're going to have to have a different one for every object unless you give a way of comparing them. is there a comparison?

 

[Integral]

What is it that I should fear? You have not presented a single coherent concept. you have repeatably demonstrated your lack of knowledge or understanding of mathematics. you simply do not have the tools to formulate a meaningful mathematical statement. Your ideas are not to be feared they are to be laughed at.

Before you can even think about fixing something you must understand how it works. You do not understand the Real Number system, therefore have no hope of "fixing" it.

lets do a bit of simple arithmetic.

.999... - .999... =

( \Sigma_{n=1}^{\infty} 9* 10^{-n}) -( \Sigma_{n=1}^{\infty} 9* 10^{-n}) =
9 * ( \Sigma_{n=1}^{\infty} 10^{-n}) -9*( \Sigma_{n=1}^{\infty}10^{-n}) =
(9-9)* (\Sigma_{n=1}^{\infty} 10^{-n}) = 0

Are you able to comprehend simple arithmetic? Now why don't you find some other misconception to share with us? Clearly 1-1=0 there is no doubt, except in your system which seems to lead you to this result. I strongly object to you attempting to tell us about the results of a system you do not understand. I will not argue the results of YOUR "system" since it is nonsense I expect nothing from it. I will continue to correct you when you misrepresent the results of standard Real Analysis.

 

[Hurkyl]

fact of the matter is you're so scared i might be right that you're unwilling to apply any effort at all to understand me.

Is that why you don't spend effort to understand the standard mathematical ideas?

numbers are just tools for describing reality.

No. Numbers are what is defined by mathematical definitions and/or axioms. Whether numbers are capable of describing reality is another question all together.

my "system" irons out a lot of kinks

Since "kink" here means "disagrees with ram's intuition", this justification doesn't particularly motivate me.

replaces a lot of false notions and assumptions with perfect logical ones.

Actually, it seems the other way around to me. Through the rigorous application of logic, I can get from the axioms to any of the statements I've made about the real numbers.

Whereas all of your arguments are simply your intuition (which the rest of us obviously don't share). You've axiomized your system enough to prove that ∞(d) - ∞(d) = 0, and that ∞(d+1) - ∞(d) = 1... (in particular, I think those were the only axioms you presented) but you haven't even said what d is or can be... and this is a far cry from the things you are asserting about your numbers.

 

[matt grime]

to echo hurkyl, all you#'ve done is state that you have a symbol that you claim is infinity, though you don't explain what it represents, or does beyond claiming it is the 'number of nines' in 0.99... (which is thus a cardinal, though why you won't accept that is a mystery)., that you can manipulate like a real number, that is you've defined an extension R[k] by adjoining k (picking a letter at random), an indeterminate, and clamining that it k is 'infinity' without explaining what that means. in what sense is k infinity, and what does the arithmetic of it mean.

 

[hello3719]

.999~ - .999~ = indeterminate then?

oh apples can't be 9's? why can't they be. numbers are just tools for describing reality. if i substitute an apple for every 9 there's no failure in logic, just the contradiction you happily supplied

k then substitute each 9 for an apple.

i see where your intuition is leading, you mean to say to me that

0.infinity - 0.infinity is indeterminate which is true in this form

( i used ridiculous notations only to satisfy your intuition)


do you know what indeterminate means? It means that we can only extrapolate the real value by means of other informations. the form in itself doesn't give ENOUGH INFORMATION BUT CAN ADMIT ANY NUMBER. The TRUE SOLUTION depends on the concept. When replacing each 9 by an apple we TOOK OUT INFORMATION

here we know that

0.infinity is a REAL NUMBER since it is between 0 and 2 , right?
We also know that we replaced .999... with 0.infnity
then IN THIS CASE the form 0.infinity - 0.infinity = 0 by properties of real numbers and the fact both represent the same number.
every number can be a solution to an indeterminate form so there is no contradiction in our definitions.
The information "REAL NUMBER" will eliminate the problem of indetermination in our present case and will give us THE ONLY CORRECT ANSWER which is 0.

You know that you can even call 4 - 3 indeterminate
but when you use the fact that both are real numbers and use their main properties you will obviously say that 4-3 = 1 ,so being indeterminate isn't at all a problem

when we say that infinity - infinity is indeterminate it is because it can yield any number and + or - infinity.
if we take the variable x and y representing numbers , we can also say that x - y is indeterminate since we don't have enough information to solve it, but is solvable if you have both values of x and y. same idea with the treatment of infinity. Hope it makes things more clear for your intuition.

 

[matt grime]

and another way of thinking:

0.9<0.99<0.999<0.9999<...<0.999~<1

is that true in ram's new world? If so, then 0<1-0.999... <1/10^n for all n, how can that be if the difference isn't zero? surely, gicen any number T, there is an n such that 10^n>T? let T be the reciprocal of 1-0.9999..., which you're claiming isn't zero.