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

[ram2048]

Windows calculator has a scientific mode!

by JOVE it freaking does.

omg omg <thanks>

 

[hello3719]

YOU PEOPLE are saying 9 = 10. in order for .999~ to be = to 1, there MUST exist a digit that is 10. since we're using base 10, a DIGIT of 10 cannot exist so that means you're wrong from the start.

what do you mean by a digit that is 10 ?
we can just have 9.999~ = 10 if you meant number
and for
1 * 0 = 0
2*0 = 0
...
will you also change the definition of zero in your system since you think that

infinity *1 = inifinity
infinity * 2 = infinity
...
is illogical?

 

[ram2048]

i might...

0^0 = 1?

hmmm... i can't even picture that but i guess it's necessary for some calculations...

we'll see

 

[Zurtex]

lol, now your threatening bodily harm because of your lack of understanding of mathematics. Basically you can even see flaws in your own argument but you seem unwilling to admit them to yourself. Anyway:

0^0 \neq 1

Can you think why?

Edit: I could be wrong on that actually, I would be interested to see if I am right.

I am fairly sure that 0^0 is undefined. But Microsoft calculator seems to disagree with me

 

[matt grime]

come one ram, you could at least pretend to explain what it is that you think that the number 0.999... plus another 9 after all of those is. because at the moment you've not managed to provide one single example of what this real number allegedly is.

Note that decimals are not exact representatives of real numbers, that is not any of the definitions of real numbers that works, as you've noticed, now why don't you look up the words cauchy sequence equivalence and class, or pssoibly dedekind and cut.


point out one sigle example where anyone but you has stated that they think 9=10. it would appear you don't understand the counter arguments to your system.

you also seem to have not noticed that we've otld you about systems of infinitesimals where there are 'numbers' smaller than 1/n for all n, and ordinals where there are infinite ordinals such as w, with w and w+1 not equal. one presumes this is because you think you are talking about real numbers alone, so let's do some analysis, prove that x^2 is a continuous function in your system. you can't as it happens, since 1.n doesn't tend to zero in your system, or it can't if you'r being consistent.

 

[Hurkyl]

In the real numbers, 0^0 is left undefined becuase no matter how you tried to define it, exponentiation would be discontinuous there. (proof based on the fact 0^x = 0 but x^0 = 1 for all positive x)

It tends to be convenient to adopt the convention that 0^0 = 1 for some applications.

 

[Zurtex]

In the real numbers, 0^0 is left undefined becuase no matter how you tried to define it, exponentiation would be discontinuous there. (proof based on the fact 0^x = 0 but x^0 = 1 for all positive x)

It tends to be convenient to adopt the convention that 0^0 = 1 for some applications.

Thanks, thought so

 

[ram2048]

Matt why do you keep hounding me about the 9?

my system works perfectly. in fact i have used your accepted conventions to prove my system.

consider x=.999~ where x possesses infinite number of digits 9.

Z) 10x - x = 9.999~ - .999~ = 9 in your system.

see:

.999~ infinite 9's
9.999~ infinite 9's +1

this is PROVEN TRUE if you accept the statement claimed in line Z) as a consequence of that calculation.

the logical conclusion to that is that in position f(n) where n=infinty+1 there existed a digit 9. (which would be false because we defined .999~ to have exactly infinite 9's but whatever)

if you accept line Z) you prove my system works for digits beyond infinity. you're also accepting infinity-infinity=0.

and every time you say .999~ = 1 you're saying 9 = 10 since in the expression .999~ there only exists digits 9. to be equal to 1, one of those 9's MUST BE a 10 somewhere in that infinite number of 9's.

in base 10 you don't HAVE digits of 10. you have 0-9. so you're wrong to even assume there might be.

 

[hello3719]

who said infinity - infnity can't be equal to 0 ? Did our system imply that?

 

[ram2048]

your system doesn't define a way for infinity - infinity to be 0. not that it couldn't be, just that it was never looked at.

my system DOES define relations amongst equal and non-equal infinities.

but that's beside the point. in THIS case you'd be accepting equal infinities to cancel out. thusly you'd be proving my point about the existence of the digit 9 in position f(n) n=infinity+1

 

[hello3719]

your system doesn't define a way for infinity - infinity to be 0. not that it couldn't be, just that it was never looked at.

in our system : lim as x->infinity of (x-x) is of type "infinity - infinity " and it does equal 0 in this case. Never looked at? very funny it seems that you didn't even take a calculus course.You don't even understand our system that's why you are trying to build a new one.

in THIS case you'd be accepting equal infinities to cancel out. thusly you'd be proving my point about the existence of the digit 9 in position f(n) n=infinity+1

IF A implies B, it doesn't mean that B implies A, your logic is flawed because of this. So we wouldn't be proving anything about your nonsense.

 

[matt grime]

i keep hounduing you about the nines because you've not explained what on Earth you mean by 'adding a 9' after the infinitely many that are there, this is the same as your intent that we ''create" a zero at the end of some string after multiplication by 10.

look back at your post 191 where you say you want to add another 9 after all of them to get a number even closer to 1, but we won't "allow" it.

 

[matt grime]

Matt why do you keep hounding me about the 9?

my system works perfectly. in fact i have used your accepted conventions to prove my system.

consider x=.999~ where x possesses infinite number of digits 9.

Z) 10x - x = 9.999~ - .999~ = 9 in your system.

see:

.999~ infinite 9's
9.999~ infinite 9's +1

this is PROVEN TRUE if you accept the statement claimed in line Z) as a consequence of that calculation.

the logical conclusion to that is that in position f(n) where n=infinty+1 there existed a digit 9. (which would be false because we defined .999~ to have exactly infinite 9's but whatever)

if you accept line Z) you prove my system works for digits beyond infinity. you're also accepting infinity-infinity=0.

and every time you say .999~ = 1 you're saying 9 = 10 since in the expression .999~ there only exists digits 9. to be equal to 1, one of those 9's MUST BE a 10 somewhere in that infinite number of 9's.

in base 10 you don't HAVE digits of 10. you have 0-9. so you're wrong to even assume there might be.

you are saying that infinity is "equal" to the "number" of digits after the decimal point in the expansion of .999..., so it is a cardinal, define the arithmetic of your cardinals then. (it can be done). for instance, take 0.99... and 0.88888... they have both an infinite number of digits, you "infinity", now i interleave them 0.989898... so i must have added the infinities together!, yet it must also be true that there are the same number digits, thus 2*infinity=infinity. so where's the arithemetic wrong there.

now, can you even remotely rigorously prove that 0.999=1 implies 9=10 using the proper definitions of addition and such? i can't see where you've done that, in fact you haven't, but then you've never even begun to understand the definition of the REAL numbers.

 

[ram2048]

you are saying that infinity is "equal" to the "number" of digits after the decimal point in the expansion of .999..., so it is a cardinal, define the arithmetic of your cardinals then. (it can be done). for instance, take 0.99... and 0.88888... they have both an infinite number of digits, you "infinity", now i interleave them 0.989898... so i must have added the infinities together!, yet it must also be true that there are the same number digits, thus 2*infinity=infinity. so where's the arithemetic wrong there.

that's absolutely wrong.

were you to take .999~ with exactly infinite 9's and .888~ with exactly infinite 8's and "interleave" them to where f(-1) is 9 f(-2) is 8 f(-3) is 9 etc each time pulling from the available 9's and 8's given, you would come up with 2x(original infinity) number of digits, BUT the sum total of those digits does NOT equal (original infinity).

yet it must also be true that there are the same number digits

very very wrong. any given infinity is equal to itself not equal to other infinities unless you clearly define the relationship beforehand, as i have been doing with digits and integers.

every time you bring something else it just gives me more and more reasons why my system is superior to the current.

in THIS case you'd be accepting equal infinities to cancel out. thusly you'd be proving my point about the existence of the digit 9 in position f(n) n=infinity+1

IF A implies B, it doesn't mean that B implies A, your logic is flawed because of this. So we wouldn't be proving anything about your nonsense.

If you're accepting that a 9 DOES exist BEYOND the number of infinite digits set forth in the initial expression such that you can multiply by 10 and a new digit 9 is brought "into play" to make the cancelling of 9's ABSOLUTELY PERFECT, then you're ACCEPTING the existence of f(n) n=infinity+1

this is a flat out consequence of YOUR logic.

that's NOT what my system believes in, it's just my system's way of interpreting the actions of your system. If you have a better way of describing to me HOW an EXACTLY infinite number of digits 9 can be multiplied by 10 and another 9 is "created" to maintain the equality you're welcome to explain it to me.

 

[matt grime]

so you are insisting that there is an infinite plus one spot in a decimal expansion which implies the existence of an infinite'th spot. none of those things exists.

we do not create any more nines who on Earth except you thinks we do? your intuition is completely wrong. stick to things you can prove.

what does the 9 in this alleged infinite'th place signify? there is no such place, stop pretending that we think there is.


but the string .98989898... must hve exactly 'infinity' digits in it - how can you dsitinguish between it and the one where i interleave two strings .9999.,.. and .888...?

so far all you've shown is that you do not understand mathematics and that you are incapable of defining a consistent notation.

 

[hello3719]

that's NOT what my system believes in, it's just my system's way of interpreting the actions of your system. If you have a better way of describing to me HOW an EXACTLY infinite number of digits 9 can be multiplied by 10 and another 9 is "created" to maintain the equality you're welcome to explain it to me.

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?

 

[matt grime]

so we've ascertained that your infinity is a cardinal, which you insisted it wasn't.

please offer formal statements about when two cardinals are equal.

 

[ram2048]

my infinity is NOT cardinal

my "default infinity" can be cardinal. it can be many things because it is a tool that i define at the onset of calculations and extrapolate meanings from that point onwards.

without definition "default infinity" has no meaning whatsoever.

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?

well then explain your infinite apples "getting eaten" such that EXACTLY none are left in 999~ - .999~. By your accounting "a never ending supply of apples" should still be "a never ending supply of apples" regardless of eating "an infinite number of apples" from that amount.

so therefore:
.999~ - .999~ = .999~ (your logic)

it is clear that your infinity cannot handle such contradictions. i don't know why you cling to it so desperately.

 

[Integral]

my infinity is NOT cardinal

my "default infinity" can be cardinal. it can be many things because it is a tool that i define at the onset of calculations and extrapolate meanings from that point onwards.

without definition "default infinity" has no meaning whatsoever.



well then explain your infinite apples "getting eaten" such that EXACTLY none are left in 999~ - .999~. By your accounting "a never ending supply of apples" should still be "a never ending supply of apples" regardless of eating "an infinite number of apples" from that amount.

so therefore:
.999~ - .999~ = .999~ (your logic)

it is clear that your infinity cannot handle such contradictions. i don't know why you cling to it so desperately.

Pure and simple nonsense.

.999... - .999... = 1-1=0

Apparently you are unable to differentiate between an infinite number of digits and a quantity of infinite magnitude. All real numbers have an infinite number of digits, no real number is infinite in magnitude.

 

[hello3719]

yea it seems he thinks that .999... is infinite LOL.

 

[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 &infin;(d) - &infin;(d) = 0, and that &infin;(d+1) - &infin;(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.