Does .999=1?

[Solo]

In my algebra 2 class a kid was saying it does. If I remeber correctly he said something about the fraction 1/3. I wasn't paying much attention because I was doing the assignment but I find the question quite intresting. I was hoping some of you math wizards could clear this up for me.

[Claverhouse]

No, not even infinitely recurring.


And I know nothing about math.



Claverhouse :ph34r:

[Vicideus]

Probably meant 1/3 = .333... and so .999... would logically be 3 * .333... or 3 * 1/3.

3 * 1/3 of course is 3/3, or 1... so .999 = 1

Im sure someone else could explain this more elequently, but I've gotta go do me an impressionist painting using triadic colors...

[Claverhouse]

3 times .333 is .999.
3 times .333333333333333 is .999999999999999.

Especially if you use a calculator: they can't do fractions mathematically.



Claverhouse :ph34r:

[SheepDog]

I've seen that trick before, Vicideus, and I think you're right that it was the basis for what was being claimed. I'm assuming that your ellipses indicate the digits repeating to infinity, since the keyboard doesn't have the symbol for a bar above the digit.

[Melody]

to repeat what has already been said

http://img.photobucket.com/albums/v259/tocca/mathy2.png




http://img.photobucket.com/albums/v259/tocca/mathy1.png

:rofl:

[Avengardh]

not, it equals .999

[file cabinet]

1/3 .. is that equal to .33333 with the little line above the last 3? if so.. .999 is = .999 (with the little line over the last night, I forget what that shit is called).

[Claverhouse]

Melody, ( & Vicideus )

When you are next hiding up from the Mob, desperately accumulating the exact $1 million you owe for assorted gambling debts, contracts you have placed on credit, and bulk shipments of avocados: don't offer them $999,000. They will take it gladly, but the debt will still run.



Claverhouse :ph34r:

[Almaviva]

Math is all about definitions, and if you're not careful with the definitions, you can get into trouble. And so you have to define exactly what you mean by "0.9999....", and this isn't that easy a thing to do.

But for many reasons, in a sensible number system, 0.9999.... needs to equal 1.

Here are some intuitive arguments:

1:
1/3 = 0.333...
(Multiply both sides by 3)
3/3 = 0.999...

2:
Let x = 0.999....
Then 10x = 9.999...
Subtract first line from second, we get 9x = 9.00000....
So x=1

3:
1 - 0.999... = 0.0000... = 0
If they weren't equal, we'd have two non-equal numbers with a difference equal to zero, which doesn't make sense.

[booyalab]

This is a dumb question. Sure for practical purposes .999....<----emphasis on repeating digits.. needs to equal 1 sometimes. But the title of the thread doesn't have an ellipsis. It looks like we need a supplemental thread to decide if .999=.999...
which it doesn't...obviously, so please make up your mind as to whether you're asking .999...=1 and in what circumstances ,or is it .999=1.

Have you learned about sig figs in your class yet? If you have, the answer is no.....if there isn't supposed to be an ellipsis

[flan2dave]

The context of the question indicates solo meant to have the ellipses. Nobody would ask if 9/10ths is equal to 1 (but perhaps Claverhouse would expect such a question to arise).

[Claverhouse]

No, not even recurring to infinity. Practical purposes be damned: logically, so long as even the minutest fraction is missing then it ( or we ) will be incomplete. Flawed, if you like.

Only by being absolutist can we find absolution.

Claverhouse :ph34r:

[Boozer]

.999... is not equal to 1

The problem is the calculator, you give it 1/3 and it thinks .33333333333... but that isn't right, it's just a very close approximation. multiply 3 approximation's together and you just get a worse approximation, hence .999999....

This whole question just shows the limitations of calculators/computers.

[Star Cannon]

That's like saying five ducks equals four.

In math, if it isn't the number in question... it is not the number in question.

[Melody]

=p u ppl ignored my beautiful equations

Starting with
http://img.photobucket.com/albums/v259/tocca/mathisdave1.png

the problem can be looked at in a few different ways

when looking at the original sum, it is as if we are starting with .9 and adding .09 and then adding .009 and so on. However, in the form

s=1-1/10^N,

it is saying something more like, "start with 1. Then subtract .001"

so like if N = infinity, it is like saying, "start with 1. Then subtract ... well, when you get to it, subtract 1/10^N. Because it is infinity, it will never get to it.

This is really the key. In this context, infinity is not really a number, it is something else...

This tomfoolery is one of the reasons why I do not trust mathematics. ^.^

In any case, a real mathematician will agree that .9999... = 1 and will show you the proof/s. This stuff I did up there is not convincing because we use infinity (by saying N = infinity) to define something that is infinite. Kinda circular because we have to then prove that 1/10^N is zero as N aproaches infinity, and this starts like another thing. In the end, I guess it just goes on what makes the mathematics of infinity most reasonable in our eyes, which is again one of the reasons I do nut trust mathematics. ^.^

[Melody]

u guys will notice that I did the same thing Almaviva did in their second example

just written differently


...and the equation from their third example is the same as s = 1-1/10^N

just written differently


...and also they pointed out that math is all about definitions, i.e. 'most reasonable in our eyes'

lol

[Hypnos]

The rationals are not Cauchy complete like the reals.

[Melody]

what does that mean?

u are saying that a rational sequence does not converge?

my math is not good cuz i trust it so little...i.e. i dunno what u mean

[Hypnos]

what does that mean?

u are saying that a rational sequence does not converge?

my math is not good cuz i trust it so little...i.e. i dunno what u mean
Roughly, yes:

http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node11.html

[Melody]

ah i see

it says rational numbers are not Cauchy complete because of the irrational numbers

but if we say we are not using the rational numbers and are instead using the real numbers (which is supposedly cauchy complete and includes the rationals,) that sounds gooder to me...

[Melody]

although they are considering the sequence as the set

so like the sequence does not converge to a member of itself...or...


@_@

*drives his VW off a cliff*

[Hypnos]

although they are considering the sequence as the set

so like the sequence does not converge to a member of itself...or...

Exactly!

[Melody]

aha! im a guyneuss

[candela]

At first I was shocked this was up for debate on an INTP forum.

But after reading this thread just before posting, it appears the debate is over whether he meant .999 or .9 repeating. Ha ha. That's awesome.

[Solo]

I did mean .999 infinetly repeating.

[Almaviva]

=p u ppl ignored my beautiful equations


No, your equations are great.



the problem can be looked at in a few different ways

when looking at the original sum, it is as if we are starting with .9 and adding .09 and then adding .009 and so on. However, in the form

s=1-1/10^N,

it is saying something more like, "start with 1. Then subtract .001"

so like if N = infinity, it is like saying, "start with 1. Then subtract ... well, when you get to it, subtract 1/10^N. Because it is infinity, it will never get to it.

This is really the key. In this context, infinity is not really a number, it is something else...

This tomfoolery is one of the reasons why I do not trust mathematics. ^.^


There's nothing to "trust" or not, just definitions.

Definition: The limit of a sequence converges to n, if, no matter how close we'd like to get, we can pick an N large enough that *all* the terms past N are at least that close to n.

Definition: By 0.9999..., what we mean is the limit of the sequence of rational numbers: 0, 0.9, 0.99, 0.999, ...

Similarily, by 1.0000 we mean the limit of the sequence of numbers 1, 1.0, 1.00, 1.000, ...

It happens that these limits are the same, and they converge to 1. This is how the decimal expansion for a number is defined (as a limit).

If you'd like to use different definitions, you can, but that's the standard way of doing things.

Going further:

Definition: A sequence of numbers is "Cauchy" if, no matter how close together I want them to get, I can pick N so that any two terms past the Nth term will be within that distance. (This is related to the idea of a limit, but not exactly.)

Definition: A "real number" is a Cauchy sequence of rationals. Two sequences are the same number if the limit of (an-bn) = 0. (This is the idea of an equivalence class.)

This is one way to construct the real numbers. Another way is to split the rationals into two subsets sets A and B, (a "Dedekind Cut") where everything in A is less than everything in B. (Example, A contains negative numbers, and all numbers so that x^2 < 2.)

A property of real numbers is that all Cauchy sequences converge to a real number. Rational numbers don't have this property. (3, 3.1, 3.14, 3.141, 3.1415, ...) does not converget to any rational number.



In any case, a real mathematician will agree that .9999... = 1 and will show you the proof/s. This stuff I did up there is not convincing because we use infinity (by saying N = infinity) to define something that is infinite. Kinda circular because we have to then prove that 1/10^N is zero as N aproaches infinity, and this starts like another thing. In the end, I guess it just goes on what makes the mathematics of infinity most reasonable in our eyes, which is again one of the reasons I do nut trust mathematics. ^.^

You've got a couple of choices:

1. Talk in the language of conventional mathematics. Meaning, don't say things like "let N=infinity", and use the idea of limits.
2. Use some form of non-standard analysis, where infinite numbers exist, and you can have numbers that are infinitely small, but still non-zero. This can actually be done, but you have to be careful, since some of the conclusions you can get are counter-intuititive. This can be done in a consistent way though.

[Xenophon]

this is just a simple question of limits.

as you keep adding 9's to the end, the number will keep gettin glarger and large. The LIMIT to how large the number will get is 1, infinity is just an artificial construction to say, keep addin ghtem on until we get close enough to the limit that we can consider it to be the same thing.

If you are questioning whether limits exist? well, calculus wouldn't work at all without them, and that has become a fairly popular form of math over the past 400 years or something.

[booyalab]

The context of the question indicates solo meant to have the ellipses. Nobody would ask if 9/10ths is equal to 1 (but perhaps Claverhouse would expect such a question to arise).

The most important thing is whether the instructor wrote it with an ellipsis, not if everyone else assumes it does. That's why I needed clarification

[booyalab]

I did mean .999 infinetly repeating.
ok...remember to indicate that. It's important for clarity and getting a good grade in that part of the class, even if it isn't important to you.

[Melody]

If you are questioning whether limits exist? well, calculus wouldn't work at all without them, and that has become a fairly popular form of math over the past 400 years or something.

i dont trust limits...

maybe i should just make my own math

itll be like

pnly prime #s

and from rotational application of these

all other "numbers"* arise

* "numbers" because only prime numbers exist

:rofl:

[Melody]

just watch this idea is whats gonna make me as popular as einstein

im such a genus

[Aryan]

OK so here it is the most important paper of the 21st century ^.^

let p=.999999999999999999...
=> 10p=9.99999999999999999...

=> 10p - p = 9.99999999999999999999... - 9.9999999999999999999...
=> 9p = 9
=> p = 1

Simple arithmetic here

[Melody]

that was already done on this topic twice :P

[jimkopelli]

I'm gonna be a punk, and say this...
.999... = .999...
and
1 = 1.

Granted, .999... is really really really close to 1, but why say 1 if you mean .999... ?

[Vicideus]

OK so here it is the most important paper of the 21st century ^.^

let p=.999999999999999999...
=> 10p=9.99999999999999999...

=> 10p - p = 9.99999999999999999999... - 9.9999999999999999999...
=> 9p = 9
=> p = 1

Simple arithmetic here

Uhh, that equation is wrong...

I can't be the only INTP who noticed that misplaced decimal point.

[MacGuffin]

Since when do you define something (p) and then solve for it?

[Melody]

Since when do you define something (p) and then solve for it?
it is defined as a series first
then the series is evaluated

we're not changing the value of anything
it is just written differently

for example

x = 5*2

evaluating it, we get

x = 10

[Melody]

I'm gonna be a punk, and say this...
.999... = .999...
and
1 = 1.

Granted, .999... is really really really close to 1, but why say 1 if you mean .999... ?

the problem can be looked at in a few different ways

when looking at the original sum, it is as if we are starting with .9 and adding .09 and then adding .009 and so on. However, in the form

s=1-1/10^N,

it is saying something more like, "start with 1. Then subtract .001"

so like if N = infinity, it is like saying, "start with 1. Then subtract ... well, when you get to it, subtract 1/10^N. Because it is infinity, it will never get to it.

This is really the key. In this context, infinity is not really a number, it is something else...

infinity is not usually considered a number in any case

i think

[Almaviva]

I'm gonna be a punk, and say this...
.999... = .999...
and
1 = 1.

Granted, .999... is really really really close to 1, but why say 1 if you mean .999... ?

It's not just really really close to 1, the difference is in fact 0.

If you want a number system where 0.9999... is not identically equal to 1, then you lose the property that X=Y is equivalent to X-Y=0, which is pretty fundamental.

(If I can't add something to both sides of an equation and have it still work, then my math is pretty messed up.)

(Unless you want to say 1-0.9999... is actually a "number" infinitely close to zero which has no decimal expansion. You actually can do this and have the math work out.)

[candela]

Since when do you define something (p) and then solve for it?He didn't solve for p really, he solved .9 repeating. It's not a variable or anything. He just used a letter so it's less confusing. You could do it like:
.9... = .9...
10 * .9... = 9.9...
10 * .9... - .9... = 9.9... - .9...
9 * .9... = 9
.9... = 1

edit: I thought the subtraction step looked awkward this way so I didn't simplify it at first.

[paladinoflunaria]

I'm surprised that someone even asked this question. I'm further surprised that after comments made by Melody and Hypnos, the thread continued (and yet more so that the continuation was either chatter or denial). Oh well.

[Almaviva]

I have no trouble accepting that 0.999... = 1. It's really just a statement about a quirk in the way we represent numbers with digits, and not anything profound beyond that.

However, the idea that 0.999.... < 1 is a false statement still bugs me a bit:)

[Hypnos]

I'm surprised that someone even asked this question. I'm further surprised that after comments made by Melody and Hypnos, the thread continued (and yet more so that the continuation was either chatter or denial). Oh well.
Hey, you know INTPs -- they are so wrapped up in their own musings that the right answer blows right by them.

Leave the creativity for the questions that don't have answers yet.

[Chill]

I think the real problem is that we don't have the approXimate symbol on easy access on the key board. because .999 is approx. 1 and often that they are close enough that they can be used interchangeably. Kinda like the 3.14 = pi or 22/7 = pi. It's approximately true, so we use it for simplicities sake.

[Aryan]

OOPs!!!
Errata:



=> 10p - p = 9.99999999999999999999... - 9.9999999999999999999...


That should be
=> 10p - p = 9.99999999999999999999... - 0.99999999999999999999...

Arrghh... i need to be careful while typing
Should have practised writing 0 before a decimal


I can't be the only INTP who noticed that misplaced decimal point.

Thanx bro, but u r not the only INTP here ;P

[MacGuffin]

Since when do you define something (p) and then solve for it?
it is defined as a series first
then the series is evaluated

we're not changing the value of anything
it is just written differently

for example

x = 5*2

evaluating it, we get

x = 10

He didn't do that. p = 0.9999999.....

There is nothing to evaluate.

[Melody]

its a series

http://img.photobucket.com/albums/v259/tocca/mathy1.png

when u evaluate the left side, which is just another way of writing .9999999...; u get the right side ( see http://www.intpcentral.com/forums/viewtopic.php?t=1135&postdays=0&postorder=asc&start=15 for specifics )

if ppl r confused it is because they arnt familiar with sigma notation

http://mathworld.wolfram.com/Sum.html

in other words, p = .99999999... is an infinite sum, and sums can be evaluated like so

x = 5+2

x = 7

[cuspuser]

Well, really it depends on what u're using it for ... say i try to put in .999 infinitely repeating into my spreadsheet with rounding off - all i get is # if i leave room for one digit ... meanwhile if there is 1 I get to see it ... its really a matter of use ... say u're measuring a length of wood, .999 repeated is good enough to be 1 ... i'd also be hard pressed to get a .999999999999 cent bill/coin ...

All the math stuff just confuses me :) (actually i got the jist which surprised the hell out of me)

[purple13]

Nights in white satin... never reaching the end.

[candela]

Melody is right, .9 repeating is just a series. Anytime you have a decimal repeating it's a series meant to represent a number that cannot be written in decimal form.

For any number system though, it doesn't make sense to have the highest value digit repeating. This topic is kind of stupid.

[Almaviva]

Is 1/3 = 0.33333....?

If you're going to say that 0.999... is not 1, then I think the answer has to be no.

Because:
0.3 < 1/3
0.33 < 1/3
0.333 < 1/3
(and so on)

Similarly, 3.14159... < Pi. Because at every step in the expansion, the partial fraction is less than Pi's true value.

So what you'd be left with is a number system where a decimal expansion can seldom equal anything:)

(At this point, the easiest thing to do is just accept that the above makes things rather over-complicated, and just live with the idea in the real numbers that if something is infinitessimally small, then it is equal to 0, and therefore 0.999... = 1.

Again, you can either do that or get tenure as a non-standard analyst.)

[MacGuffin]

I don't think 0.3333... is 1/3. That is just an approximation.

[Melody]

whatever u say ^_^

[jimkopelli]

Melody, you can't change your avatar!
If you have to change it, could you just draw a new one, but with the same theme?

[HairlessBluetick]

I think the real problem is that we don't have the approXimate symbol on easy access on the key board. because .999 is approx. 1 and often that they are close enough that they can be used interchangeably. Kinda like the 3.14 = pi or 22/7 = pi. It's approximately true, so we use it for simplicities sake.

What about ~?

[KoDeN]

Well, there is always the age old paradox of needing to go through an infinite amount of distances in order to get somewhere. This concludes one of two things, either we never get anywhere, or that the universe does not have an infinite number of degrees. I believe the latter. I believe that it is pure imagination that catapults math and numbers into the infinitesimle. I believe that there is a basic structure to which physical matter is limited too. Which would eliminate alot of these impossibilities.

[jimkopelli]

Ok then... find that limit.
We aren't really discussing thirds of physical objects or distances... and we can split numbers as thin as we want, and thinner besides. Pure nmbers don't have to obey little things like reality, they can be as detailed as they want.

[KoDeN]

numbers are simply symbols and ideas that we use to represent groups or values. ie there is no 1. 1 can mean anything, 1+1 = 2, but what is 2 in that equation can be 1 in another. The problem we have is that our imagination sometimes goes places where reality does not.

[Sackanaka]

Sorry to interrupt, but just wanted to ask how many others have been both intrigued at this subject but quickly lost interest in hearing the argument? I think everything from "proving" it works to how it doesn't work has been said, and nothing's been agreed upon. Then again, it is more interesting than a typical club/board meeting.

[KoDeN]

you know, in my experiece, if you can settle a discussion, you aren't looking into it deep enough.

[jimkopelli]

That depends.
"What should we have for lunch?"
"I could do with a sandwich."
"Sounds like a plan."

Some things don't really need depth gone into overmuch.

[KoDeN]

first of all what kind of sandwich?

[SheepDog]

See, you're the picky one. That means YOU should decide what to have for lunch.

[Crazy]

for my tow cents worth (cause I just can't resist)

1/3 does not equal .333....

but .333... can be used as an approximation of 1/3 in certain cases where the exactness of that particular # is irrelevant.

Therefore, 3*.333... does not equal 3*1/3
Therefore, .999... does not equal 1

however, 1 can be used in the place of .999.... for the sake of simplicity when the degree of exactness is irrelevant.


and furthermore, Tuna on toasted sourdough

[Crazy]

oh, and in relation to the original arguement about .999... and the other remark about decimal places being in the right spot, didn't anyone see the movie "Office Space"?

[candela]

for my tow cents worth (cause I just can't resist)

1/3 does not equal .333....

but .333... can be used as an approximation of 1/3 in certain cases where the exactness of that particular # is irrelevant.

Therefore, 3*.333... does not equal 3*1/3
Therefore, .999... does not equal 1

however, 1 can be used in the place of .999.... for the sake of simplicity when the degree of exactness is irrelevant.


and furthermore, Tuna on toasted sourdoughWHAT.

[jimkopelli]

oh, and in relation to the original arguement about .999... and the other remark about decimal places being in the right spot, didn't anyone see the movie "Office Space"?
Didja get the memo?

[Almaviva]

1/3 does not equal .333....
how so

It's so if either you're not using the real numbers, but instead using something like infinitessimals and backing up your argument or... just using some intuitive arguments without knowing precisely what you're doing or talking about. (I.e. not doing math. Not that doing things beside math isn't fun.)

With the real numbers, it's an assumption that any bounded sequence has a least upper bound. (There are other ways to define the real numbers, but they all imply this.)

(This is how math works. You assume things. If you reach a contradiction, you've assumed too much. If you can show one assumption is implied by another, you can get rid of it. There's no "I feel this is right.")

We see that 0.3, 0.33, 0.333, 0.3333, ... is bounded (it's never bigger than 4), so we know there's a least upper bound in the real numbers (by assumption). In this case, the least upper bound is the same as the limit. (There's a definition for what limit is too, usually with deltas and epsilons.)

1/3-0.3 = 1/3-3/10 = 10/30-9/30 = 1/30
1/3-0.33 = 1/3-33/100 = 100/300-99/300 = 1/300
1/3-0.333 = 1/3000
...

We see that the limit of (1/3-0.3, 1/3-0.33, 1/3-0.333, ...) is 0. We can prove that the limit of a difference has to be the difference of the limits, so then we have shown that the limit of 0.3, 0.33, 0.333, ... is 1/3.

And limits are how we define the decimal expansion.

You don't often see "0.999... != 1"-ers define exactly what they mean by 0.999..., probably because if you know enough math to know that it's important to do so precisely, you realize that the whole line of discussion just isn't that interesting compared to what else is out there:)

[SheepDog]

Pi are round.

[jimkopelli]

Cobbler are square.
(mmm... blackberry...)

[Mealz]

do you have some pi for me?

[jimkopelli]

http://www.weebl.jolt.co.uk/pie.htm
Here ya go.

[Vylence]

Here is my proof that .9... repeating doesn't equal 1

Which also happens to be my proof that .7... repeating equals 1

NOTE we are doing our math in OCT today

x = .7...
10(x) = 7.7...
10x - 1x = 7.0
7x / 7 = 7.0 / 7
1x = 1

Now it should be fairly obvious that .9... is some problem with the decimal system. Or that any numbering system that shifts places or reuses symbols have flaws.

[Network Alchemy]

if you are working in octal of course .7777.... is 1 and if instead of writing .7777.... in octal the corresponding decimal form 7*(8^-1 + 8^-2 + 8^-3 + 8^-4 + ....) is used the summation is also 1 so there is no inconsistency and perhaps the difficulty is in confusing .7777.... with an octal 7*(8^-1 + 8^-2 + 8^-3 + 8^-4 + ....) with .7777.... with a decimal radix 7*(10^-1 + 10^-2 + 10^-3 + 10^-4 + ....) as they are of different values

[Vylence]

My point is that math doesn't always show the right answers.

[Almaviva]

I think math always shows the right answers (provided the math is correct, of course.)

It's the study of what patterns you can find, based on (and only on) what you define and assume to be true from the start. This is both its main strength and main limitation.

But when you're talking about math, you can say things with certainty. If you don't like the things that math says with certainty (say, that in the real numbers, the limit 0.9999... must equal one), then you can either change the definitions you want to use, or become a philosopher:)

(There's the claim that nearly all philosophical dilemas can be "resolved" by asking what do you mean by that.)

[Arcades]

I dont relly care about .999
I like .0(inf.)1 ___
The little candy coating between .333 x 3 and 1
The smallest possable number in existance.
that is truly a cool number.

[Vylence]

Math does always show the right answers from within the system. Though as a system to show reality it does not. Now where the .9... repeating could come up in reality, I have no clue. I just think that we should be aware that you can't always rely on math to properly describe truth, in the instance of reality. It might be true that there are no .9..., therefore math is describing an illusion.

Now Network there is no 8 in OCT. Maybe you are getting confused. You don't ever do math with 8 in OCT.

[Almaviva]

Math does not reflect reality, true. It has no pretenses of doing so. If you're trying to use math to model reality, you're now doing physics or something like that, and the idea of a "real number" is no longer relevant.

But, in response to your octal post, the system of representing numbers with digits doesn't have faults. It has limitations in that of course you can't actually write out an infinite sequence. But any real number can be written as a sequence of digits. It's not quite unique because of cases like 1 and 0.99999... which represent the same real number. Further, there is no real number between what would be represented by the infinite sequence 0.3333... and 1/3.

(What I mean by "real number" is a fairly advanced concept that few on this thread understand, but it is a precise concept nonetheless.)

[blue_eye]

Of course it doesn't equal 1 because there's a zero infront of it. Besides if it's like 1/3 or something, that means that 1/3 can't be satisfied and so we just keep adding a 9.

[Almaviva]

First, "of course" is probably not appropriate when there is nine pages of posts on it, and many thousands of posts on sci.math arguing about this.

Second, your idea of "equals" is pretty shallow if it depends only on the typographical representation. We'd like to say that 0.5 + 0.5 = 1, for instance.

In the sense of the real numbers (which are defined precisely) and equality (which is also definined precisely) and the idea of the decimal representation as a limit, 0.999... is equal to one, and there is no ambiguity about it. You can challenge the definitions, but it's hard to do that without understanding the subtlety of real numbers. And once you do, the question isn't that interesting.

(Kind of like asking, is 1/2 + 1/2 REALLY 1?? Maybe there's something small lost when I break up 1 into two pieces and them put them back together...)

[Vylence]

It could be that some fractions in no way can ever be turned into a real number.

[Almaviva]

Any fraction is a rational number, since that is the definition of a rational number. Real numbers include the rational numbers, which also follows from their definition.

[Vylence]

If a real # is one that can be graphed, how do you graph .999... realistically? To what precision do you go? Would your number be off if we were to go to a higher precision? Would you graph it as 1? Would you graph .777... in OCT as 1?

[Almaviva]

A number isn't something that can be represented in the real world. That isn't what math is about.

So you can't "graph" 0.999... realistically any more than you can graph 2.

[indie]

I think the appropriate mathematical representation is .999 ≈ 1, but not "equal" to 1. :sombrero:

[Almaviva]

If you mean 0.999 without the ellipsis (...), then it's not equal, obviously.

With the elipsis, 0.999... IS most definitely equal to 1, in the usual sense of "real number". 0.999... is defined as a limit, and "equals" is defined under real numbers as a=b if (a-b), as a limit, goes to 0. And the limit of (1-0.9), (1-0.99), (1-0.999), ... does go to zero, in that eventually it will be less than any positive number. In math definitions are all that matters. It doesn't pre-suppose any connection with reality or intuition.

[Edmond Zedo]

1:
1/3 = 0.333...
(Multiply both sides by 3)
3/3 = 0.999...

False! Use logic.

If 1/3=x, then 3x=1.
If 1/3=0.333..., then 3*0.333...=1 (Not 0.999...!)

So firstly, the original question is flawed and therefore pointless. Secondly, 0.999... doesn't equal 1, to answer said question.

[Vylence]

I refute that 0.9... is a limit. It won't actually be less than any positive # either, considering I could take 0.f... which would be smaller.

The easier way of saying it is that if 1 and 0.9... are not equal there must be a difference between them.

If you subtract 0.9... from 1 you get 0.0..., or repeating zeros. Which means that you will always have a zero difference.

[Edmond Zedo]

It never equals 1, no matter what. If it equaled 1, it wouldn't be stated as 0.999..., but as "1."

Let me simplify. If you have a (what do you call it, it's been so long) "real" number, as a fraction, like 1/3, and you convert it to decimal, it gets ugly, but that's not important: 0.333...

If you want to convert the "fraction" 1/1 to decimal, it's not 0.999..., it's 1, or 1.0 if you like.

[Dman]

Forgive me for stating the obvious, but .999 does not equal 1. It equals .999. This is an amazingly long thread for such a question. You'd think a bunch of INTP's hung out here.

[Almaviva]

False! Use logic.

If 1/3=x, then 3x=1.
If 1/3=0.333..., then 3*0.333...=1 (Not 0.999...!)

So firstly, the original question is flawed and therefore pointless. Secondly, 0.999... doesn't equal 1, to answer said question.


First off, things with different typographical representation can be "equal" in the sense of mathematics. Otherwise math wouldn't be very useful!

For example, 0.5 + 0.5 = 1

Second, that heuristic example is based on wanting to be able to use the decimal expansion to do arithmetic.

3 * 0.3 = 0.9
3 * 0.33 = 0.99
3 * 0.333 = 0.999
(And so on.)

If 0.333... is defined as the limit of 0.3, 0.33, and so on, then we know that limit of a product is the product of the limits (this can be proven using the definition of a limit... Take a first year university calculus class for how.)

So this is what I mean when writing 3*0.333... = 0.999...

That three times the limit of (0, 0.3, 0.33, 0.333, ...) is equal to the limit of (0, 0.9, 0.99, 0.999...)



I refute that 0.9... is a limit. It won't actually be less than any positive # either, considering I could take 0.f... which would be smaller.


What do you mean by "refute"? If it's not a limit, what would you like it to be?

Are you saying it dosen't converge?

I didn't say that 0.999... is less than any positive number. I said that (1 - 0.999...) is less than any positive number.




The easier way of saying it is that if 1 and 0.9... are not equal there must be a difference between them.

If you subtract 0.9... from 1 you get 0.0..., or repeating zeros. Which means that you will always have a zero difference.


But I did say this! But I said it in terms of limits. I also said it at the beginning of the thread.



It never equals 1, no matter what. If it equaled 1, it wouldn't be stated as 0.999..., but as "1."

Let me simplify. If you have a (what do you call it, it's been so long) "real" number, as a fraction, like 1/3, and you convert it to decimal, it gets ugly, but that's not important: 0.333...

If you want to convert the "fraction" 1/1 to decimal, it's not 0.999..., it's 1, or 1.0 if you like.


Again, different statements of numbers can be equal in the sense of math! (Example: 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + ... = pi^2 / 6) Stuff like this is part of what makes math fun, actually.

The hang-up here is that the decimal representation of a real number is not always unique. The other hang-up is that the decimal representation is just a bunch of symbols representing sums of fractions of powers of ten. There's no magic or deeper meaning behind it than that.

Infinite geometric sequences can have a finite sum. This is a foundation for a lot of other math actually. A lot of the field of combinatorics is sort of built around the sum forumula 1 + r + r^2 + ... = 1/(1-r), for r < 1.

That's exactly what 0.999... is: an infinite geometric series. 9*1/10 + 9*(1/10^2) + 9*(1/10^3) + ... You can plug that into the formula S=a/(1-r), so a=9/10 (first term), r=1/10 (ratio), and a/1-r = (9/10)/(9/10) = 1.

Read Melody's posts if you like sigma notation for this, and really big, heh.

I find this thread therapeutic, by the way:)

[Dman]

0.999... is not equal to 1 and never will be. It can try to conform as much as it likes to be equal to 1, but no matter how hard it tries, it can never quite get there. It needs to accept what it is and get on with the rest of its life.

Think of it in terms of percentage. Check out this example: if 99.99999% (i.e. 0.9999999) of all US citizens were not space aliens, that means 29 citizens are space aliens. That is not the same as if 100% (i.e. 1) of people were not space aliens. Thus, .9999999 does not equal 1.

[Vylence]

I don't see 0.9... as a limit. Furthermore there are numbers smaller than 1 - 0.9..., to the tune of 1 - 0.f..., which is what I was trying to say.

[Almaviva]

Okay, if 0.999... is not a limit, then what the heck is it? How do you define it? As it goes, these problems tend to resolve themselves if we explain what we're talking about.

Conventionally, infinite decimal expansions express a real number which is the limit of the fractions of powers of 10. You could be using some definition of your own, but you have to explain what it is.

What is 0.f? Are you switching to hexadecimal here?



0.999... is not equal to 1 and never will be. It can try to conform as much as it likes to be equal to 1, but no matter how hard it tries, it can never quite get there. It needs to accept what it is and get on with the rest of its life.

Think of it in terms of percentage. Check out this example: if 99.99999% (i.e. 0.9999999) of all US citizens were not space aliens, that means 29 citizens are space aliens. That is not the same as if 100% (i.e. 1) of people were not space aliens. Thus, .9999999 does not equal 1.


No indeed it doesn't. And any finite fraction of 0.9999's isn't going to equal one either. But the "..." represents the idea of infinity. We *define* what we mean by an infinite series in terms of limits. That's what the symbols mean, in standard analysis. If you don't like what the symbols mean, you at least have to explain what you really do mean.

[Edmond Zedo]

Almaviva, I honestly don't know, because I know little about "higher" math, and have never been that fond of it, but I suspect you're intentionally over-complicating the issue as a sort of red herring. Again, I may be wrong!

I have to say, using only logic, I don't believe 1 - 0.999... is zero.

What would you say 1.999... divided by 2 is?

[Edmond Zedo]

0.999... is not equal to 1 and never will be. It can try to conform as much as it likes to be equal to 1, but no matter how hard it tries, it can never quite get there. It needs to accept what it is and get on with the rest of its life.

Think of it in terms of percentage. Check out this example: if 99.99999% (i.e. 0.9999999) of all US citizens were not space aliens, that means 29 citizens are space aliens. That is not the same as if 100% (i.e. 1) of people were not space aliens. Thus, .9999999 does not equal 1.

But if the remainder is infinitesmal, there's no room for a whole person.

[Almaviva]

Intentionally over-complicating things?? Who, me?:)

Are you sure you're using logic to reach your conclusion? To me, logic means you're explaining exactly what you mean, and every step is made using clear rules. This is what I'm attempting to do in this thread.

I'd say 1.999.../2 is 1. (Or 0.999... if you like, given that they represent the same thing.)

[Edmond Zedo]

Intentionally over-complicating things?? Who, me?:)

Are you sure you're using logic to reach your conclusion? To me, logic means you're explaining exactly what you mean, and every step is made using clear rules. This is what I'm attempting to do in this thread.

I'd say 1.999.../2 is 1. (Or 0.999... if you like, given that they represent the same thing.)

Let us define the word logic as "deductive reasoning," for that's how I meant it. Join me in the playground of pure reason for a moment.

x is approaching zero, So:
x is not zero.
1-x=0.999..., So:
x>0, So:
(1-x)<1
0.999...<1

[SheepDog]

Almaviva, what's the function of the limit you refer to when you ask, "if 0.999... is not a limit, then what the heck is it?" Doesn't a limit require a function? ;)

Goodness, this is rusty, and I'm attempting some mind-reading here, but I think the limit would be 1, for the series that you are imagining. The limit defines the value of the function as it approaches a value, but has nothing at all to say about the incident at the value.
<pulls out old calculus book>

[Almaviva]

Let us define the word logic as "deductive reasoning," for that's how I meant it. Join me in the playground of pure reason for a moment.

Okay.



x is approaching zero, So:


What do you mean by x? Is it a sequence now? Or a pure number? If it's a pure number, it isn't approaching anything, it has a value, and that's that.



x is not zero.


This is tangential, but because something approaches zero does not mean it is not zero. The sequence 0,0,0,0,0,... "approaches zero" in the mathematical sense, and it is zero. This isn't relevant here, though.



1-x=0.999..., So:


Again, what do you mean by x? What I meant by it was the limit of the sequence 1-0.9, 1-0.99, 1-0.999, 1-0.9999, and so on. This limit exists, and the limit is zero.

If a limit isn't what you mean, then again, exactly what do you mean?



x>0, So:


If we're saying x is a limit (which is what I'm saying) it's not greater than zero. It is zero.



(1-x)<1
0.999...<1




Almaviva, what's the function of the limit you refer to when you ask, "if 0.999... is not a limit, then what the heck is it?" Doesn't a limit require a function?


I'm talking about the sequence 0.9, 0.99, 0.999, 0.9999... A sequence is really just a function over the natural numbers, if you like to speak that way. Then I'm talking about limit, as n goes to infinity, of the function
f(n)=0.9 + 0.09 + 0.009 + ... + 9/10^n

If you want to see that in sigma notation, go to page 1 or 2 of this thread to Melody's posts.

[Edmond Zedo]

x is the infinitesmal difference between 0.999... and 1, of course. Theoretical (As in having no bearing on the real world) and immeasurable.

[Almaviva]

x is the infinitesmal difference between 0.999... and 1, of course. Theoretical (As in having no bearing on the real world) and immeasurable.


But whatever does "infinitesmal difference" mean? Can you define it in a mathematical sense? (This is the difference between math and something like philosophy. We can define precisely what we're talking about, and derive properties precisely.)

I have defined things in a mathematical sense, for what I mean by "0.999...", "limit", "equals", and so forth. I also know that this is the usual way of doing things.

(Note, it *is* possible to define infinitessimals and make the concept precise, but it is very hard to get right. If you want to lay down exactly how you're working, and then argue that under your system, what "0.999..." means in your system is not 1, then I wouldn't argue with you.)

I'd also be curious how you would represent (0.999... + 1)/2 in your system. It has to be something between the two quantities, right?

And also what is 0.999... * 10?
What is 9.999... - 0.999...?
How do you get around the 10x - x = 9 thing?

[Edmond Zedo]

infinitesimal

\In`fin*i*tes"i*mal\, n. (Math.) An infinitely small quantity; that which is less than any assignable quantity. (From dictionary.com)

In this case and many others, the reason to use the term (or synonyms) is merely to state that something indeed is, or that something indeed separates.

You ask if 1.999.../2 would be between 0.999... and 1. It is not necessarily between them, but occupies the same theoretical point as 0.999..., as the answer is 0.999... To use the opposite of infinitesimal to clarify: There is an infinite amount of space on one side of a plane, and an infinite amount on the other. The sum of the "two spaces" is also infinite.

[Almaviva]

infinitesimal

\In`fin*i*tes"i*mal\, n. (Math.) An infinitely small quantity; that which is less than any assignable quantity. (From dictionary.com)

In this case and many others, the reason to use the term (or synonyms) is merely to state that something indeed is, or that something indeed separates.


That's a definition, but it's not very mathematical. (What's an infinitely small quantity? Is 0 an infinitely small quantity? Am I supposed to be able to use my imagination here?)

Is an infinitessiman an "assignable quantity"? If so, does this definition mean that it's less than itself? (It's less than any assignable quantity, right?)



You ask if 1.999.../2 would be between 0.999... and 1. It is not necessarily between them, but occupies the same theoretical point as 0.999..., as the answer is 0.999...


In the normal number system, we can do this:
Suppose:
(y+x)/2=x
=> y+x=2x (If two things are equal, then their doubles will be equal too.)
=> y=2x-x (Subtract x from both sides.)
=> y=(2-1)x (Distributive property of multiplication.)
=> y=1*x
=> y=x (Definition of "1" as multiplicative identity.)

But it looks like in your number system, this isn't true, right? You said (1+0.999...)/2=0.999..., so that's an instance of (y+x)/2=x.

So, in your system, what part of the above "proof" fails?

Also, why didn't you answer my question about how you resolve the 10x-x issue?


To use the opposite of infinitesimal to clarify: There is an infinite amount of space on one side of a plane, and an infinite amount on the other. The sum of the "two spaces" is also infinite.

[Edmond Zedo]

That's a definition, but it's not very mathematical. (What's an infinitely small quantity? Is 0 an infinitely small quantity? Am I supposed to be able to use my imagination here?)
Well it says "Math." in the definition. I didn't write it, but I understand it. Zero is nothing. Infinitesimal is something. If you know math, you're familiar with asymptotes, right? The curve never reaches the line, but it gets infinitely close. The eventual difference is infinitesimal. If the line = zero, the curve never = zero.


Is an infinitessiman an "assignable quantity"? If so, does this definition mean that it's less than itself? (It's less than any assignable quantity, right?)
Infinitesimal: No assignable quantity. I believe that means you can't assign a numerical value to it, and rightly so. It's not less than itself.



In the normal number system, we can do this:
Suppose:
(y+x)/2=x
=> y+x=2x (If two things are equal, then their doubles will be equal too.)
=> y=2x-x (Subtract x from both sides.)
=> y=(2-1)x (Distributive property of multiplication.)
=> y=1*x
=> y=x (Definition of "1" as multiplicative identity.)

But it looks like in your number system, this isn't true, right? You said (1+0.999...)/2=0.999..., so that's an instance of (y+x)/2=x.

So, in your system, what part of the above "proof" fails?
It's as I said. Two things which are each infinitesimal or infinite aren't required to be equal to one another in any practical sense. I don't believe those equations are capable of dealing with infinites. For example, I'll plug "infinity" in for x, and 1 in for y, solving each one.

Suppose (1+infinity)/2=infinity (that's true)

1=0, or 1=infinity (false)
1=0, or 1=infinity (false)
1=infinity (false)
1=infinity (false)
1=infinity (false)



Also, why didn't you answer my question about how you resolve the 10x-x issue?
I didn't see it. Sorry.

10*0.999...=9.999...
9.999...-0.999...=9

Mathematically, 0.999... always = 0.999... It's the same number.

[Vylence]

Edmond, the problem with infinity is that you can't add anything to it... mainly because how do you add something on to a never ending end?

And for all purposes 0.9... does equal 1 in the decimal system. Is this a flaw? Yes it is. How can you assign an infinitasmal difference to something when you can't even be sure what it is? In the case of 1 - 0.9... you never get a infinitly small number, you never get any number. All you will ever see is zeros. In fact its zeros stretching off into forever. How can you get to a 1 that doesn't exist? If it did exist yes I guess we could say its an infinitsmal number.

[Dman]

Edmond, the problem with infinity is that you can't add anything to it... mainly because how do you add something on to a never ending end?

And for all purposes 0.9... does equal 1 in the decimal system. Is this a flaw? Yes it is. How can you assign an infinitasmal difference to something when you can't even be sure what it is? In the case of 1 - 0.9... you never get a infinitly small number, you never get any number. All you will ever see is zeros. In fact its zeros stretching off into forever. How can you get to a 1 that doesn't exist? If it did exist yes I guess we could say its an infinitsmal number.

0.9....does not equal 1 in the decimal system; it is ROUNDED to 1 in the decimal system.

Just because you can't "see" an infinitely small number does not mean that it does not exist. For purposes of materiality, 0.999... certainly can be treated as 1, because it's close enough on a human scale, but it still does not equal 1.

[Edmond Zedo]

0.9....does not equal 1 in the decimal system; it is ROUNDED to 1 in the decimal system.

Just because you can't "see" an infinitely small number does not mean that it does not exist. For purposes of materiality, 0.999... certainly can be treated as 1, because it's close enough on a human scale, but it still does not equal 1.
I couldn't have said it better myself.

[Vylence]

Its not rounded. The problem is obviously with you guys not being able to imagine infinity, it doesn't stop guys. If you were giving 0 for the rest of your life, you would always have zero. 0.0... works out to be zero repeating forever, there is no rounding. I'm not getting rid of the one, I'm not rounding down, there is no one. How can I round down when I will never see a 1 to round down from?

[SheepDog]

Its not rounded. The problem is obviously with you guys not being able to imagine infinity, it doesn't stop guys. If you were giving 0 for the rest of your life, you would always have zero. 0.0... works out to be zero repeating forever, there is no rounding. I'm not getting rid of the one, I'm not rounding down, there is no one. How can I round down when I will never see a 1 to round down from?
I can imagine an infinitely small difference between 1 and .999...

Is that what you meant?

[Vylence]

Can you? Describe it to me, because there is no infinitly small difference between 1 and .9...

See the problem is for there to be a infinitly small difference, there would have to be a 1 somewhere. Or there would have to be a 1 at the end of infinity. There isn't because there is never an end to infinity. The fact that all this math points to that I would think prove this to you guys. Infinity doesn't end, as long as it doesn't there is no one. There is no infinitly small difference, there is no difference.

[SheepDog]

First of all, I'm not taking this as serious as I think you may be. I'm just saying that to be fair, and to let you know I'm not trying to attack anyone. This has gone round and round for a long time, and...

Now, having said that, what is 1/infinity ?

[Network Alchemy]

i would have thought the dead horse being beaten severely would have stopped this thread but it does not seem to be having any effect so i shall join in further harassing the overbeaten corpse and as was said .9999.... is close to 1 on a human scale what about on the scale of an ant or the scale of a molecule or the scale of a quark etc the sequence does not end and regardless of how small you get the sequence will be the same as if you were viewing it from a human scale ie a quark will say .9999.... is 1 just like a human will and a corollary is that you are probably viewing the sequence from a perspective which is ~traveling~ with it but a reliance on this destroys the basis of a global frame which is needed for mathematics to work for example if i say car 1 is driving x kph on a specific road from germany to japan and car 2 is driving y kph relative to the kitten owned by fabio then you cannot simply place these two cars in a relation and find the speed of one relative to the other because they were given in a different form eg car 1 was given with an absolute velocity and position but car 2 was given by a relative velocity and position and the problem is we do not know how these two are connected maybe if we knew the absolute position of the kitten owned by fabio we could work something out but in general mixing them up is much pain on and in the ass which is what happens when you say something like .9999.... + .7777.... because simply having more than one sequence suddenly adds another observer going a different speed if you will and compensating methods would have to be used and in general it is best to stay away from this .9999.... does not equal 1 methodology in the first place but i certainly am not a mathematician and almaviva could probably rewrite massive portions of this post but that is my 1 + .9999.... cents oh the hilarity

[Edmond Zedo]

SheepDog: 1/infinity=infinity.

Network: Car 1 road kph is not absolute velocity as you've failed to consider the motion of der Cosmos.

Vylence: There is a difference. The function which displays it is asymptotic (I assume. I'm not going to learn how to graph again right now). There is no 1 at the end of the zeros. It must be displayed by other means such as:

(1-0.999...)

That was easy. We can call (1-0.999...) Krunk. Krunk is infinitesimal.

[Vylence]

I would think 1/infinity = 0
Which would also mean that infinity(0) = 1
And infinity = 1/0

[Network Alchemy]

Network: Car 1 road kph is not absolute velocity as you've failed to consider the motion of der Cosmos.so

[Vylence]

Edmond, we are not graphing a curve here. 1 - 0.9... is not asymptotic because it is a point, not a line or a curve.

[Edmond Zedo]

I would think 1/infinity = 0
Which would also mean that infinity(0) = 1
And infinity = 1/0

"I thought I was having trouble with my adding...But it's all right now."--For a Few Dollars More

I had it wrong before. 1/infinity=(1-0.999...)

[Edmond Zedo]

Edmond, we are not graphing a curve here. 1 - 0.9... is not asymptotic because it is a point, not a line or a curve.

I, uh, don't start calculus until next semester.

I think the function for Krunk would be (1-0.9-0.09-0.009-...), and the limit would be zero. It's a curve approaching zero, and asymptotic. But I'm in way over my head, and could be wrong.

[Vylence]

I actually haven't taken calculus either, but I think its odd to define 0.9... as 0.9 + 0.09 + 0.009... because there is no end therefore if you limit it it will equal different things. Its fine to take the limit of 1 as Almiviva did because where ever you do it equals 1, but this isn't true with 0.9...

Now you can on the other hand take the limit of 1 - 0.9..., which will be 0 because no matter where you take it once again it will equal 0. I personally don't know how you would graph this as a curve.

[SheepDog]

I would think 1/infinity = 0
Which would also mean that infinity(0) = 1
And infinity = 1/0
1/infinity is NOT zero, though that's still part of this argument.

1/0 does not exist. This is true in any form of mathematics that I'm familiar.

While we're at 1/infinity cannot be reduced. It's simply "an infinitely small number".

[Vylence]

You can't conceptulize how many times 0 would go into 1?

[SheepDog]

You can't conceptulize how many times 0 would go into 1?
I know you're just being rhetorical, but you're not helping your case by trying to refute the argument "1/0 does not exist". Seriously.

[Almaviva]

Well it says "Math." in the definition.

I know, but I'd venture to say that the person who wrote it doens't know much math:)



I didn't write it, but I understand it. Zero is nothing. Infinitesimal is something. If you know math, you're familiar with asymptotes, right? The curve never reaches the line, but it gets infinitely close. The eventual difference is infinitesimal. If the line = zero, the curve never = zero.


A number isn't a curve, it's a set quantity.

The function f(n)=1- (1/10^n) is always less than 1 for positive n.

The limit, as n "approaches infinity" of f IS 1.

0.999... is, *by definition*, the limit of the sequence of partial sums 0.9, 0.99, 0.999, and so on, which is actually the same as f above.

Your intuition clearly wants to say that 0.999... and 1 are two different things. Which is why I'm harping on the fact that you're going to have to use a non-standard number system.

What I don't think you realize, not having studied math much, is that for this to be true, the number system you're going to need is going to be very weird and counter-intuitive. The decimal representation is not going to be good enough, trust me:)

For an introduction, look up "nonstandard analysis" on wikipedia.com. You'll probably get bogged down really quickly if you don't have a math degree, heh.



Infinitesimal: No assignable quantity. I believe that means you can't assign a numerical value to it, and rightly so. It's not less than itself.


So you're arguing math based on your beliefs now? Do you expect computer programs to run because you believe in them enough too, or do you actually run them and see?



It's as I said. Two things which are each infinitesimal or infinite aren't required to be equal to one another in any practical sense. I don't believe those equations are capable of dealing with infinites. For example, I'll plug "infinity" in for x, and 1 in for y, solving each one.

Suppose (1+infinity)/2=infinity (that's true)

1=0, or 1=infinity (false)
1=0, or 1=infinity (false)
1=infinity (false)
1=infinity (false)
1=infinity (false)


This is an indication that you can't just plug infinities into formulas at a whim and expect them to work.

But one benefit of the conventional number system is that algebra works when values of variables are real numbers, and you don't try to do a few things like dividing by zero.

So, your number system is limited in a way that the conventional one is not, because the series of steps I gave can't be used. It *can* be used for any real numbers.

You also didn't answer my question about which step fails for 1 and 0.999...

Plugging in nfinity fails because the idea that you can substitute infinity into a variable doesn't work right from the start.



I didn't see it. Sorry.

10*0.999...=9.999...
9.999...-0.999...=9

Mathematically, 0.999... always = 0.999... It's the same number.

So is it true then that if x=0.999...., then 10x-x = 9? and that 9x = 9?



I actually haven't taken calculus either, but I think its odd to define 0.9... as 0.9 + 0.09 + 0.009... because there is no end therefore if you limit it it will equal different things. Its fine to take the limit of 1 as Almiviva did because where ever you do it equals 1, but this isn't true with 0.9...


This is exactly why you *can't* define 0.999... as 0.9+0.09+0.009+..., because you can't perform an infinite number of operations!

The way to define an infinite sum is with the idea of a limit. If you don't want to define it as a limit, well please tell me how you do want to define it!

And you have to be careful. For example, what is 1 -1 +1 -1 +1 ...?
Well, it is (1-1) + (1-1) + (1-1) + ... = 0 + 0 + 0 + ... = 0
Or is it 1 + (-1 +1) + (-1 + 1) + ... = 1 + 0 + 0 + ... = 1??

You can't say that this infinite sum equals anything.

[Vylence]

I think its very unprecise to take a limit of 0.9..., if you do it will equal a 1 somewhere and that kinda defeats the purpose.

[Vylence]

I can refute that 1/0 does not exist! It happens all the time.

[SheepDog]

I can refute that 1/0 does not exist! It happens all the time.
I'm sure that you do.

[Almaviva]

The thing about 1/0, if you want to say it exists, is: Is it a really big positive "infinity", or a really big negative "infinity"?

Look at the graph of y=1/x. Just to the right of zero, it gets really really big, but just to the left it gets really really negative.

[Dman]

I think you have to be careful when dealing with 0. It is not a number; it is the lack of a number. So 1/0 is a pointless argument - it's like saying I have $47 million, let's share it and divide it up between zero people. It's nonsense.

[SheepDog]

This thread is like the metaphorical cat. It keeps dying, but it must have multiple lives.

[Dman]

Ok, maybe this will end the thread. According to "Dr. Math", .999... does equal 1. For those of us who disputed, we appear to have been wrong (D'oh!).

Vindication for those of you who believed .999..=1! Here's one of the links that tries to describe it:
http://mathforum.org/library/drmath/view/55746.html

[Edmond Zedo]

Zero can't divide anything because it isn't anything.

I don't care what Dr. Math says. I do agree that 1 can be substituted for 0.999... in nearly any conceivable situation or calculation though. 0.999... is infinite so its function must be too, and graphed the curve never meets 1. 0.999... isn't 1.

PS I looked at the Dr. Math Q&A. I would almost stay my course just to disagree with this clown.

[Dman]

Never give up! I love it. Ok, you twisted my arm - I don't agree with Dr. Math either. The explanation that (to paraphrase) 'since you can't move away from one towards .999... then they must be the same' is bogus.

For all intents and purposes you can treat it like 1, but it still isn't the same as 1, in the literal sense of things, simply by definition. Maybe that's the real argument here - is it effectively equal to 1? Yes. Is it literally equal to 1? Of course not.

[Network Alchemy]

it is as if you are saying ( 1 + 2 ) and ( 3 ) are different of course you can say .9999.... is whatever you want it to be but it is meaningless if the results you acquire are inconsistent with the rest of your mathematics

[Edmond Zedo]

so you are saying ( 1 + 2 ) and ( 3 ) are different

Are you talking to me? If so: No. Them two are the same.

[Network Alchemy]

all of this reminds me of the continuous nondifferentiable functions of weierstrass which he created just to prove such functions were possible and i believe the main problem here is that we do not grasp how specific mathematical definitions have to be and in particular how intuition has to be shown its place

[Network Alchemy]

Are you talking to me? If so: No. Them two are the same.i was not but if ( 1 + 2 ) and ( 3 ) are the same then clearly one cannot argue that .9999.... is not 1 just because of notation

[Dman]

So are you saying that in the world of Math, .999.... is in fact equal to 1, regardless of what our logical intuition wants us to think?

Ok, the "literal" statement I made was a poor one (1+2 is not literally the same as 3). Yes, I was thinking intuitively that .999... cannot be equal to 1, because it is not 1, it is .999.... but I suppose in the world of Math, if .999.... always behaves exactly the same as 1, then yes, it must equal 1. Eh?

[Network Alchemy]

yes

[Dman]

Alright, I'm convinced. Now I remember why I didn't major in math.

[Edmond Zedo]

but I suppose in the world of Math, if .999.... always behaves exactly the same as 1, then yes, it must equal 1. Eh?

Not exactly, but it can usually be treated as 1. 1-1=0, but 1-0.999...=krunk.

A friend of mine provided some insight on this. I'll restate what he said. This example, more real in a sense, is that 0.999... of light speed is theoretically attainable by a massive object, but light speed itself (1) is not.

[Dman]

Not exactly, but it can usually be treated as 1. 1-1=0, but 1-0.999...=krunk.

A friend of mine provided some insight on this. I'll restate what he said. This example, more real in a sense, is that 0.999... of light speed is theoretically attainable, but light speed itself (1) is not.

But .999...of light speed is NOT theoretically attainable, because .999... is equal to 1. So it would have to be something just less than .999...of light speed to be theoretically attainable. Couldn't be .999...infinite 9's, it would have to have an "8" somewhere on the end.

[Edmond Zedo]

Ein minuten, bitte! I'm pretty sure you've just gone goofy, and it was no accident. Stop that.

[Dman]

Ein minuten, bitte! I'm pretty sure you've just gone goofy, and it was no accident. Stop that.

Nope, was already goofy in the first place!

Anyways, in Math-land, .999...behaves exactly the same as 1. So .999...of the speed of light behaves exactly the same as (1) the speed of light. Which is theoretically impossible to achieve. So you theoretically can't achieve .999...the speed of light.

[Edmond Zedo]

Weren't you on my side yesterday? Anyway...

I think the function of an object approaching light speed could be the same function representing 0.999... But you'll need to have one of the pro mathematitians here represent it for you.

[Vylence]

I thought we had gotten away from this curve business.

0.9... repeating does equal 1. Though only in the decimal system. Since you can do math in BINARY or OCT or HEX, each of which 0.9... doesn't equal 1, why don't you guys just switch the way you count then you don't have to worry about it anymore.

0.9... is a flaw with counting in decimal, which either has to do with numbers changing places or repeating symbols that represnt different numbers. There is actually a flaw I'm sure in each of the counting systems, because 0.7... is equal to 1 in OCT.

Almiviva I don't think its fare to limit infinity to only half of the #s on the line. So it would make sense to me that its both negative and posative. It could be that in reality there are no negative numbers, only posatives heading in different directions. Negative numbers could be a falsity of math as we know it.

[SheepDog]

There is actually a flaw I'm sure in each of the counting systems, because 0.7... is equal to 1 in OCT.
Please explain.

My understanding is that 1 in OCT is 1 in BIN is 1 in HEX is 1 in DEC.

[Almaviva]

A friend of mine provided some insight on this. I'll restate what he said. This example, more real in a sense, is that 0.999... of light speed is theoretically attainable by a massive object, but light speed itself (1) is not.


The kinetic energy of an object under relativity is:
KE=m0*c^2*(1/sqrt(1-v^2/c^2)-1)

(I could derive this for you, but it would take quite a lot of discussion.)

Where m0 is the rest mass, v is the velocity as a fraction of c, which is the speed of light. We choose units so that the speed of light is 1 (very very commonly done in relativity problems.) We suppose v=0.999... of c. I.e. the object is moving at 0.999... of the speed of light.

Then
KE=m0*(1/sqrt(1-0.999...^2)-1)
(We have issues here because I don't know how your math works. I'm going to assume that 0.999...^2 = 0.999...?
If you were using the actual, mathematically consistent version of infinitessimals, you can't just ignore squaring like that. (1-(infinitessimal)) squared would actually have to be (1-2*infinitessimal), basically because (1-x) squared is (1-2x+x^2), and the x^2 is insignificant.)
=m0*(1/sqrt(1-0.999...)-1)
=m0*(1/sqrt( (infinitessimal) ) - 1)

Now the 1 over the square root of an infinessimal quantity is infinity. (I.e. larger than any amount of energy in the universe.)

So this speed is not theoretically attainable. (Just as if you'd plugged 1 into the equation.)

[Vylence]

One in OCT, BIN, HEX, and DEC is the same. I actually posted about this a few pages back.

If we are to do our math in OCT...
x = 0.7...
10x = 7.7...
10x - 1x = 7.7... - 0.7...
7x = 7
x = 1

The other numbering systems are interesting, like whats OCT 1.7 in DEC? We know its 0.1 off 2.0 in OCT. We know that 2 in either is equal. Since there is only 8 places in OCT and 10 in DEC we know we are going to be somewhat off here. The OCT fractions must be worth more. I think it would come down to 1.875 in DEC. Since we have 7/8 of a whole.

Now its even more confusing to consider a repeating # in OCT. Like 0.7..., since we know that .7 in OCT is .875 in DEC, it means that each place of 7 is worth .875 of the place preceding it. Which I would think is a case of 0.875 + 0.0875 + 0.00875 ... and so on. Which turns out to be 0.9722... with the 2 repeating. Of course thats if I did all that correctly...

Now what this means is that you could theoretically have any # as equal to 1, you would just have to change how you numbered things...

[Almaviva]

Not exactly, but it can usually be treated as 1. 1-1=0, but 1-0.999...=krunk.


First, why didn't you answer my questions in my last post? I'm trying to show you that your way of thinking about numbers is internally inconsistent.

Second, if you go by the usual definitions, it can't just usually be treated as 1, it can always be treated as one.

Again, the question you never answer: If 0.999... is not defined as the limit of the sequence 0.9, 0.99, 0.999, ..., how are you going to define it?

A dictionary isn't good enough here, you're going to need to define this in mathematical terms. And you also need to be able to explain how to deal with operations like multiplication and addition, and basic algebra.

I've shown you at least one reason why basic algebra is going to have problems in your system.

This is why mathematicians don't commonly use the concept of infinitessimals: They are hard to work with! And the usual system, where decimal expansions are defined as limits is consistent.

One thing to understand is that mathematicians don't start with a decimal and call it "a number" and then work from there. It's the opposite: they start with the idea of a number as being a thing that has properties we think numbers ought to have:
1. For a, b, and c, a+(b+c)=(a+b)+c
2. There is a number 0 so that a+0=0+a=a
3. For every a, there is -a so that a+(-a)=(-a)+a=0
4. For all a,b: a+b=b+a
5. For all a,b,c: a*(b*c)=(a*b)*c
6. There exists a number 1 so that: a*1=1*a=a
7. For every a<>0, there exists a^(-1) so that a*a^(-1)=a^(-1)*a=1
8. For all a,b: a*b=b*a
9. For all a,b,c: a*(b+c)=a*b + a*c
10. For every a, either a=0, a>0, or -a>0
11. If a>0, b>0, then a+b>0
12. If a>0, b>0, then a*b>0
13. If a set of numbers is bounded above, then it has a least upper bound.
(This is from the book Calculus, by Michael Spivak)

Now there is NOTHING here about it being base 10, or octal, or binary, or anything about how we actually write numbers. Other than that 1 and 0 exist.

From these properties, we can prove everything about real numbers that can be proven, and we can also prove that real numbers are unique, in the sense that the construction is exactly like any other construction that has the same properties. (There are no additional properties that are true for one construction and not for the other.)

So if you are going to use infitessimals, numbers that are less than 1/n for any integer n, then you are going to lose at least one of these properties. (You're going to lose #13 for sure, and maybe a couple others, I'm not sure.)
---

Now, the idea of infinite sums is something that is very useful in mathematics.

We want to be able to say things like
1/2 + 1/4 + 1/8 + 1/16 + ... = 1

This is like Zeno's paradox: In order to get there, I have to go halfway there, then halfway again, then halfway again, so it should take an infinite amount of steps. But I DO get there! So in this sense, it is intuitive to say that an infinite series has a finite sum.

The concept also allows equations like:
pi = 6*sqrt(1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ...)
e = 1/0! + 1/1! + 1/2! + 1/3! + ...

Working with these things is nice. And we don't have to worry about the fact that that first equation "never actually reaches" pi. We don't know the decimal expansion infinitely anyway!!

---

The point here of all this is that the decimal number system is ARBITRARY.

So 0.999... is just a way of saying 9/10 + 9/100 + 9/1000 + ...
It's no more profound or natural than 1/2 + 1/4 + 1/8 + 1/16 + ...
Or 7/8 + 7/64 + 7/512 + 7/4096 + ... (this is 0.7777... in octal)
Or 1.1 - 0.11 + 0.011 - 0.0011 + 0.00011 - ...
All of these converge to 1. None is more special than the others in that they are all just series of sums of fractions.

You seem to want to treat the decimal number system with special magic.

Note, a lot of this is quite advanced, and I don't expect all readers to have a clue what I'm talking about:)

[Vylence]

Oh yeah, your right, my math is off. Need more math classes obviously.

[melancholeric]

funny, i was just going to mention zeno's paradox.
assume a short distance runner (let's call him achilles) and a tortoise in a running race. Achilles runs, say, 10 times faster than the tortoise. for the sake of competition, tortoise gets 10 meters advance. Achilles starts running when the tortoise is 10 meters away. Achiles never gets past the tortoise.

Achilles reaches the 10 meter point, where the turtle was when when he started running. But now the turtle is 1 meter ahead of achilles. achilles runs 1 meter. turtle has moved 10 cm. achilles gets there, the tortoise is 1 cm ahead. etc.

(if the turtle had 9 meter advance, achilles would "not reach" it at 9.999... meters. Obviously Achilles will get past the turtle, at 10 m.)

[MacGuffin]

I think people are thinking at cross purposes here. Almaviva is defining 0.999.... mathematically. As a limit that converges to one.

Those arguing against this (like myself) do not approach it from a mathematic standpoint. More a philosophical viewpoint.

In the real world, Zeno's paradox converges, because we obviously get to our destination at some time. I believe this is because everything (space, time, energy) is quantitized on some level. Eventually you can't divide something in half, you have reached the smallest amount possible (like Planck's constant).

Therefore, mathematicians have defined 0.0999.... as converging to 1. Us on the other side see it as a purely philosophical question. 0.999.... does not exist. It cannot exist. It is an imaginary construct as the closest lesser value to 1, without actually being 1. In this imaginary world there are no limits or convergences.

[Dman]

Weren't you on my side yesterday? Anyway....

It's the INTP thing. I often argue both sides of the story, so that I can better understand the opposing arguments. Especially when I'm not well versed in the subject matter.

[Dman]

by the way, if there's nothing else gained from this enormous thread, you all make math actually really interesting & fun - I hate doing the legwork involved in math, which is what turned me off of it, but the theory & concepts are awesome. In case anybody cared.

[SheepDog]

It's the INTP thing. I often argue both sides of the story, so that I can better understand the opposing arguments. Especially when I'm not well versed in the subject matter.
When we really seek understanding, which side we represent doesn't necessarily matter. When we're fighting for our ego, it does. ;)

[Edmond Zedo]

We'll just see about that shit when I finish my fusion engine.

But anyway, in case you've forgotten, the original question was confined to the decimal system, and it was "Does 0.999...=1." It still doesn't.

[knome]

This question is like asking if 3.14159265... is equal to PI. This isn't an issue of equality, it is a matter of the system being unable to properly represent nonterminating fractional entities.

.999 does not equal 1. .333 does not equal 1/3. You _cannot_ solve 1/3 to a terminating decimal sequence, hence, it loses precision and meaning. It becomes an approximation. The appropriate way to multilpy it by three and get one would be to leave it alone convert three to fractional notation and then mutliply.

Using the ad infinitum symbol to infinitely extend .999 is an inglorious hack if there ever was one. While it technically solves the problem of nontermination for any given repeating sequence, you are then forced to attempt to work with infinite sets of numbers, thus creating problems like this. Just leave it in fractional form.

[Almaviva]

This question is like asking if 3.14159265... is equal to PI. This isn't an issue of equality, it is a matter of the system being unable to properly represent nonterminating fractional entities.

.999 does not equal 1. .333 does not equal 1/3. You _cannot_ solve 1/3 to a terminating decimal sequence, hence, it loses precision and meaning. It becomes an approximation. The appropriate way to multilpy it by three and get one would be to leave it alone convert three to fractional notation and then mutliply.

Using the ad infinitum symbol to infinitely extend .999 is an inglorious hack if there ever was one. While it technically solves the problem of nontermination for any given repeating sequence, you are then forced to attempt to work with infinite sets of numbers, thus creating problems like this. Just leave it in fractional form.


It's not that bad. You use the idea of a limit, and say that an infinite sum of fractions is, by definition, equal to its limit. This is rigorous and quite useful.

This way, I don't have to say I can express e approximately, I can say:
e = lim(n->infinity)1/0! + 1/1! + 1/2! + ... + 1/n!
This equality is EXACT, because of the idea of the limit.

Or, in short form
e = 1/0! + 1/1! + 1/2! + ...
The "..." here MEANS take the limit. It's not a hack, it's really the only sensible thing it could mean. It can't mean add it up forever. That would take too long:)

[Warrior413]

I'm tired and I'm sure this has already been said, but:
.999 = .999
1 = 1
;P

[Edmond Zedo]

This is exactly why you *can't* define 0.999... as 0.9+0.09+0.009+..., because you can't perform an infinite number of operations!

The way to define an infinite sum is with the idea of a limit. If you don't want to define it as a limit, well please tell me how you do want to define it!

And you have to be careful. For example, what is 1 -1 +1 -1 +1 ...?
Well, it is (1-1) + (1-1) + (1-1) + ... = 0 + 0 + 0 + ... = 0
Or is it 1 + (-1 +1) + (-1 + 1) + ... = 1 + 0 + 0 + ... = 1??

You can't say that this infinite sum equals anything.
I have some time to go back and try to analyze this further.

In the last example, I agree. There can be no answer, and the function simply is, i suppose, unless some event stops it. Is that what a limit is? If so, I don't see why one must be inserted unless you need to make calculations or show results using the function, and that wasn't a requirement of the orig. question. It's the same reason why 0.999... can never equal 1 autonomously.

[Edmond Zedo]

The kinetic energy of an object under relativity is:
KE=m0*c^2*(1/sqrt(1-v^2/c^2)-1)

(I could derive this for you, but it would take quite a lot of discussion.)

Where m0 is the rest mass, v is the velocity as a fraction of c, which is the speed of light. We choose units so that the speed of light is 1 (very very commonly done in relativity problems.) We suppose v=0.999... of c. I.e. the object is moving at 0.999... of the speed of light.

Then
KE=m0*(1/sqrt(1-0.999...^2)-1)
(We have issues here because I don't know how your math works. I'm going to assume that 0.999...^2 = 0.999...?
If you were using the actual, mathematically consistent version of infinitessimals, you can't just ignore squaring like that. (1-(infinitessimal)) squared would actually have to be (1-2*infinitessimal), basically because (1-x) squared is (1-2x+x^2), and the x^2 is insignificant.)
=m0*(1/sqrt(1-0.999...)-1)
=m0*(1/sqrt( (infinitessimal) ) - 1)

Now the 1 over the square root of an infinessimal quantity is infinity. (I.e. larger than any amount of energy in the universe.)

So this speed is not theoretically attainable. (Just as if you'd plugged 1 into the equation.)

You know, that language is basically foreign to me, but I understand concepts themselves. Let me restate.

If an object is constantly accelerating, at any instant its speed will be precise. But there is no theoretical limit to how close it can get to light speed. The difference decreases forever, but light speed is never equaled. Intuition tells me the answer to the question "What is the maximum speed attainable by the object." is 0.999...

[Aryan]

Never thought INTPs would be so screwd up so as to continue this thread so long on a simple algebra question

http://intpcentral.com/forums/images/smilies/ng_mad.gifhttp://intpcentral.com/forums/images/smilies/ranting.gifhttp://intpcentral.com/forums/images/smilies/BangHead.gifhttp://intpcentral.com/forums/images/smilies/spam_laser.gif


Well what can i say
Just enjoy your stay
And dont talk so loud
and dont draw more crowd
becoz i'm an INTP

[Almaviva]

If an object is constantly accelerating, at any instant its speed will be precise. But there is no theoretical limit to how close it can get to light speed. The difference decreases forever, but light speed is never equaled. Intuition tells me the answer to the question "What is the maximum speed attainable by the object." is 0.999...



Interestingly, you can ask the question of what the maximum speed in the universe could be, and put a limit on it.

If all the matter in the universe except one electron was converted into energy, and that energy was used to accelerate that electron, that speed would be finite. It would be 0.999999999(some number of nines) of the speed of light, but the number would end.

You can't theoretically get faster than that. (Well maybe if there's a less massive particle you could accelerate, but the same method stands.)



I have some time to go back and try to analyze this further.

In the last example, I agree. There can be no answer, and the function simply is, i suppose, unless some event stops it. Is that what a limit is? If so, I don't see why one must be inserted unless you need to make calculations or show results using the function, and that wasn't a requirement of the orig. question. It's the same reason why 0.999... can never equal 1 autonomously.


A limit basically just says that if I choose how close I want a function to get, I can find a place where eventually *all* terms are at least that close. For example, for any number (usually called Greek letter epsilon) greater than zero, we can find some series of 0.9999's, so that (1- that series of nines) is less than epsilon.

The series 1-1+1-1+1... never narrows down on a single value, so it has no limit.

Don't let anyone fool you though: The reason mathematicians say that 0.999... =1 is that they cheat. They define the sum of an infinite series of fractions to be equal to its limit. Basically, "It's equal because I say so, and there's no mathematical inconsistency in doing so."

The reason for doing this is that
1: It makes sense, and it useful, and leads to many interesting parts of mathematics.
2: The alternatives, including studying infinitessimals, are very hard, not intuitive in some ways, and lead to all kinds of difficulties that have to be resolved.

(For instance, I don't think you can resolve this:
Let x=0.999...
Then 10x - x = 9.999... - 0.999... = 9+(9/10+9/100+...) -(9/10+9/100-...)
=9
So 9x=9)

If you can resolve that, explain where it goes wrong, please.

[Edmond Zedo]

Interestingly, you can ask the question of what the maximum speed in the universe could be, and put a limit on it.

If all the matter in the universe except one electron was converted into energy, and that energy was used to accelerate that electron, that speed would be finite. It would be 0.999999999(some number of nines) of the speed of light, but the number would end.

You can't theoretically get faster than that. (Well maybe if there's a less massive particle you could accelerate, but the same method stands.)
I realize you need matter, but I don't consider that a limiter when we're discussing theoretical physics. For all we know there could be an infinite number of universes, and matter could be transferrable among them. I've read theories about new universes being created by events in others.

I wonder if a new universe is created each time an object of mass actually reaches light speed due to quantum fluctuation? :)



Don't let anyone fool you though: The reason mathematicians say that 0.999... =1 is that they cheat. They define the sum of an infinite series of fractions to be equal to its limit. Basically, "It's equal because I say so, and there's no mathematical inconsistency in doing so."

The reason for doing this is that
1: It makes sense, and it useful, and leads to many interesting parts of mathematics.
2: The alternatives, including studying infinitessimals, are very hard, not intuitive in some ways, and lead to all kinds of difficulties that have to be resolved.
I believe we've reached the point where we're not in disagreement, without either conceding.


(For instance, I don't think you can resolve this:
Let x=0.999...
Then 10x - x = 9.999... - 0.999... = 9+(9/10+9/100+...) -(9/10+9/100-...)
=9
So 9x=9)

If you can resolve that, explain where it goes wrong, please.
10x-x=9, yes. 9x=8.999..., however. I can't see how you got from here:

9+(9/10+9/100+...) -(9/10+9/100-...)=9

to here:

So 9x=9

[Edmond Zedo]

It occured to me that

10x-x=9

would normally imply that x=1, but as I said before, some normal rules don't apply to infinites and infinitesimals (Reference "infinity" plugged into those equations).

[Dman]

So in summary:

.999...does =1 when applying mathematical formulas (i.e. in Math Land), due to the esoteric characteristics of infinite progressions; although there are avenues that can be explored in Math Land that attempt to define these characteristics but as of yet have no practical application;

but .999... does not = 1 by strict (logical) definition.

Have I got it right yet?

[MacGuffin]

So in summary:

.999...does =1 when applying mathematical formulas (i.e. in Math Land), due to the esoteric characteristics of infinite progressions; although there are avenues that can be explored in Math Land that attempt to define these characteristics but as of yet have no practical application;

but .999... does not = 1 by strict (logical) definition.

Have I got it right yet?

That's how I see it.

[melancholeric]

So by strict logical definition, aforementioned Achilles never reaches the aforementioned turtle? (Poor turtle, having to spend rest of its life running with Achilles on its heels...)

Any occurence of 0.9999.. or similar sequence (sequence approaching a certain number) is always a result of something similar to zeno's paradox. Now I have to find that high school math book that explained this.

[Almaviva]

So in summary:

.999...does =1 when applying mathematical formulas (i.e. in Math Land), due to the esoteric characteristics of infinite progressions; although there are avenues that can be explored in Math Land that attempt to define these characteristics but as of yet have no practical application;

but .999... does not = 1 by strict (logical) definition.

Have I got it right yet?


There's no logic involved with wanting 0.999... to not equal 1. To me, the intuitive arguments are compelling the other way. (The main one being that 10x-x=9 ought to imply that x=1).

But, traditionally, math has been expanded by assuming the existence of new things:

1. Start with the counting numbers 1, 2, 3, 4, ... We have addition and multiplication and so forth.
2. Is there a number b so that a+b=a?? Suppose there is! Now we have 0.
3. Does a-b make sense when b>a? Suppose it does, now we have negative numbers.
4. What about a/b when b does not divide a? If we allow this, we have fractions (rational numbers.)
5. What about things like the square root of 2, and the cube root of 3, and so forth? In general, roots of equations like x^3+x^2=3? If we assume these exist, we have algebraic numbers. (Which can be "irrational").
6. Does every sequence that converges (is "Cauchy") converge to some number in particular? If we assume that it does, we have the real numbers. We can define a number like "pi" now. (The usual way of saying this is that every bounded sequence has a least upper bound and greatest lower bound. It is the "completeness" property of real numbers).
7. Are there square roots of negative numbers? If we assume there are, we have the complex numbers.

8. Is there a number, x, that is bigger than 0, but less than 1/n for any integer n??

If we want the completeness property, #6, then the answer has to be no! Because 1/n is bounded, so it has to have a greatest lower bound. That number has to be zero. If it weren't zero, (call it x) we couldn't find the greatest lower bound, because 2x still would have to be less than 1/n for all n. (x < 1/n for all n implies that 2x < 1/(n/2) for all n, so it has to be less than 1/n as well.) This reasoning is hard to understand if you haven't studied stuff like it a lot.

But we don't have to require the completeness property.

Apparently, some mathematicians were looking for about a hundred years for a consistent way to contruct a system that has this property. So then there is an x where x < 1/n for all integer n under this system.

You can see some of the problems. You'd like to do basic algebra, like saying that 10x - x = 9 implies that x=1. If you can't rely on basic algebra, your system isn't very useful, and very very hard to deal with.

But it can be done if you're perverse, and want to do post-doctorate math research, heh.

[Dman]

Not sure what you mean by 10x-x = 9 implies x=1 thing. Does it not imply that x = 1?

If x = .999..., then 10x-x = 8.999....but not 9 (under the intuitive logic model). Don't see how that is compelling the other way, it's the same thing..??

Again, I see that for math purposes it works to have .999...=1, but still don't see it that way by intuitive logic. And perhaps I am misinterpreting what's being said, but just because it's easier and it works in math to have .999...= 1, I don't see that this makes it absolutely true.

[Almaviva]

Not sure what you mean by 10x-x = 9 implies x=1 thing. Does it not imply that x = 1?


It should, if you want math to be useful, in my opinion.



If x = .999..., then 10x-x = 8.999....but not 9 (under the intuitive logic model). Don't see how that is compelling the other way, it's the same thing..??


If a=b, and b=c, then a=c. This is really fundamental to "equality". (It's called the "transitive property".)

10x-x = 9.9999.... - 0.9999... (Agree here?)
9.9999... - 0.9999... = 9. (Agree here?)

Therefore, 10x-x=9, by transitive property. (Do you want a world where a=b, b=c, but a!=c ?)

If your intuition and "logic" can deal with a way to know when you have to magically use 8.999... instead of 9, then it's different from mine.



Again, I see that for math purposes it works to have .999...=1, but still don't see it that way by intuitive logic. And perhaps I am misinterpreting what's being said, but just because it's easier and it works in math to have .999...= 1, I don't see that this makes it absolutely true.

If you can describe your intuition it would be interesting. To me, the nature of infinities is that you can "cycle" them without changing them. Sliding people over in the infinite hotel freeing up a room, if you like. So cycling 0.9999... gives you 9.9999.... so it makes it 10x=9+x.

Also, my intuition says that if two things are infinitely close, they are equal. If there is nothing between two quantities, they are equal as well. And intuitively, 1/infinity=zero, to me.

[Dman]

Ok, I get the 10x-x=9 thing (be easy on me, I flushed most of my math education a long time ago. Obviously).

I guess what I'm saying is that I don't necessarily agree that "if it sounds like a duck, and it looks like a duck, then it is a duck". I believe that even if for all intents and purposes .999... behaves like 1 that is fine, but does not make it ABSOLUTELY = 1. Just because I can't provide a real-world example does not mean it is or isn't true.

It's the "close enough" thing. If 1 is a point, then .999... gets right up snuggled next to point 1, but is still never 1. Intuitively.

I understand (on a very fundamental level) that math can prove that it equals 1, but I'm intuitively thinking of it in the manner above - it never quite gets there, so it isn't absolutely 1, even if it behaves just like 1.

But I'm getting the impression that in higher math, there really can't be a point 1, because it is on a continuum, like time. On an infinitesimal scale, there is a fuzzy area that can be called 1, that both .999... and 1+(1-.999...) are both essentially 1.

But wait, if you have 1 apple, do you also have .999.... of an apple? It seems (intuitively again) that there would be some missing piece of 1 apple vs. .999....of an apple.

See what you've done, now I'm arguing with myself.

[Edmond Zedo]

It should, if you want math to be useful, in my opinion..
We're discussing theory, not building a spacecraft. Use practical math for that.


1/infinity=zero, to me.
There's our fundamental disagreement.

[Robespierre]

1/infinity only approximates zero. It gets very close, but never gets there. In an engineering application, one might simply round it over to zero and never know the difference. However, if you are going to take the definitions of these terms literally, 1/infinity does not equal zero, 1/infinity > 0. Always.

[Network Alchemy]

going far off topic i want 1/infinity to be equivalent to a pink light bulb how does that sound

[melancholeric]

interesting. that would have to be quite a small (infinitely small) light bulb

[Network Alchemy]

but i mean just a regular pink light bulb and if this requires the birfurcation of the existing mathematical school then so be it one will be the existing mathematics and the other the one where 1/infinity is a pink light bulb which will have implications within the rest of the plb mathematical framework and bring about renovations in physics but i am thinking i should calm my mind as of now

[melancholeric]

:rofl: :rofl:
You, my friend, are what i would consider a genius.

[Dman]

but i mean just a regular pink light bulb and if this requires the birfurcation of the existing mathematical school then so be it one will be the existing mathematics and the other the one where 1/infinity is a pink light bulb which will have implications within the rest of the plb mathematical framework and bring about renovations in physics but i am thinking i should calm my mind as of now

Assuming you meant bifurcation.

This is a good idea, because when we confine ourselves to only using what appears to work best, rather than explore other avenues (no matter how bizarre or esoteric), we eventually reach a point where we cannot progress. Think of Newton vs. Einstein. Newton was the existing mathematics, and Einstein was the pink light bulb. Of course, now Einstein is the existing mathematics.

Or maybe I just read more into your metaphor than was really there, and this post will confuse people!

[indie]

0100100101110011001000000011000000101110001110010011100100111001001000000110010101110001011101010110000101101100001000000111010001101111001000000110111101101110011001010011111100100000001000000000110100001010000011010000101001010000011001010111001001101000011000010111000001110011001000000110100101110100001000000010101001101001011100110010101000101100001000000110101001110101011100110111010000100000011011010110100101100011011100100110111100101101011011100110000101101110011011110010110101110011011001010110001101101111011011100110010001110011001000000110001001100101011001100110111101110010011001010010000001110100011010000110010100100000011001010110111001100101011100100110011101111001001000000111001101110100011000010111010001100101001000000110100101110011001000000111001101110111011010010111010001100011011010000110010101100100001000000110011001110010011011110110110100100000011011110110011001100110001000000111010001101111001000000110111101101110001110110010000001100110011100100110111101101101001000000111010101101110011000110110100001100001011100100110011101100101011001000010000001110100011011110010000001100011011010000110000101110010011001110110010101100100001011100010000000100000

happppy

[Almaviva]

But I'm getting the impression that in higher math, there really can't be a point 1, because it is on a continuum, like time. On an infinitesimal scale, there is a fuzzy area that can be called 1, that both .999... and 1+(1-.999...) are both essentially 1.


What you're describing here is like the idea of an "equivalence class", which is used very very much in higher math.

Basically, an equivalence class is a set of all the stuff that looks and quacks the same. So you could say, in the real number system, that (0.5+0.5), 1, 1/2+1/4+1/8+..., 0.999... are all members of the same equivalence class, called {1}. The "number" we're trying to represent isn't the symbol 1, as we write it, it's a mathematical concept. (With properties like x*1=x)

So yeah, the Tao you can write down isn't the real Tao:)



1/infinity only approximates zero. It gets very close, but never gets there.


Infinity never gets there either. If we're willing to conceptualize "forever", I think its inverse ought to be "never", not "almost never".


In an engineering application, one might simply round it over to zero and never know the difference. However, if you are going to take the definitions of these terms literally, 1/infinity does not equal zero, 1/infinity > 0.


What are the definitions of the terms that you're taking literally? Can you answer this?

In an engineering application, if infinity comes up, you're in some serious trouble. In some applied math calculations involving quantum mechanics, they do somewhat arbitrarily pick a really big number (say the length of the universe) sometimes, but it's kind of a hack. There's also the idea of the Dirac Delta function, which is a curve that goes infinitely high, but has a finite area, and this can be made rigourous. All this is complicated though, and much more subtle than what we're talking about.


Always.

Do you really mean always, or do you mean an increasing probability that is infinitely close to always, but never quite gets there?:)



going far off topic i want 1/infinity to be equivalent to a pink light bulb how does that sound


I think that's the most insightful thing that's been said in this thread.

[Dman]

Infinity never gets there either. If we're willing to conceptualize "forever", I think its inverse ought to be "never", not "almost never".

Wouldn't the inverse of forever remain forever, just change the "direction"? In other words going forever to the left instead of to the right. But it would still be "forever". Like the inverse of time would be time going perpetually backwards, and it would still be time - time would not cease to exist.

[Almaviva]

Inverse in the math sense, although admittedly the concept of taking an inverse of "forever" is strange.

But look at the graph of y=1/x for positive numbers. What is the y value at x=0? Intuitively, infinity. What is the x value at y=0? Intuitively, it has to be symmetric, so infinity. This "means" that 1/0 is infinity, and 1/infinity is zero. Mathematically, you run into problems working with infinity, but it makes intuitive sense to me here.

[Edmond Zedo]

Inverse in the math sense, although admittedly the concept of taking an inverse of "forever" is strange.

But look at the graph of y=1/x for positive numbers. What is the y value at x=0? Intuitively, infinity. What is the x value at y=0? Intuitively, it has to be symmetric, so infinity. This "means" that 1/0 is infinity, and 1/infinity is zero. Mathematically, you run into problems working with infinity, but it makes intuitive sense to me here.
Gotcha!
1. y=1/x is an asymptotic curve
2. You can't divide by zero, once again, because zero can't divide. Infinity and Krunk can both divide in theory, because they're something. Zero doesn't exist at all. ERR DIV BY 0. END OF LINE.

[Robespierre]

What are the definitions of the terms that you're taking literally? Can you answer this?

Yeah, infinity. If you really mean infinity, and not some numerical approximation, the solution will never equal zero.



In an engineering application, if infinity comes up, you're in some serious trouble. In some applied math calculations involving quantum mechanics, they do somewhat arbitrarily pick a really big number (say the length of the universe) sometimes, but it's kind of a hack. There's also the idea of the Dirac Delta function, which is a curve that goes infinitely high, but has a finite area, and this can be made rigourous. All this is complicated though, and much more subtle than what we're talking about.

Sure, we can throw fractals into the mix and talk about infinite perimeter within a finite area, Koch Snowflake.

The point is, all of these works are idealized representations of the real world. And in these perfect systems, 1/infinity will never = zero.



Do you really mean always, or do you mean an increasing probability that is infinitely close to always, but never quite gets there?:)

Always as in Always the Low Price.

[Almaviva]

Gotcha!
1. y=1/x is an asymptotic curve
2. You can't divide by zero, once again, because zero can't divide. Infinity and Krunk can both divide in theory, because they're something. Zero doesn't exist at all. ERR DIV BY 0. END OF LINE.

Zero can't divide. And infinity doesn't exist. But if zero could divide, intuitively, to me anyway, it seems like infinity would be the answer you would get. And also that if you divide by anything non-zero, you just get a really big number, not infinity. Of course all this is fuzzy and imprecise.




What are the definitions of the terms that you're taking literally? Can you answer this?


Yeah, infinity. If you really mean infinity, and not some numerical approximation, the solution will never equal zero.


Did you noticed I asked you to define it twice, and you didn't do it?

If you want to argue in "perfect systems", you ought to be able to state precisely what you're talking about, no?



Sure, we can throw fractals into the mix and talk about infinite perimeter within a finite area, Koch Snowflake.

The point is, all of these works are idealized representations of the real world. And in these perfect systems, 1/infinity will never = zero.


Who says these works are in any way representitive of the real world?

Where does the Koch Snowflake occur in the real world? Presumably not in Antarctica.

The Mandelbrot Set came about by Mandelbrot (and some other guy many years before him that didn't have computers) screwing around with the iterative function over complex numbers Z(n)=Z(n-1)^2 + C. Not from the real world.

[Robespierre]

Did you noticed I asked you to define it twice, and you didn't do it?

If you want to argue in "perfect systems", you ought to be able to state precisely what you're talking about, no?

Yeah, I saw that. A "perfect system" is a idealized model of the real world. Geometry is an example. There are no zero dimensional points or 2 dimensional lines or planes in the real world, these are ideal constructions which help us model the real world.


Who says these works are in any way representitive of the real world?

Where does the Koch Snowflake occur in the real world? Presumably not in Antarctica.

The Mandelbrot Set came about by Mandelbrot (and some other guy many years before him that didn't have computers) screwing around with the iterative function over complex numbers Z(n)=Z(n-1)^2 + C. Not from the real world.

Thank your for restating my point.

[Almaviva]

Yeah, I saw that. A "perfect system" is a idealized model of the real world. Geometry is an example. There are no zero dimensional points or 2 dimensional lines or planes in the real world, these are ideal constructions which help us model the real world.


What I wanted was your definition of infinity within a "perfect system". E.g. is it something I can assign to a variable?


The other thing I was arguing, which is pretty trivial and semantic, is that I don't think 'perfect systems' have to be representations of the real world in any way at all. Some, like Euclidean Geometry, clearly do have connections with the real world. Even fractals are sort of connected with the real world, e.g. geological and biological structures, maybe the structure of the galaxies in the universe even. But some areas of math are just people playing around, with no motivation from reality at all. For example, studying functions that can't be integrated normally (Riemann's sums) and need Lesbegue integration, or even yet more pathological functions. Or weird set theory concepts like the Continuum Hypotheis and Zorn's Lemma/Axiom of Choice. These don't apply to the real world in any way at all (I don't think!)

In other words, I agree with "idealized", but I don't agree with "model of the real world", although many times these things are models of the real world in some way.

[Edmond Zedo]

Zero can't divide. And infinity doesn't exist. But if zero could divide, intuitively, to me anyway, it seems like infinity would be the answer you would get. And also that if you divide by anything non-zero, you just get a really big number, not infinity. Of course all this is fuzzy and imprecise.

You don't seem to understand how impossible and nonsensical it is to say zero could even theoretically halfway divide anything. It's like: "Well, if I subtract 0 from 1, wouldn't it even be just a little less?" Zero is nothing! Infinity, on the other hand, is mathematically, and theoretically possible

[Dman]

Let's just say that yes, in MATH CLASS, .999...=1, because no method currently exists to distinguish between the two.

But in the real world, where 1 is a whole number, and .999... is NOT a whole number, it does not equal 1.

And if zero could divide, it would obviously = a pink light bulb.

[Almaviva]

You don't seem to understand how impossible and nonsensical it is to say zero could even theoretically halfway divide anything. It's like: "Well, if I subtract 0 from 1, wouldn't it even be just a little less?" Zero is nothing! Infinity, on the other hand, is mathematically, and theoretically possible

Well yeah, impossible and nonsensical is what we're talking about when we introduce the idea of infinity as a number.

"Here's a number that's bigger than any number." "Okay, what happens when I add 1 to it?" So it's nonsense, mathematically.

"Well if I subtract 1 from infinity, wouldn't it even be just a [i]little[i] less?

If I count to 10, how much progress have I made towards counting to infinity? Basically, none. If I count to 10 million, how much progress has been made toward infinity? Still none. If I count up the number of electrons in the universe, and then put that many zeroes after a 1, how close is that to infinity? Still no closer than counting to 10 is. That's the intuitive concept of infinity to me: As a proportion, 1 out of infinity is zero chance. 10 trillion out of infinity is still zero chance.

[Edmond Zedo]

From http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Infinity.html#s85

Recently, however, evidence has come to light which suggests that not all ancient Greek mathematicians felt constrained to deal only with the potentially infinite. The authors of [32] have noticed a remarkable way that Archimedes investigates infinite numbers of objects in The Method in the Archimedes palimpsest:-

... Archimedes takes three pairs of magnitudes infinite in number and asserts that they are, respectively, "equal in number". ... We suspect there may be no other known places in Greek mathematics - or, indeed, in ancient Greek writing - where objects infinite in number are said to be "equal in magnitude". ...

The very suggestion that certain objects, infinite in number, are "equal in magnitude" to others implies that not all such objects, infinite in number, are so equal. ... We have here infinitely many objects - having definite, and different magnitudes (i.e. they nearly have number); such magnitudes are manipulated in a concrete way, apparently by something rather like a one-one correspondence. ... ... in this case Archimedes discusses actual infinities almost as if they possessed numbers in the usual sense ...

Even if most mathematicians accepted Aristotle's potentially infinite arguments, others argued for cases of actual infinity, others argued for cases of actual infinity. In the first century BC Lucretius wrote his poem De Rerum Natura in which he argued against a universe bounded in space. His argument is a simple one. Suppose the universe were finite so there had to be a boundary. Now if one approached that boundary and threw an object at it there could be nothing to stop it since anything which stopped it would lie beyond the boundary and nothing lies outside the universe by definition. We now know, of course, that Lucretius's argument is false since space could be finite without having a boundary. However for many centuries the boundary argument dominated debate over whether space was finite.

It became largely theologians who argued in favour of the actual infinite. For example St Augustine, the Christian philosopher who built much of Plato's philosophy into Christianity in the early years of the 5th century AD, argued in favour of an infinite God and also a God capable of infinite thoughts. He wrote in his most famous work City of God:-

Such as say that things infinite are past God's knowledge may just as well leap headlong into this pit of impiety, and say that God knows not all numbers. ... What madman would say so? ... What are we mean wretches that dare presume to limit his knowledge.

Indian mathematicians worked on introducing zero into their number system over a period of 500 years beginning with Brahmagupta in the 7th Century. The problem they struggled with was how to make zero respect the usual operations of arithmetic. Bhaskara II wrote in Bijaganita:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

It was an attempt to bring infinity, as well as zero, into the number system. Of course it does not work since if it were introduced as Bhaskara II suggests then 0 times infinity must be equal to every number n, so all numbers are equal.

There's more regarding the idea of infinite there, a little on Newton and others.

[Doug]

Zzzzzzzzzzzzzzzzzzzzzzz

Anyone who really thinks 3 times 0.3333 recuring is not equal to 1 needs to stop thinking in finite terms. If your just doing it to promote argument then fair enough, I got bored of this argument when I was 8.

[Edmond Zedo]

Zzzzzzzzzzzzzzzzzzzzzzz

Anyone who really thinks 3 times 0.3333 recuring is not equal to 1 needs to stop thinking in finite terms. If your just doing it to promote argument then fair enough, I got bored of this argument when I was 8.

Who does? since 1/3=0.333..., then 3*0.333...=1. There isn't any crap out there you can divide anything by to get 0.999...

[QrioCT]

i didnt have the time to read everyone elses response so sorry if im repeating something that someone already thought of.

but i think .999 is a really strange number. like say, if you're trying to cut a 0.999(goes on forever) piece of an apple, how do you know when to stop counting and insert the knife since the number goes on forever? wouldnt you rather be counting for ever and ever? (pretend you have a magical knife that can divide very very well and you can cut through atoms too)

so wouldnt you rather have had the whole apple? or maybe this is a bad way to think of it.

[melancholeric]

Well, with 20 pages and 197 replies, no one has covered that particular aspect of the number yet.





Sorry.
:rofl: :rofl:

[Edmond Zedo]

so wouldnt you rather have had the whole apple? or maybe this is a bad way to think of it.
That's the old kick in the nuts, isn't it. You never get to have that last bit of apple.

[HackerX]

Its no doubt been stated before... but 1/3 != 0.3333 etc

The process of recursion is an attempt to get close to 1/3, but the idea is that each "3" you go, you get closer, but never reach 1/3

[QrioCT]

Its no doubt been stated before... but 1/3 != 0.3333 etc

The process of recursion is an attempt to get close to 1/3, but the idea is that each "3" you go, you get closer, but never reach 1/3

yes. nice, it works. so the idea is that 0.99999... tries to get closer to 1 but never reaches it? meh, i guess so.

ps. i'll never get the last bit of my apple no matter how hard i try now...:D

[Network Alchemy]

0.3333.... is equal to 1/3 and it is not recursion which would be a process it is a single number but i think i shall do what almaviva did and let this argument go

[Dman]

Coming from a non-math major, I think the problem with this formula is that there are different schools of thought trying to reconcile the issue. One school, mostly coming from the non-math people, sees the number 1 as a finite, specific point on the number scale. A whole number.

However, introducing the more advanced schools of continuance and infinity, 1.0 ceases to exist as simply a specific point. That’s the thing that blows your mind, because intuitively we’re inclined to believe that if it’s not 1.0, it’s not 1.0. But throw in this crazy infinity thing and you have to throw intuition out the window.

The fact is that there’s no way to distinguish between .999…. and 1.0, other than what it looks like written down here. If you have this concept of something that is infinitely closer to 1.0, it is impossible to measure how it isn’t 1.0. Using the apple analogy, you could never know that .999…. of an apple was any different than 1.0 of an apple. Of course, the analogy is nonsense anyways, because you can’t apply infinity of some tangible object in this manner.

Am I right?

[QrioCT]

yeah, the apple thing was just an analogy. in real life i dont think it would work. but thinking at that train of thought, 0.9999... of an apple does not equal 1 apple because no matter how close you get to 1, you'll never get there.

on the other hand, i question that 0.99999.... can be a real number measured by tangible means. imagine an inch ruler, and imagine moving your finger to the point 0.99999... of an inch. since 0.9999... continues forever, you must keep moving your finger every millisecond toward the 1 inch mark. (pretend you are superman and can move as precisely as you want)

how can you keep moving something toward a direction forever and never get there?

[Edmond Zedo]

moving something toward a direction forever and never get there?
Why is this difficult for so many people? Never hit with the Common Sense club?

Because, figuratively speaking, your speed keeps decreasing.

[QrioCT]

yeah, but you keep going on forever. so even if ur speed decrease, wont you get there sometime? cuz u have to keep moving.

[Dman]

yeah, the apple thing was just an analogy. in real life i dont think it would work. but thinking at that train of thought, 0.9999... of an apple does not equal 1 apple because no matter how close you get to 1, you'll never get there.

on the other hand, i question that 0.99999.... can be a real number measured by tangible means. imagine an inch ruler, and imagine moving your finger to the point 0.99999... of an inch. since 0.9999... continues forever, you must keep moving your finger every millisecond toward the 1 inch mark. (pretend you are superman and can move as precisely as you want)

how can you keep moving something toward a direction forever and never get there?

That was my point.

In this context you can't apply infinity to a tangible amount of something. To put another way, imagine a piece of the apple that is (1 - .999...) in size. Not possible. Can't apply infinity to a tangible object like this.

[Dman]

yeah, but you keep going on forever. so even if ur speed decrease, wont you get there sometime? cuz u have to keep moving.

No - your speed decreases infinitely.

[QrioCT]

No - your speed decreases infinitely.

hmmm...you might be able to do that. but can you really infinitely decrease speed? the question repeats itself here. this time, you go down in speed: 0.0....1. so does that equal 0? ok, speed is distance/time. so in order for speed to infinitely go down, the distance you travel per second has to go down infinitely. so it's almost like what you said about the apple analogy again. keeping approaching 0 in distance on a ruler, wouldnt you eventually get there? or maybe not. im not a math/physics major either.

[Dman]

Yep, .999... is an abstract, and it equals 1 in an abstract way. You can't think about it like you think about 1 as a tangible object. Think about it like time, like saying when is it exactly 1:00. Using the best instruments possible, you could never determine exactly when you reached 1:00. It would always be possible to get a more and more precise measurement of when you reached 1:00, infinitely so. In other words you can carve out smaller and smaller increments of time leading up to 1:00, but you would always be limited by the ability of your measuring devices. You can only estimate a fuzzy point where it reaches 1:00 and passes it.

[QrioCT]

i guess...our time system is determined by the change of earth spinning around the sun. so, u mean there is no exact point where the earth is at 1:00 position. yeah, i get what ur saying about the measuring device thing. u cant get exactly at a certain point. wait...but what if u first put it there and then say that's the exact point? wait...no... i need to think.

[QrioCT]

i got a system running here.

time is actually just change in distance that's constant(refer to the "time doesnt exist" thread if you dont get what im saying).

the speed equation distance/time is just (change in distance of one object)/(a constant and consistent change in distance of another object, like earth around the sun) speed is just a comparison of two objects changing in distance together.

so speed can only infinitely decrease if distance can infinitely decrease.

and by the way, does 0.000...1=0? because if it does, then 0.99... would equal 1. thats because 0.00..1 + 0.999.. =1, and 0+1=1, so if 0.00..1=0, then 0.999..=1.

i was thinking, since the 0's in 0.00...1 keeps going on forever, if you were to actually write the number out, you would be writing 0.000.... and never write the 1. so does that make it equal 0? because if it does, then we can conclude 0.99... equals 1.

[Edmond Zedo]

You're barking up the wrong tree, silly, but I will at least tell you that I came up with an easy way to notate that.

(1-0.999...)

I call it "krunk."

[MacGuffin]

and by the way, does 0.000...1=0? because if it does, then 0.99... would equal 1. thats because 0.00..1 + 0.999.. =1, and 0+1=1, so if 0.00..1=0, then 0.999..=1.

i was thinking, since the 0's in 0.00...1 keeps going on forever, if you were to actually write the number out, you would be writing 0.000.... and never write the 1. so does that make it equal 0? because if it does, then we can conclude 0.99... equals 1.


Using the same intuition that to me says 0.9999..... does not equal 1 - I'd say 0.000...1 does not exist.

Why?

Because 0.000...1 says there is an infinity of zeros before you get to the one. But since infinity is unbounded, you can never get to the end to place a one.

But for practical purposes, 0.000...1 does equal zero, the same way 0.999... equals one.

[Dman]

But for practical purposes, 0.000...1 does equal zero, the same way 0.999... equals one.

Nah, I don't think so. .999... is an infinite progression, whereas .000....1, although tiny, has a finite end. There is something there, although extremely small. Zero is the absence of anything. By attaching a 1 to the end of .000… it effectively brings it into existence.

[MacGuffin]

Nah, I don't think so. .999... is an infinite progression, whereas .000....1, although tiny, has a finite end. There is something there, although extremely small. Zero is the absence of anything. By attaching a 1 to the end of .000… it effectively brings it into existence.

I don't think that 0.000...1 had a finite end, which is why I don't think it exists.

[Dman]

I don't think that 0.000...1 had a finite end, which is why I don't think it exists.

I must have missed something...

If it has a one on the end of it (emphasis on "end"), by definition it would be finite, no?

[MacGuffin]

I must have missed something...

If it has a one on the end of it (emphasis on "end"), by definition it would be finite, no?

How many zeros are in front of that 1? If 0.000...1 finite, the answer cannot be "infinity".

[Dman]

How many zeros are in front of that 1? If 0.000...1 finite, the answer cannot be "infinity".

Ok, I see what's going on. The problem is QrioCT said ".000...1 going on forever" which is nonsense (no offense Qrio). That's like saying imagine infinity but it's finite.

[QrioCT]

so does 0.0...1 even exist at all if its "imagine infinity but its finite?"(Dman's quote) it has to be either finite or infinite....i think.

this whole 0.0...1 doesnt make sense, like Dman said.

But if that's the case, what adds to .99... to create 1?

[Edmond Zedo]

Qrio we've been over all this until it was as clear as it's ever gonna get. Read!

[MacGuffin]

so does 0.0...1 even exist at all if its "imagine infinity but its finite?"(Dman's quote) it has to be either finite or infinite....i think.

this whole 0.0...1 doesnt make sense, like Dman said.

But if that's the case, what adds to .99... to create 1?

Nothing.

[Dman]

Qrio we've been over all this until it was as clear as it's ever gonna get. Read!

No! About every 3 or 4 pages, let's go over it all again!

Sorry, the smart-ass thing coming out again. I believe Edmond is referring to his "krunk".

BTW - Brilliant signature E.Z.

[HackerX]

so does 0.0...1 even exist at all if its "imagine infinity but its finite?"(Dman's quote) it has to be either finite or infinite....i think.

this whole 0.0...1 doesnt make sense, like Dman said.

But if that's the case, what adds to .99... to create 1?

The easy way out is:
1-0.99999....

does 0.0...1 (or 0.9999.. for that matter) exist.. no, that's the point of infinite recursion, they're concepts.

[Claverhouse]

So, to wrap it up: with an infinite system .999... is the same as 1.0 in your imagination; however, in a finite world of reality infinity does not actually exist, therefore the question cannot arise.

Or something.

'Of Souls' is no longer the official thread of evil...



Claverhouse :ph34r:

[Dman]

So, to wrap it up: with an infinite system .999... is the same as 1.0 in your imagination; however, in a finite world of reality infinity does not actually exist, therefore the question cannot arise.

Or something.

'Of Souls' is no longer the official thread of evil...

Claverhouse :ph34r:

Not so fast...

Infinity may indeed exist - for instance it is generally believed that gravity's effects are infinite. Therefore .999… is the same as 1.0 in reality – but again referring to a continuum (since infinity is involved).

[Almaviva]

0.3333.... is equal to 1/3 and it is not recursion which would be a process it is a single number but i think i shall do what almaviva did and let this argument go

You give me too much credit for self-control, lol. I haven't replied here because I haven't been on the forums at all.

Anyway, there are three approaches to this "issue".

1. Use mathematics. Make sure you have precise definitions for everything you are speaking about. (E.g. what does 0.00...1 mean? Is it like a square circle?) Learn about the way real numbers are treated mathematically. Realize that the written decimal expansion most reasonably means a limit. Conclude, based on definitions as they are used, that 0.999... describes the real number 1.

2. Try to resolve the issue using only intuition. This way, you don't have to worry about contradictions, or even knowing exactly what you are talking about. Conclude whatever the heck you want. If you want 0.999... to be something different than 1, conclude that. If you want them equal, conclude that instead! This approach is very popular. If you've had experience doing things with approach 1, rigorous mathematics, this starts to become less fun and interesting.

3. Work precisely as in #1, but reject the usual construction of real numbers! (This requires fully understanding what you are rejecting.) Contruct another system for working with things, maybe involving "infinitesimals". This is an extremely hard problem, in that some mathematicians worked with this goal for over a hundred years before it could be done rigourously. Once you do it rigourously, it isn't all that useful.

[IceMasterDLarge]

Well .333333333333333333333333333 isn't exactly 1/3, just an estimation. I've always pondered about the digits of the TRUE 1/3, but 1 will always equal 1, on the other hand it's like the third way point between .33 and .34, and 1 is the third way point between .333 and .334 and so on, which at the end would lead to .333333333333333333333333333333333333333333, but that times 3 should equal .999999999999999999999999999999999.
8O
This something to ponder, but on the other hand, the more nines added after the decimal, it gets closer and closer to one. So maybe eventually it would actually equal (1/3)*3, (a.ka.: 1).

[Network Alchemy]

perhaps this thread should be closed

[Ascending]

Perhaps, but here is a reference I found some time ago. Forgive me for not reading all 23 pages but this seemed definitive.
Repeating Sequence (http://mathforum.org/library/drmath/view/64296.html)

[Almaviva]

By the way, this question, whether 0.999...=1, is one of the math issues that gets the most argument between mathematicians and people trying to argue intuition. I think this is because there is a pretty appealing intuition people want to cling on to that you can't take a bunch of things less than 1 and make something equal to 1. (But this is exactly the key thing about the concept of a limit.)

What you can graduate to after you get sick of arguing this, is another problem which also stretches the intuition, that is also extremely popular to argue:

There are rational numbers and irrational numbers. Rational numbers can be represented by fractions, and have repeating decimals. E.g. 1/7=0.142857142857...

Irrational numbers have non-repeating decimals, and can't be represented by fractions, e.g. pi, sqrt(2), 0.1234567891011121314...

Now, according to mathematical argument, there are only as many rational numbers as there are natural (counting) numbers, 1,2,3,4,... Because you can pair them up. But you can't pair up irrationals with natural numbers. So, there are "many more" irrationals than rationals. In fact, "essentially all" numbers are irrational.

But, between any two rationals is at least one irrational, and between any two irrationals is at least one rational. (Infintely many, in both cases.) So how can there be "more" irrationals, intuitively?

Some people don't like this, and try to poke holes in the mathematical arguments. The most famous one is the "diagonal argument" showing that there are "more" real numbers than natural numbers.

[Claverhouse]

PLACE-HOLDER FOR CLASSIC STATUS

[file cabinet]

just thougth I'd let everyone know.......
http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

[GreyDeath]

I suck bad at maths but I wonder if maybe, if we don't throw .999 into a math problem and just look at practicality, .999 is effectively 1. If I'm making cookies (why cookies...cuz they're tasty) and it calls for 1 stick of butter but I only have .999 a stick of butter...hell, its still a whole (1) stick of butter for all intents and purposes.

Maybe be a backwards ass way to look at it, but like I said, thats with throwing the actual math applications out the window.

0.999 stick o' butter vs 1.000 stick of butter, its effectively the same thing.

[Rozza]

Yeah, why not? Come on guys, live a little... ;)

[Limey]

I measure in man inches (tm)
So .501=1 no problem.
It aint lyin' unless yer flyin' !

Honestly, who doesn't measure from the bezel.

Watch out yo! I getz my bezel in yo veszel and muzzle yo puzzle!

EDIT: I don't know wtf I just said either, so don't ask.

[Rozza]

I don't know wtf I just said either, so don't ask.

It stimulated my imagination anyhow. ;)

[PlayerOfGames]

But, between any two rationals is at least one irrational, and between any two irrationals is at least one rational. (Infintely many, in both cases.) So how can there be "more" irrationals, intuitively?

I'm surprised noone mentioned this before, but the corollary of that fact is another reason why 0.999... = 1 - they are both rational (repeating) numbers, and there can be no other numbers between them since 0.999 is infinitely close to 1.

So, since there can be no irrational number between these two rationals, but there must be if they were distinct rationals, they must be the same number.

[ben from below]

I measure in man inches (tm)
So .501=1 no problem.


=))

[sandwich]

I'm surprised noone mentioned this before, but the corollary of that fact is another reason why 0.999... = 1 - they are both rational (repeating) numbers, and there can be no other numbers between them since 0.999 is infinitely close to 1.

Okay, this is the problem. .999 is not infinitely close to 1. It is .001 close to 1. What do they teach in schools these days?

[puzzled-observer]

Okay, this is the problem. .999 is not infinitely close to 1. It is .001 close to 1. What do they teach in schools these days?

it's meant to be .999 repeating
Whereby it's infinitely close to one.

[sandwich]

it's meant to be .999 repeating
Whereby it's infinitely close to one.

But that isn't what they wrote, and this thread MUST continue until people stop making the clumsy mistake of .999 = .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...

[puzzled-observer]

But that isn't what they wrote, and this thread MUST continue until people stop making the clumsy mistake of .999 = .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...

I think you forgot a few. Allow me:
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999... k i'm bored

[centerofthesun]

1/3 is not .333 or even infinitely repeating.. that's why they call it 1/3.

[PlayerOfGames]

1/3 is not .333 or even infinitely repeating.. that's why they call it 1/3.

Actually, I believe 1/3 = .333... for the exact same reasons given above that .999... = 1.

[PlayerOfGames]

Okay, this is the problem. .999 is not infinitely close to 1. It is .001 close to 1. What do they teach in schools these days?

You win

[Autumn]

Sweet Jesus! I can't believe that this thread still continues...

Actually 0.9999... is equal to 1.0 just as 0.3333... is equal to 1/3 .
Deduction:
0.9999.... * 10 = 9.9999...

10x - x = 9x

9.9999... - 0.9999... = 9.0 = 9x => x = 1.0
Simple as that.

[LongSilence]

Jesus Christ, if no-one used calculators for their sums we wouldn't even be having this discussion. Mathematics uses language that can be misunderstood and doesn't always quite literally say what it means. Big deal.

[LowEnd]

I suck bad at maths but I wonder if maybe, if we don't throw .999 into a math problem and just look at practicality, .999 is effectively 1. If I'm making cookies (why cookies...cuz they're tasty) and it calls for 1 stick of butter but I only have .999 a stick of butter...hell, its still a whole (1) stick of butter for all intents and purposes.

Maybe be a backwards ass way to look at it, but like I said, thats with throwing the actual math applications out the window.

0.999 stick o' butter vs 1.000 stick of butter, its effectively the same thing.

ugh, this made my head hurt more than any amount of math problmems