infinity in numbers

[MAss]

Hi there,

looks a little off topic, but its rooted in philosophy....
I'm no math expert here, but I was looking at proofs that 0.9 recurring = 1.

x=0.999...
10x = 9.999...
10x-x = 9
therefore x = 1

couldn't the same logic be used to prove that x <> 1?

ie:

x= 0.999...
y= 1 - x
x + y = z

z can't then be equal to x... can it?

i mean... x + y = 1, but 1 + y can't equal 1... right??? right???

and ah... 1 - y - y - y - y - y - y - y recurring equals 0 right?? so it must have SOME value... I dunno about that last part actually.. i just put it in there on a whim.... :P

ahm.. yeah any thoughts would be appreciated.. this one's wrapped its tentacles around my brain quite tightly ....

MAss

[Durin]

your second example only proves that 0,999 recurring is actually just 1. Just as the first is proof of that. For your example to work, x would have to be one, which is what was trying to be proved.

[MAss]

ok.. i'll get all a little heisenbergs cat here..... throw some big fat quantification into the scenario...

you've got a half moon cookie..
(of infinite detail... I'll be scuffling over who has the larger number of chocolate muons & quark crumbs when it comes time to eat it..)
half is black, half is white...
the number is the angle on the cookie that a point resides... (Point G)
0 degrees and up would be white, anything below zero would be black....

if point K is at 1 degree (i'm in the white side of the cookie) and then...

I move point G thusly.... G = (K - 1) - (1 - (0.333" x 3)) am i on the black part of the cookie, or the white part of the cookie....

..Sorry for overly complicating things... basically if i put G on 0 - (1 - 0.999")

i'm partial toward the chocolate side of the cookie... but we can share... thats what the half moon cookie is all about

MAss

[Symptom777]

C'mon MAss - ypou just proved that 0.99999... = 1, what more do you want?
1-0.99999... =0


also 1/3+1/3+1/3 =1 (no surprise)
1/3 = 0.3333....
3*0.3333.... = 3*1/3 = 1

0.3333333... is just a symptom of 1 divided by 3 not being a non-recurring decimal! Its a perfectly good fraction, and in duodecimal (base 12) it would be 0.4.

It doesn't contain any mystical property!!

[MAss]

so if i go at 99.9999... percent of the speed of light, i'm actually going at the speed of light?

I'm looking at an exponential curve closer and closer, and still never seeing it touch the roof.. theres never a point where it 'snaps' to 1

MAss

[Symptom777]

Originally Posted by MAss

so if i go at 99.9999... percent of the speed of light, i'm actually going at the speed of light?

I'm looking at an exponential curve closer and closer, and still never seeing it touch the roof.. theres never a point where it 'snaps' to 1

MAss

Yes. 0.999 recurring =1

BUT remember if at any time you STOP recurring the 9s, then its not 1! You have to recur to infinite number of decimal places!! son if you have 99.9999% to one million decimal places, thats not 100%.

Any asymptotic curve hits the line at infinity.

[MAss]

Thanks for all your responses, I know you're not necessarily up for a big debate on this, but it started out as an innocent question, and the proofs are just seeming to cement my dubiousness...

I believe it's correct. (in a sense) but i believe it goes a little deeper than that (i'll explain...)

I'm going to go out on a limb here, and suggest that the term '1' actually indicates that the data required to created a 'whole' has been met. (read on.. its not that long)
I believe that a 'whole' goes up to and includes 0.999... but doesn't extend to the number 1

==1==

Take a year (just to paint a pretty picture)...
ok.. so the year 0 begins at 0.000...% and continues, through 20% of the year, through 33.333...%, through 80% of the year.. all the way up to 99.999..% of the year.... the year zero ends at that 'point'. After it reaches one, it is a separate whole. It becomes the year 1 (which is back at 0%)
the year 0 and the year 1 both exist independently, and without overlap..

3 x 33.333... = 99.999... because that includes every aspect of that year without touching or overlapping any point in the neighboring years... it includes every aspect of that single "whole".
so in that sense 1 whole is actually made up of 0.999... This is, in my opinion, a different thing to saying that the number 1 is made up of 0.999... because the number 1 is an entirely new set.. the number 1 belongs to the second 'whole' in the series.

==2==
On a slightly different aspect of it:
Measure the smallest distance between two points.... 0 and 1....
actually.. 10% of 1 is a smaller distance.... actually 10% of that is a smaller distance:
0.1... 0.01... 0.001...0.0001...0.00001 etc etc (q)
call it 'q'.
It's an exponential curve, never touching zero. It's the imaginary remainder left at the end of 0.999... It is the difference between 0.999... and 1 (imaginary or otherwise) It's also essentially the smallest 'increment' between any two numbers.
so using the logic that makes 0.999... = 1, doesn't it follow that the distance between any two numbers is actually 0?
...making every number the same number? (well.. I don't think it does atleast.)

To paraphrase a proof that 0.999... = 1: There is actually no positive number that can be written that is the distinction between any two numbers (there is always one smaller, and definitely not a negative number), so therefore if the number is not positive, it must be zero
According to that logic, it would demonstrate that theres no distinction between *any* numbers...

==3==
Ignore this point really as it only complicates the issue... but...

I might even say that 1/2 of 1 whole is actually 0.4999... etc etc
The logic here is that, (perhaps if only to keep with our current clean system of maths) at the heart of it the problem lies not at 0.999... but at the point at which 0 (nothingness) becomes somethingness.... (boy.. doesn't it ever)

Assuming we add q (refer to the earlier part of this doc for what q is) to 0, we can then go ahead and add "wholes" ie 0.999... to it to give us the natural numbers: 1, 2, 3, 4, etc. and their respective even decimals: ie 1/2 = 0.5

====

So to sum up.. i'm suggesting that 1 'whole' is made up of 99.999...% , while the *number* 1 in whatever form it takes, is made up of 100% (which is actually 0% if you consider it to be a new set or 'whole'.. which you should...)

You count one orange, if you say you have 1.000... oranges, you've finished counting to 0.999... of the orange and moved on to the next set... you're sitting there staring at the empty spot on the table waiting for the next orange.... you're not even looking at the original orange when you look at the 100th percent

what are your thoughts?

MAss

[Symptom777]

MAss

I'm not explaining this good am I?
you are talking about two different (sets of) numbers here, one is 0.999999... (and numbers like it) that have recurring decimals. The other is 0.99..9 (and numbers like it) that have a finite set of recurring digits.
1=0.99999...
1=/=0.99..9

1/3 = 0.33333... simply because 3 goes into 10, 3 times with 1 left over! There is nothing mystcial about this. if we used bas 3, 1/3 = 0.1 exactly. If we use base 12, 1/3 = 0.4 exactly, and 1/3 (in base 10) = 0.33333... exactly. BUT 1/3 =/= 0.333..3

[MAss]

every time i type 0.999... I am refering to 0.99999............................ for ever... forever! ad infinitum.. i don't care about the non recurring 0.999...9 not at all! :P

I understand the premise all too well, what I don't understand however is how an exponential curve... something which is used so frequently is maths, is denied its right to exist?

For example... I'll never get a vacuum by removing 99.999...% of all matter contained within a region...

If there exists 1kg of matter in a region,
and then I remove 90% of all matter contained within the region...
and then I remove 90% of all matter contained within the region...
and then I remove 90% of all matter contained within the region...
etc etc ad infinitum
I will *never* get a vacuum... simply never...

At what arbitrary point are you going to declare the exponential curve i'm following to be equal to 0?

never? thought so...

to suggest any different would be deceptive, and to deny the existence of a number known as "approaching 1" is to deny the destinction between ANY numbers...

Every number is separated by an infinitely small value..... so infinitely small that it is essentially 0... (but not zero... approaching 0)

MAss

[Symptom777]

This is simply Zeno's paradox in a different setting!!
See http://www.mathacademy.com/pr/prime/articles/zeno_tort/

[MAss]

yeah... its zeno's paradox in a different setting... simply...

the point remains.

[Symptom777]

Yeah, but Zeno's paradox isn't a paradox, it's based on false reasoning.

[DoctorDick]

MAss,

In response to your private note!

You are confusing two very different issues. First there are mathematical concepts! These are concepts which are internally consistent. They allow us to sumarize the consequences of things we can do in terms of things we can not do. The concept "infinity" and the concept "continuity" are very valuable concepts which allow us to sumarize the consequences of procedures which are otherwise difficult to express.

You are confusing these valuable concepts with reality and what we can actually do which is another matter entirely (an issue pointed out by Zeno many centurys ago). Infinity is defined by the fact that, no matter how many times we have done something, it can be done more times. It is definately not a "number".

When we refer to a result which is proved as a limit as we approach infinity, we are essentially saying that, by increasing the number of times the process is continued, we can make the difference between that limit and what we actually achieve as small as is desired. When one says that .9999... is equal to 1 one is actually saying that the "operational" difference between the two can be made as small as is desired.

This is a "mathematical" definition, not a consequence of any actual real proceedure. Now, what you are actually bringing up is a slightly different matter. When one defines a region (suppose the line between zero and one on the real axis), one is allowed to either define the boundry itself as part of the region or as not part of the region (the common term used to specify which you mean is the "open" region or the "closed" region). The open region does not include "0" or "1" whereas the closed region includes those values.

Mathematics is an internally consistent structure of definitions. One has to be careful to make sure you define what you mean; otherwise it is very easy to get inconsistent results. A lot of people do very sloppy reasoning with mathematics. It is not a simple subject.

Have fun -- Dick

[Symptom777]

MAss - I was thnking about this some more. Now I see DrD has put in some stuff which is right, but I'm not sure it's convincing to you?? Maybe it is. Anyway, this was my thought to try and shed some light:

Consider the function y=x. We can plot this function as a graph, with x on the horizintal axis and y on the vertical axis. So, get some graph paper, with say 1cm squares on it. On the vertical axis (y), mark each square as 1,2,3,4,5 (starting with 0 where the axes cross), on the horizontal axis (x), mark each square as 1,2,3,4,5 (starting with 0 where the axes cross). Now plot the function x=y. You'll get a straight line bisecting the axes.

Now get a second piece of graph paper. This time mark the vertical axis (y) as before - mark each square as 1,2,3,4,5 (starting with 0 where the axes cross), however on the horizontal axis (x), mark each square as 0.9, 0.99, 0.999, 0.9999, 0.99999, etc. (starting with 0 where the axes cross). Now plot the same function. You'll get an asymptotic curve pissing off to the right and never quite getting to the value y=1.

But it's the same function! This is the same as Zeno did, only he compressed time.

Did this help?

[MAss]

hey thanks doc dick and symptom,

I see the mathematical concept quite clearly now (and it makes alot of sense, but still would be a long way off telling anyone as a fact that 0 = 0.999... for fear of misleading them)

I think perhaps the problem here comes from the fact that:
Conceptually 0.999... can be considered in an instant as being already complete, whereas any real world example requires that at any given instant (when someone wants to compare for example) the recursion is halted and the number becomes 0.999...9

also, the idea that 0 through to 0.999... would contain every bit of information within 1 whole (1) is something i would consider legitimate, but, so would 0 back through to negative 0.999, giving an arbitrary preference to 0 (which would be fictional.. making it illogical to draw a line from 0 to 0.999... in preference to 1 to (1-0.999...) if you know what i mean)

also also... 0.999... could be considered 1 because if you were to go at 99.999...% the speed of light, you would have to use an infinite amount of energy to continue to accellerate infinitely (so it would hypothetically be allowed, because the assumption of 99.999... assumes infinite available energy)

so thanks ey...

I still maintain that 99.999...% contains all the information within a whole unit, but that may be the same point...

Aspects of it remain a little unclear in my mind, (the differing values of infinity? I may be back... :P) so if you have anything to add, feel free, otherwise thanks alot for the time.

MAss