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  #1  
Old 08-15-2001, 06:42 PM
Kevin Partida Kevin Partida is offline
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My geometry teacher used to always tell my class that .99999999 with a repeating bar over it is equivalent to 1. He also said that he could prove it mathematically, but he never gave the proof. Could someone out there give the proof that .99999999 with a repeating bar is exactly equal to 1?
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  #2  
Old 08-15-2001, 07:22 PM
AwSnappity AwSnappity is offline
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All I know is that fractions with a denominator of 9 are shown as decimals with the numerator repeating. (I know, that was poorly worded. Allow me to demonstrate.)

1/9 is the same as 0.1111...
2/9 is the same as 0.2222...
3/9 is the same as 0.3333...
...and so on.

9/9 would be the same as 0.9999...
Since a fraction with the same number as the numerator and denominator equals one, we can conclude that 0.9999... = 9/9 = 1.

I hope that was understandable.
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  #3  
Old 08-15-2001, 07:31 PM
andros andros is offline
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Yup, that's the easiest proof. Well, actually that's a common sense shorthand. The mathematical proof (in thirds for brevity) goes more like:

1/3 * 3 = 1

1/3 = 0.333...

0.333 * 3 = 0.999...

Therefore, 0.999... = 1
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  #4  
Old 08-15-2001, 07:38 PM
JS Princeton JS Princeton is offline
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Here's another proof:

Given that:
x = 0.9999999999 etc.
Then:
10 x = 9.999999999 etc.
So, then
10x - x = 9.999999999 etc. - 0.999999999 etc. = 9
Which implies...
=> 9x = 9
Which implies...
=> x = 1

but x = 0.9999999999 etc.
so 0.999999 etc. = 1

QED
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  #5  
Old 08-15-2001, 07:40 PM
ModernRonin2 ModernRonin2 is offline
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Another proof, the one that I like best...

We want to prove that .9rep = 1.

First, let's assign a symbolic name for .9rep so it's
easier to deal with. Let's use "x" - it's as convenient as
any other letter.

So,

x = .9rep

Now, multiply both sides of the above equation by 10:

10x = 9.9rep

Now, subtract x from each side:

9x = 9.9rep - x

But since "x" is just .9rep:

9x = 9.9rep - .9rep

Or

9x = 9

Divide both sides by 9, and...

x = 1

Q(uickly).E(nds).D(at).

Slick, eh?


-Ben
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  #6  
Old 08-15-2001, 07:40 PM
barbitu8 barbitu8 is offline
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This is .9 repeating ad infinitum, but never reaching 1.0 since it would take infinity to get there. I believe, in calculus, that that is considered 1.0.
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  #7  
Old 08-15-2001, 07:41 PM
andros andros is offline
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Thanks, JS Princeton . . . I was trying to think of that one. I always prefer an algebraic proof to an arithmetic one.
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  #8  
Old 08-15-2001, 07:42 PM
JS Princeton JS Princeton is offline
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By the by... by using the method I posted you can convert all repeating decimals into whole number fraction. Just instead of 10x use whatever power of 10 is required so that your repetition is cancelled upon subtraction of (10^n)x - x

As a boss of mine at a science museum used to say whenever a visitor was exposed to something educational (even if it was something like a magnet and they had clearly seen it before): "Pretty cool, huh?"
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  #9  
Old 08-15-2001, 07:44 PM
andros andros is offline
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You, too Ben. #@$@ simulposts.
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  #10  
Old 08-15-2001, 07:45 PM
SPOOFE SPOOFE is offline
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So does this mean that it's impossible to have a true .9999..., that is, having the 9 repeat itself all the way into infinity? Or, mathematically, do we always refer to .9999... as 1?

In other words... is .9999... ALWAYS equal to 1?
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  #11  
Old 08-15-2001, 07:50 PM
andros andros is offline
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Well . . . yeah, SPOOFE. I mean, if you could manipulate an infinitely repeating number, 0.999... would always behave as though it were equal to 1.

It would be different if we were talking about a converging series, in which something approaches 1, but never can be said to reach it. But 0.999... is a constant.
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  #12  
Old 08-15-2001, 07:50 PM
JS Princeton JS Princeton is offline
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As long as you're certain the 9s go on forever (iterative function, I believe it's called) it's equal to one.

If it stops or changes numbers ANYWHERE, it's not.

So, this is a semantics question: how do we know that it repeats forever?

Well... I could give you an inductive proof of that (which isn't inductive at all, but deductive... just utilizes a theorem in mathematics that certain things always happen the same for any number on up to infinity)... but we'd have to start with some assumptions.

What would those assumptions be?

Well, the number would have to be zero followed by an infinite number of nines.

But there you see, we really already have the proof (though not rigorous).
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  #13  
Old 08-15-2001, 07:51 PM
JS Princeton JS Princeton is offline
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of course, don't forget the all important decimal point that I blatantly left out of my assumption.
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  #14  
Old 08-15-2001, 07:55 PM
astro astro is offline
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Related thread


"Stupid math question"

http://boards.straightdope.com/sdmb/...threadid=71810

" How can 1/3+1/3+1/3=1 but .3333...+.3333...+.3333... never truly equal 1? I believe a recent thread tried to explain this, but truthfully I didn't understand the answer. Can someone explain this in plain English. Pretend I'm your six year old. "

"Why Daddy, Why?"
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  #15  
Old 08-15-2001, 07:57 PM
ModernRonin2 ModernRonin2 is offline
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It's a big of a boggler at first...

To think that a number you've known so well may not be what it seems.

The big difficulty that most people encounter with this, I think, is that their internal assumption that every number can only be represented with one syumbol has been violated.

But that's is not so strange if you think about it. What is the fraction 2/2? Isn't that just another symbol for "1"? Same with 5/5 and 32,973/32,973. They're all just different symbols for the same number. As is the symbol "1". As is the symbol ".9999...". Etc.

Maybe it'll help if you think of numbers like people. I'm Ben. On the boards I am represented by the symbol "ModernRonin." To the people who maintain the computers at my work, I am represented by my login, "bcantric". Some of my friends use my last name because they know more than one Ben. To these people, I am simply "Cantrick." These are all different symbols, but they mean the same person. Similiarly, "1" and ".999..." and etc are just different symbols for the same number.

It's a little easier to accept if you can think about it that way.


-Ben
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  #16  
Old 08-15-2001, 08:56 PM
MrDeath MrDeath is offline
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Quote:
Originally posted by SPOOFE In other words... is .9999... ALWAYS equal to 1? [/B]
Short answer: No! Long answer: In any conceivable real-world application, yes.

I could tell you why .99999... is not equal to one, but then I'd have to bore you to death.
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  #17  
Old 08-15-2001, 09:03 PM
JS Princeton JS Princeton is offline
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what on earth does "any conceivable real world application" mean?

I love it when mathematicians talk dirty ;-s)
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  #18  
Old 08-15-2001, 09:15 PM
Manduck Manduck is offline
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Quote:
Originally posted by MrDeath

I could tell you why .99999... is not equal to one, but then I'd have to bore you to death.
At the risk of being bored to death, could you give us a hint? Because I find it diffult to imagine .999... being anything but 1.
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  #19  
Old 08-15-2001, 09:24 PM
ultrafilter ultrafilter is offline
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Quote:
Originally posted by MrDeath
I could tell you why .99999... is not equal to one, but then I'd have to bore you to death.
I'm curious. Or are you talking about the series representation below?

A slightly more sophisticated answer to the OP: .9999... can be represented as nine times the sum of (1/10)n for n = 1 to infinity. This is a convergent geometric series, and the formula for the limit of such a series (9 * r/(1-r), where r is the common ratio) gives us 1. This is not the same formula you get if you start the sum from n = 0; in that case, you'd get 9/(1-r), which gives you 10. But it's not the right formula, so we don't care if it gives the right answer.

While this is more abstract, it would satisfy pretty much anyone. I've always felt that the argument that starts with x = .999.... is a little lacking in style, so I prefer this one.
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  #20  
Old 08-15-2001, 09:35 PM
mangeorge mangeorge is offline
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So .999.... is the same as .999999999999..... ?
Or .9..... ?
Wierd, but I think I get it. It's just a convention.
Except when your .9's go into a black hole.
Peace,
mangeorge
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  #21  
Old 08-15-2001, 09:35 PM
JS Princeton JS Princeton is offline
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Quote:
Originally posted by ultrafilter
I've always felt that the argument that starts with x = .999.... is a little lacking in style, so I prefer this one. [/B]
Wow, talk about your mathematical snobbery. (I'm always getting into arguments with mathematicians on these points, because they INSIST they know what the best way to say something always is).

Really, our two statements (yours of teh series, mine with the decimal) are ENTIRELY equivalent. Honestly. If I define a repitend to be infinitely repeating digits, by God, that's what it is.

I like your proof, in any case. This has become sorta fun, looking at all the ways you can come up with one.

By the by, your "starting with zero" series is the subject of another thread that's been zipping around tonight. I'd parse a URL, but I'm lazy. Look for it yourself all ye with curiousity.
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  #22  
Old 08-15-2001, 09:46 PM
ultrafilter ultrafilter is offline
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Quote:
Originally posted by JS Princeton
Quote:
Originally posted by ultrafilter
I've always felt that the argument that starts with x = .999.... is a little lacking in style, so I prefer this one.
Wow, talk about your mathematical snobbery. (I'm always getting into arguments with mathematicians on these points, because they INSIST they know what the best way to say something always is).[/b]
Course, I've never said that was the "best way", just the way I prefer. The algebraic answer is easier for most people to digest, and in a lot of cases that's what matters. I just thought I'd chime in with this as an alternate version.

Quote:
Really, our two statements (yours of teh series, mine with the decimal) are ENTIRELY equivalent. Honestly. If I define a repitend to be infinitely repeating digits, by God, that's what it is.
Yup. I was just a bit more explicit about it.
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  #23  
Old 08-15-2001, 10:01 PM
Saltire Saltire is offline
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I hope that these proofs have been enough to convince everyone, but I'm gonna add one anyway. Not really anything rigorous, but just a logical argument.

In order for two real numbers (however they may be represented) to be different, you should be able to describe some number that would exist between them. Stated another way, there should be some value you can add to one of the numbers that will increase it, but not push it higher than the other number. You should also be able to subtract something from one number without going below the other.

Any number you add to .9rep makes it larger than 1. Any number subtracted from 1 makes it smaller than .9rep.
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  #24  
Old 08-15-2001, 10:02 PM
kinoons kinoons is offline
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so I'm curious -- mathmatics is supposed to be an absolute. 1=1 -- therefor wouldnt 0.9999999... = 0.9999999... ?

in any pratical application of the numbers it would be so close to 1 that there would be no point to call it anything but 1, but, it still is not absolutely completely equal to one?
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  #25  
Old 08-15-2001, 10:06 PM
hansel hansel is offline
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Here's the way I first wrapped my head around the concept when it was presented to me:

What's 1 - 0.9999999...? It's 0.000000....1, where the one is preceded by an infinite number of zeros. Effectively, you never reach the 10-n that is the difference between 1 and 0.9999999999. So the difference is an infinite decimal expansion of 0.00000..., which is zero.

If x - y = 0, then x = y.
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  #26  
Old 08-15-2001, 10:14 PM
Race Bannon Race Bannon is offline
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This has been beaten to death, I don't know why I'm jumping in but I always want to...

Consider the axiom of completeness for the real numbers (that is, look it up if you don't know what it means). Any convergent sequence of real numbers converges to something that is a real number. .9rep is just a convienient way to represent a series, which has a sequence of partial sums, that converges.

So, to what real number does it converge? Use the axiom that says that for all real numbers A and B, one and only one is true: A>B, A<B, or A=B. Since there's no number in between .9rep and 1, then the first two can't be true. That leaves the last, .9rep=1.
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  #27  
Old 08-15-2001, 10:18 PM
Race Bannon Race Bannon is offline
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Quote:
Originally posted by kinoons
so I'm curious -- mathmatics is supposed to be an absolute. 1=1 -- therefor wouldnt 0.9999999... = 0.9999999... ?

in any pratical application of the numbers it would be so close to 1 that there would be no point to call it anything but 1, but, it still is not absolutely completely equal to one?
Well, it is absolute. You wouldn't hesitate to say 3-2=1 would you? No, because (3-2) and 1 are the same thing. Saying .999... = 1 is no different.
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  #28  
Old 08-15-2001, 10:20 PM
mangeorge mangeorge is offline
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Thanks, kinoons



Well I thought I had it, but;
How would you notate the difference between .9999... and one? In other words, The difference between .9999 and 1 is .00001, right?
How would one write the difference between .9999... (to infinity)and 1?
.0000....1 ?
Peace,
mangeorge
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  #29  
Old 08-15-2001, 10:32 PM
kinoons kinoons is offline
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Quote:
Originally posted by kellymccauley
Quote:
Originally posted by kinoons
so I'm curious -- mathmatics is supposed to be an absolute. 1=1 -- therefor wouldnt 0.9999999... = 0.9999999... ?

in any pratical application of the numbers it would be so close to 1 that there would be no point to call it anything but 1, but, it still is not absolutely completely equal to one?
Well, it is absolute. You wouldn't hesitate to say 3-2=1 would you? No, because (3-2) and 1 are the same thing. Saying .999... = 1 is no different.
but 0.999... and 1 are not the same thing... they cannot be. Any two values cannot at the same place in a number line. As soon as you deviate from the value of 1.000... you are at a different number that may be so incredibly close to one that it is pratically the same, but it is not exactly the same.
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  #30  
Old 08-15-2001, 11:16 PM
Manduck Manduck is offline
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Quote:
Originally posted by kinoons


but 0.999... and 1 are not the same thing... they cannot be. Any two values cannot at the same place in a number line. As soon as you deviate from the value of 1.000... you are at a different number that may be so incredibly close to one that it is pratically the same, but it is not exactly the same.
They are the the exact same thing. .999... does not deviate from 1 by even the slightest amount, as has been proven several different ways in this thread. They are dead-on, precisely the exact same number.
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  #31  
Old 08-15-2001, 11:21 PM
ElvisL1ves ElvisL1ves is offline
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Quote:
Originally posted by kinoons
but 0.999... and 1 are not the same thing... they cannot be. Any two values cannot at the same place in a number line. As soon as you deviate from the value of 1.000... you are at a different number that may be so incredibly close to one that it is pratically the same, but it is not exactly the same. [/B]
True only if you stop at a certain number of decimal places and look where you are. But the repeating bar means you NEVER stop. After an INFINITE number of decimal places, the difference from 1 DOES vanish.
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  #32  
Old 08-15-2001, 11:31 PM
kinoons kinoons is offline
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guys, your applying a mathematical proof to a philosophical debate -- the number line is not only a mathematical concept, but a philosophical one as well. Every number on the number line has its own position on the number line. No matter how you manipulate the math to try and prove that two different values can occupy the same location on the number line cannot be true. It is akin to saying that two different particles of matter can occupy the same space. It is simply not possible. The two pieces of matter can get so absolutely close to one another that the difference cannot be measured, but the two particles cannot be at the same place at the same time.
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  #33  
Old 08-15-2001, 11:33 PM
kinoons kinoons is offline
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Quote:
Originally posted by ElvisL1ves
Quote:
Originally posted by kinoons
but 0.999... and 1 are not the same thing... they cannot be. Any two values cannot at the same place in a number line. As soon as you deviate from the value of 1.000... you are at a different number that may be so incredibly close to one that it is pratically the same, but it is not exactly the same.
True only if you stop at a certain number of decimal places and look where you are. But the repeating bar means you NEVER stop. After an INFINITE number of decimal places, the difference from 1 DOES vanish. [/B]

please, please do keep adding the 9 to the end of 0.999... and I'll keep adding the 0 to 1.000 -- the two numbers will never be the same, they can not be. It is fundamentally not possible for any two numbers to occupy the same location on the number line.
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  #34  
Old 08-16-2001, 12:16 AM
Manduck Manduck is offline
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Quote:
Originally posted by kinoons

please, please do keep adding the 9 to the end of 0.999... and I'll keep adding the 0 to 1.000 -- the two numbers will never be the same, they can not be. It is fundamentally not possible for any two numbers to occupy the same location on the number line.
They are not two numbers, they are the same number, just written differently. They do occupy the same point on the number line because there is no difference between them.
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  #35  
Old 08-16-2001, 12:23 AM
kinoons kinoons is offline
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Quote:
Originally posted by Manduck
Quote:
Originally posted by kinoons

please, please do keep adding the 9 to the end of 0.999... and I'll keep adding the 0 to 1.000 -- the two numbers will never be the same, they can not be. It is fundamentally not possible for any two numbers to occupy the same location on the number line.
They are not two numbers, they are the same number, just written differently. They do occupy the same point on the number line because there is no difference between them.
Okay, this arguement is not getting anywhere...you seem to have compeletely ignored my above analogy -- two numbers cannot occupy the same point on the number line just as two particles of matter cannot occupy the same space. Every time you say that 1.000... is the same as 0.999... there is an additional 0.000...01 difference between the two numbers.
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  #36  
Old 08-16-2001, 12:32 AM
Manduck Manduck is offline
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Quote:
Originally posted by kinoons
Every time you say that 1.000... is the same as 0.999... there is an additional 0.000...01 difference between the two numbers.
Would it help to point out that 0.000...01 is just another way to write 0?
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  #37  
Old 08-16-2001, 12:33 AM
kinoons kinoons is offline
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Quote:
Originally posted by Manduck
Quote:
Originally posted by kinoons
Every time you say that 1.000... is the same as 0.999... there is an additional 0.000...01 difference between the two numbers.
Would it help to point out that 0.000...01 is just another way to write 0?
no 0=0 0 does not equal 0.000...01 -- like I said, one point on the number line, one number, and only one number.
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  #38  
Old 08-16-2001, 12:40 AM
Evno Evno is offline
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Hmm...

I'm not a math person, but I play one on T.V. And I have a question:

If .99999999..... = 1, then does -.11111111..... = 0?
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  #39  
Old 08-16-2001, 01:12 AM
John Kentzel-Griffin John Kentzel-Griffin is offline
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Quote:
Originally posted by kinoons
Okay, this arguement is not getting anywhere...you seem to have compeletely ignored my above analogy -- two numbers cannot occupy the same point on the number line just as two particles of matter cannot occupy the same space. Every time you say that 1.000... is the same as 0.999... there is an additional 0.000...01 difference between the two numbers.
Your analogy is flawed. You are taking the fermi-exclusion principle and trying to apply it where is does not apply. There is nothing in philosophy nor in mathematics that prevents two different names from referring to the same thing. Just as 2 + 2 equals 4, two representations can refer to the same number. One and 0.999. . . are two representations for the exact same number.
Quote:
Originally posted by Manduck
Would it help to point out that 0.000...01 is just another way to write 0?
It might help. If he realizes that zero can have more than one representation (like zero, 00.000...01, 0, 1-1, etc.) , mabye he'll realize that one can have different representations too. You know, like 0.999. . . and 1 both represent the same number.
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  #40  
Old 08-16-2001, 01:14 AM
douglips douglips is offline
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Quote:
Originally posted by kinoons Every time you say that 1.000... is the same as 0.999... there is an additional 0.000...01 difference between the two numbers. [/B]
And when you say that, you demonstrate a misapprehension of the concept of 'infinite'. If the ellipsis in your 0.000...01 stands for 'an infinite number of zeroes', then there is no '1' at the end. Where would you put it? Go ahead and walk out to the end of that infinite line of zeroes to put the one out there.

Another proof:
If .99999... is not exactly 1, then there is some number epsilon such that 1 - epsilon = .9999....
But, no nonzero epsilon can be demonstrated to satisfy this.

Kinoons would say epsilon is 0.000...01, but that is only do to a bad definition of "..."

Quote:
Evno gets into the act with
[b]I'm not a math person, but I play one on T.V. And I have a question:

If .99999999..... = 1, then does -.11111111..... = 0?
Surely you can't be serious.
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  #41  
Old 08-16-2001, 01:22 AM
Evno Evno is offline
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humor me.
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  #42  
Old 08-16-2001, 01:27 AM
kinoons kinoons is offline
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okay, you want to prove to me that 0.999... is equal to 1.000... then do this for me...


draw a line. Label one point on the line 0.9. one inch later on that line mark a point 1.000... (add as many 0's as you would like.

now, draw a line for 0.99. Now draw a line for 0.999. Now draw a line for 0.9999. Now one for 0.99999. Continue to keep drawing lines please.

Call me when you draw an additional line on 1.000
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  #43  
Old 08-16-2001, 01:30 AM
Evno Evno is offline
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The inability for the thickness of the ball in my pen that applies the ink to create thinner and thinner lines as I draw them does not justify your point.
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  #44  
Old 08-16-2001, 01:40 AM
kinoons kinoons is offline
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Quote:
Originally posted by Evno
The inability for the thickness of the ball in my pen that applies the ink to create thinner and thinner lines as I draw them does not justify your point.
okay, then change the scale -- make the two points 10 feet apart, and start drawing lines...

not convinced -- okay 100 feet apart...1000 feet? How much space do you need to be convinced that a line will never be placed on the exact same locaton?
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  #45  
Old 08-16-2001, 01:54 AM
Evno Evno is offline
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wait wait wait.......I reread your first post. I'm on your side here, I thought you said "Okay you want me to prove that 0.99999 = 1?" Sorry. Pretend that's what you were trying to say, then reread my post, then I'll make sense.


0.99999.....(to infinity) does not = 1.

If you say they are the same, then you are just saying "infinity is = 1". And that's just not true. Infinity is a concept ONLY, and not to be considered a number. "1" is just a number.

Apples are fruits. Oranges are round, like apples. Does that mean Oranges are Apples?

Saying an infinite number (like .99999...) is like 1 is comparing apples to oranges. They are two different figures in a huge way.
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  #46  
Old 08-16-2001, 02:09 AM
Manduck Manduck is offline
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kinoons, you're being silly. Obviously nobody has the time to draw the infinite number of lines that your demonstration would require.

Look at the proofs in the 2nd, 3rd, 4th and 5th posts in this thread. They are very simple and should erase all doubt. I mean, this is not a point of controversy in mathematics. It's gradeschool stuff.
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  #47  
Old 08-16-2001, 03:01 AM
ryoushi ryoushi is offline
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Quote:
Originally posted by kinoons
okay, you want to prove to me that 0.999... is equal to 1.000... then do this for me...


draw a line. Label one point on the line 0.9. one inch later on that line mark a point 1.000... (add as many 0's as you would like.

now, draw a line for 0.99. Now draw a line for 0.999. Now draw a line for 0.9999. Now one for 0.99999. Continue to keep drawing lines please.

Call me when you draw an additional line on 1.000
And for your task:

Present us with a list of 10 numbers between 0.9999... and 1. There should be an infinite number there for you to choose from, if they are really different numbers. Or, to go back to you number line - if they occupy different places, then there must be some space between them. Draw 10
lines in that space that are on neither 0.999... nor 1.

Yeah, Saltire brought this one up first, I know...
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  #48  
Old 08-16-2001, 05:39 AM
SPOOFE SPOOFE is offline
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I would just like to mention that, by defition, "0.000....01" does not go on for infinity. It has an end (the "...01"), and thusly, you cannot say that .999... has a difference of .000...01 from 1.

Infinity cannot be confined between two points.
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  #49  
Old 08-16-2001, 05:48 AM
Burnt Toast Burnt Toast is offline
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But wait a minute, if 0.999...= 1 then 0.999...8= 0.999... then 0.999...7= 0.999...8 then 0.999...6= 0.999...7 etc. etc. Then if you did it infinite times, you'd soon come to 0 so are you saying that 1=0?
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  #50  
Old 08-16-2001, 06:47 AM
Wraith Wraith is offline
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Burnt Toast- Read SPOOFE's post above. You can't put infinity between those two numbers. If you're putting it between those two numbers, that means that it has two ends and you've restricted an infinite number.

kinoons - You can put two numbers on the same point on the number line if they are equivalent. For example, I could put 100/100 on the same point on the number line as 1, because they are the same number. Because .9rep = 1, (the proofs shown are the ones I was taught), then .9rep CAN go on the same spot on the number line as 1.
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