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Other Discussion Boards => Technology, Science & Alt Science => Topic started by: Masterchef on March 09, 2007, 05:28:29 PM

Title: .9999... equals 1?
Post by: Masterchef on March 09, 2007, 05:28:29 PM
There are a few idiots on a different forum trying to claim that .9999... is actually equal to 1, which anyone with half a brain knows is false. There is no way that any number can equal a different number. Please, can anyone who isn't a fucking moron back me up on this?

http://www.kirupa.com/forum/showthread.php?p=2080661&posted=1#post2080661
Title: Re: 1 = .9999... ?
Post by: Erebos on March 09, 2007, 05:36:21 PM
1 ≈ 0.999∞

It isn't equal, but it is close enough to not matter to anything human. We are finite creatures, hence have a margin of error that can be seen as "correct" and be basically correct; nothing is absolute, fact & truth is relative. So 1=1 is a myth in reality, as there is never exactly "1"; it is always approximate. This means to us, finite beings, an infinitely small difference (as that between 1 and 0.999∞) makes no difference at all, therefore they can be seen as the same.

Basically: correct, they do not equal each other, but it is pointless, and this is a stupid question spawned from boredom at having nothing better to do.  ;)

And the proof that I've seen for it is flawed in one key area:
9.999∞ - 0.999∞ ≠ 9

You can't subtract or add or anything infinite quantities, as that is applying a rule of finite math to infinite. They are basically untouchable in objective mathematics. Saying that "9.9∞ - 0.9∞ = 9" is like saying  "5.5∞*∞ - 4.4∞*∞ = 1.1∞*∞"; it is retarded ... you can't apply finite rules of mathematics to infinite quantities. Its the same thing as dividing a number by zero: if you think it results in realistic mathematics, you are retarded.

∞ - ∞ =
9.999∞ - 0.999∞ =
X∞ - Y∞ =

Simple.
Title: Re: .9999... equals 1?
Post by: Ubuntu on March 09, 2007, 05:46:29 PM
Masterchief2219, I'm sorry, you're wrong this time.

Quote from: Wikipedia
In mathematics, the recurring decimal 0.999 denotes a real number. Notably, this number is equal to 1. In other words, "0.999" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

http://en.wikipedia.org/wiki/0.999

This issue is dealt with extensively here as well: http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html



Title: Re: .9999... equals 1?
Post by: Masterchef on March 09, 2007, 05:55:43 PM
How is that even possible?
Title: Re: .9999... equals 1?
Post by: Ubuntu on March 09, 2007, 05:58:43 PM
How is that even possible?

They are the same number. 1/2 = 0.5 for the same reason 1 = 0.999...

If they are different numbers, name a number in between them.

1/3 = 0.333...
+2/3 = 0.666...
=3/3 = 0.999...
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 05:59:43 PM
Masterchief2219, I'm sorry, you're wrong this time.

Quote from: Wikipedia
In mathematics, the recurring decimal 0.999 denotes a real number. Notably, this number is equal to 1. In other words, "0.999" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

http://en.wikipedia.org/wiki/0.999

This issue is dealt with extensively here as well: http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html


Wow, a completely false wikipedia article. Interesting. No, he is correct. These proofs are bullshit, basically going in circles with obvious gaps in logic (if it can be called logic).

(http://upload.wikimedia.org/math/3/3/1/33190b5d742626c840d4653650ccb472.png)

0.333∞ = 1/3 since when? It's almost as if these morons got a first grade calculator and typed in the digits to make up this proof.
0.333∞ ≈ 1/3! Approximately equals! That means there is an infinitely small difference, but there is a difference.

So rewording this "proof" to be correct:
0.333∞ ≈ 1/3
3 x 0.333∞ ≈ 3 x 1/3 ≈ (3 x 1)/3
0.999∞ ≈ 1

Wow, so basically we just gave a bunch of examples of what the approximately equal sign is for ...


(http://upload.wikimedia.org/math/a/a/7/aa7c4f22b682312e877215f27a0ac3c7.png)

This one goes wrong with the:
10c - c = 9.999∞ - 0.999∞
9c = 9

You simply cant subtract 0.999∞ from 9.999∞. Just as ∞ - ∞ ≠ 0, 9.999∞ - 0.999∞ ≠ 9.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 06:02:42 PM
How is that even possible?

They are the same number. 1/5 = 0.5 for the same reason 1 = 0.999...

If they are different numbers, name a number in between them.

1/3 = 0.333...
+2/3 = 0.666...
=3/3 = 0.999...

Um, 1/5 = 0.5? I don't think I really even need to comment here.  :p

And a number between them? That is like saying "God exists! If he doesn't, prove it." You can't make a positive claim without support. Anyway, a number between them would be 0.999∞. Learn to comprehend infinite quantities.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 06:05:46 PM
Did I win yet?
Title: Re: .9999... equals 1?
Post by: Masterchef on March 09, 2007, 06:06:46 PM
The entire argument for this all comes down to the human mind not being able to comprehend infinity.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 06:11:16 PM
The entire argument for that all comes down to the human mind not being able to comprehend infinity.

Nah, it's not having to do with comprehend. Math simply isn't infinite. We can comprehend the concept of infinity (we simply can't grasp its scope), but we can't deal in it. Math can't deal with infinite quantities, as it is not a finite science; therefore 9.999∞ - 0.999∞ = (no solution). We can comprehend that 1 is different from 9.999∞ by an infinitely small quantity, we simply can't use that quantity for anything as we are not capable of dealing in the infinite. We can represent the quantity with something like 0.0...1, just simply can't use or deal with it in any manner that makes sense. We're dealing with apples & wrenches here, two different things entirely, and not compatible; therefore the proofs are retarded as they pretend we can logically deal in infinite quantities.
Title: Re: .9999... equals 1?
Post by: Masterchef on March 09, 2007, 06:13:25 PM
Yeah, I actually thought of that after I posted that. :P
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 09, 2007, 06:38:54 PM

This one goes wrong with the:
10c - c = 9.999? - 0.999?
9c = 9

You simply cant subtract 0.999? from 9.999?. Just as ? - ? ? 0, 9.999? - 0.999? ? 9.

And why not? Surely you must agree that .999... is equal to some number, correct? And that if you subtract that number from .999... you will have 0, correct? And therefore if you subtract that particular number from 9.999... you will have 9. It's the same as subtracting, say, .5 from 2.5. Surely you agree that subtraction would leave 2?
Title: Re: .9999... equals 1?
Post by: Ubuntu on March 09, 2007, 06:40:02 PM
Sorry Erebos, you're mistaken:

http://www.cut-the-knot.org/arithmetic/999999.shtml
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://descmath.com/diag/nines.html
http://www.newton.dep.anl.gov/askasci/math99/math99167.htm
http://qntm.org/pointnine
http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 06:42:06 PM

This one goes wrong with the:
10c - c = 9.999? - 0.999?
9c = 9

You simply cant subtract 0.999? from 9.999?. Just as ? - ? ? 0, 9.999? - 0.999? ? 9.

And why not? Surely you must agree that .999... is equal to some number, correct? And that if you subtract that number from .999... you will have 0, correct? And therefore if you subtract that particular number from 9.999... you will have 9. It's the same as subtracting, say, .5 from 2.5. Surely you agree that subtraction would leave 2?

You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999∞ equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 06:42:57 PM
Sorry Erebos, you're mistaken:

http://www.cut-the-knot.org/arithmetic/999999.shtml
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://descmath.com/diag/nines.html
http://www.newton.dep.anl.gov/askasci/math99/math99167.htm
http://qntm.org/pointnine
http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html

The repeat the same flawed proof over and over; I'm not wrong.
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 09, 2007, 06:51:55 PM
You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999? equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?

.999... does not equal infinity. It may be an infinite string of symobols, but it still represents a finite amount. Any special rules that may apply to adding, subtracting, multiplying, or dividing infinity do not apply to a finite amount.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 07:00:10 PM
You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999? equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?

.999... does not equal infinity. It may be an infinite string of symobols, but it still represents a finite amount. Any special rules that may apply to adding, subtracting, multiplying, or dividing infinity do not apply to a finite amount.

It has an infinite aspect; there are no "special rules" per say, it's basically just mathematic common sense. It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

9.999... - .999... simply does not equal 9

It has no solution.
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 09, 2007, 07:07:31 PM
It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

But .999... represents neither an infinitely small quantity nor an infinitely large quantity. It represents a definite, finite quantity, and can therefore be treated as such.
Just as 2+2, 8-4, 164, (?4)^2 all represent 4, .999... is just another way of representing 1.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 09, 2007, 07:12:25 PM
Masterchief, are you serious, or are you joking? The identity .99999...=1 is a well known fact. Every so often people doubt it, but such people don't even know the definitions.

First question: what is meant by the representation .98604, or .33333..., or .9999....?

Answer: each of these is shorthand for a sum, where the nth digit is the multiple of 10-n that we are adding. So .98604 is simply shorthand for 9*10-1+8*10-2+6*10-3+0*10-4+4*10-5, and .3333... is shorthand for 3*10-1+3*10-2+3*10-3+3*10-4+....

This immediately raises a second question: what is meant by an infinite sum? You can't add infinitely many things, can you?

Answer: yes, you can, in some situations. You can add 1+0+0+0+0+0+..., because after the first term you aren't changing things at all. You can add 1/2+1/4+1/8+1/16+..., because such a sum can be represented "physically" on the number line as follows: first you go half way from 0 to 1, then you go half way from where you are to 1, then you go half way from where you are now to 1, etc. The sum is 1, because that's exactly where you get when you move half way to one infinitely many times (this is not, of course, a proof, but acts as motivation for the idea of limits.) You cannot, however, take the sum 1-1+1-1+1-..., because that sum alternates between 1 and 0, and never settles on any one number (anyone who tells you that sum exists is either completely nuts or has been staring at the zeta function too long). The definition that makes this work is the definition of limits. A sequence of numbers x_n is just that: a sequence x_1, x_2, x_3, x_4, .... Such a sequence is said to converge to a limit x if for any e>0, there is an N>0 so that the differences between x and x_n are less than e for n>N. An infinite sum is just the limit of the partial sums of the initial terms, provided that limit converges. So the decimal number .999999... really is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc.

They don't teach you all this when they teach you decimals in school, because it's complicated, so they usually just gloss over it, but it is what decimal representations actually are, and so any demonstration that .999...=1 has to use this fact (of course, there are some tricks people sometimes use to "cheat", but those involve taking things on faith, which are true, but which are exactly the things that Erebos is denying above). The lack of such actual definitions is why so many people are confused, or try to deny that .9999...=1. But what's true is that decimal numbers are really just representations of real numbers, and are not always unique. The fact that 1 can also be written as .9999.... is just one example.

So, without further ado-

Proof that 1=.99999...:
We know that .999... is, by definition, the sum 9/10+9/100+9/1000+... which is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc. Simply finding common denominators tells us that 9/10+9/100+9/1000+...+9/10n=99...9/10n=(10n-1)/10n=1-1/10n, and since for any e>0, 1/10n is eventually smaller than e, this limit is 1.

p.s. I decided to change my avatar in honor of this thread.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 09:21:40 PM
Sigh. I'd like to know why the fuck there is an "approximately equal" sign in math, then; because apparently it has no use, now that something not equal to one (but approximately equal) miraculously now "equals" one.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 09:36:17 PM
It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

But .999... represents neither an infinitely small quantity nor an infinitely large quantity. It represents a definite, finite quantity, and can therefore be treated as such.
Just as 2+2, 8-4, 164, (?4)^2 all represent 4, .999... is just another way of representing 1.
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 09, 2007, 09:41:58 PM
Sigh. I'd like to know why the fuck there is an "approximately equal" sign in math, then; because apparently it has no use, now that something not equal to one (but approximately equal) miraculously now "equals" one.
There isn't an "approximately equal" sign in math. It only exists in physics, statistics, and other subjects where precision in definitions is frequently given up for the sake of stating things simply. In math, people are careful to define what they mean by an approximation. For example, a physicist would say,
(http://upload.wikimedia.org/math/7/9/f/79f8c1fba6542599762dc28c36449631.png)
or even just nn/en, and leave it at that. A mathematician, on the other hand would say that,
(http://upload.wikimedia.org/math/6/c/2/6c28c3a2de95dafcfb144d784ea11fa0.png), and might additionally mention the bounds on the ratio, and the rate at which the ratio converges.

You are completely wrong in saying, "something not equal to one (but approximately equal) miraculously now 'equals' one". As I have explained, the real number represented by .999... is exactly the same as the number represented by 1. There is no approximate equality here; the two quantities are identical.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 09, 2007, 09:42:49 PM
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 09:48:41 PM
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.

Go figure; did I ever say it was a "real" number?
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 09, 2007, 09:49:41 PM
If decimals don't represent real numbers, what the hell are they?
Title: Re: .9999... equals 1?
Post by: Dioptimus Drime on March 09, 2007, 09:52:43 PM
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
I really don't know enough to demand authority on this, but out of curiosity, couldn't you approximate a number as such: 0.000...1
Wouldn't that be an infinitely small real number?

~D-Draw
Title: Re: .9999... equals 1?
Post by: Erebos on March 09, 2007, 09:55:21 PM
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
I really don't know enough to demand authority on this, but out of curiosity, couldn't you approximate a number as such: 0.000...1
Wouldn't that be an infinitely small real number?

~D-Draw

According to his reasoning that would equal zero. In math, that is accepted because it makes no difference, in reality it's retarded to say it is actually true.
Title: Re: .9999... equals 1?
Post by: beast on March 09, 2007, 10:10:22 PM
I think this topic really just demonstrates to what level people have studied maths.  There is no doubt that in the formal system of mathematics, 0.9 recurring is the same number as 1.  It is not almost the same, it is the same.

I also wanted to comment on this statement;

Quote
Math simply isn't infinite. We can comprehend the concept of infinity (we simply can't grasp its scope), but we can't deal in it. Math can't deal with infinite quantities, as it is not a finite science;

This is completely false.  There are whole fields of maths that deal with infinity and infinity is a crucial concept in maths.  In fact many proofs rely on dealing with infinity.  Your arguments are clearly based out of an ignorance of maths and philosophical understanding, rather than a factual understanding.  I suggest you actually take an advanced maths, and you'll clearly see how wrong you are.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 09, 2007, 10:13:49 PM
D-Draw, what you have just given me is not a decimal number. Decimal numbers are infinite sequences of digits. What you gave is something that has infinitely many digits, and then after all of those digits, there's another one. One can make sense of such objects, but not as decimal representations of real numbers. You can also define systems where .999... and 1 are not the same object. However, if you look at the actual definition of real number, and the actual definition of a decimal representation, then .999...=1. To see all the definitions in detail, it would be best to read a book on real analysis - such as Rudin's Principles of Mathematical Analysis (Amazon (http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/0070856133)) - or take a class on the subject.

Quote
In math, that is accepted because it makes no difference, in reality it's retarded to say it is actually true.
It's not accepted because it "makes no difference", it's accepted because it follows from the definitions of real number and decimal representation. That makes it true. If you refuse to accept it, then you either mean something completely different by decimal than the rest of the world means (including all of the educated mathematicians), or else you're simply clueless.
Title: Re: .9999... equals 1?
Post by: beast on March 09, 2007, 10:49:25 PM
Here are 4 maths based websites that all claim that 0.9r = 1.  I challenge all those who do not believe that this is the case to present a genuine maths based website with evidence supporting your beliefs.


http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/
http://mathforum.org/library/drmath/view/57035.html
http://sprott.physics.wisc.edu/Pickover/pc/9999.html
http://www.purplemath.com/modules/howcan1.htm
Title: Re: .9999... equals 1?
Post by: Phentos on March 10, 2007, 02:33:23 AM
Erebos--

you lose.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 10, 2007, 07:22:56 AM
This argument was suggested to me on another forum, so let's see if it helps at all.

Decimal representations are not the same thing as numbers. They are representations for numbers. For example, 1/2, 2/4, 34/68, 0.5, and 0.49999... are all representations for the same real number. They are not all representations for the number "0.5", they are representations of the number represented by "0.5". Just because two representations are different does not mean that the things they represent are different. Does that help?
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 10, 2007, 10:13:08 AM
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.

Maybe it would be easier to picture this using a number line. We can both agree that .999..., no matter what amount it represents, is a real number, and therefore has a definite position on the real number line, correct? To find this place on the number line, you would simply count .999... spaces to the right of zero. To subtract this amount from any place on the number line, you would simply start at whatever place represents the number you wish to subtract it from, and count .999... spaces to the left. If you start at the place that represents .999... and count .999... spaces to the left, you will be on zero. If you start at 9.999... and count .999... spaces to the left, you would be on 9.
The place on the number line that represents .999... is the exact same place that represents 1. It's not almost the same place, it's not just before 1, it is the exact same place.
Title: Re: .9999... equals 1?
Post by: Masterchef on March 10, 2007, 11:10:19 AM
I give up. :D
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 10, 2007, 12:09:55 PM
[sarcasm]I'm sorry I'm so dense and mathematically uneducated that I stubbornly refuse to accept that .999...≠1. Maybe I should consider a change of profession.[/sarcasm]

Seriously masterchief, why are you continuing to refuse to concede the point when you haven't even seen rigorous definitions of a real number and of a decimal representation? Do you have any reasons for your belief other than blind faith? If you haven't seen the definitions, what makes you think you have any idea what you're talking about when you claim two things are different? It's not like the representations "1" and "0.9999..." are up to user interpretation; they actually have a meaning which you clearly don't understand. Mathematics doesn't run on democratic principles, where everyone's opinion counts. Once you have formulated the definitions, there's only ever one right answer, and I've got news for you: yours is not it.
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 12:37:40 PM
Ok. I failed to get through with the mathematics, now I'll try to reason.


Are you all denying that infinitely small quantities/numbers exist? Do you think that 0.0∞1 = 0?

I'll assume that is yes, as you think 0.999∞ = 1, and 0.999∞ = 1 - 0.0∞1 (1 = 1 - 0).


So, if you go magnify something infinitely (so that you get to an infinitely small quantity), there is nothing?

This would mean that everything is composed of an infinity of nothing.


If everything is composed of an infinity of nothing, do you realize what that would mean?

I think you'll agree that we are made of something, but according to your logic, we are made of an infinity of nothing (which I think you'll agree is nothing).


If 0.999∞ = 1, we don't exist. I don't know about you guys, but I'm quite sure that I exist.
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 10, 2007, 12:54:49 PM
Are you all denying that infinitely small quantities/numbers exist? Do you think that 0.0?1 = 0?

That number is not possible. You can't have an infinite number of zeros if you put an end to them. By sticking the 1 on the end, you've now got a very, very long string of zeros that ends with a 1, but you do not have infinite zeros.
Like Skeptical already said, there is no such thing as an infinitely small real number.

Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:00:42 PM
Do you think that 0.0?1 = 0?

That number is not possible. You can't have an infinite number of zeros if you put an end to them. By sticking the 1 on the end, you've now got a very, very long string of zeros that ends with a 1, but you do not have infinite zeros.

Nice job completely ignoring the point; and you are wrong. According to mathematic "rules" you are correct (as such a number has no use), but logically you are wrong -- such a number is possible. It is simply the best representation of an infinitely small number; yes, there are infinite zeros, just as there are infinite 9's in 0.999r. According math 0.999r is the same thing as 1, as well, but technically that is false. They are different by an infinitely small number, and that is pivotal. They do not truly equal each other, they simply are as close as anything can come to equaling without equaling.

My point was that everything is made up of infinite infinitely small quantities, and if those infinitely small quantities were zero then nothing exists; this is retarded as things do exist. Therefore the infinitely small quantity is not zero, but is a quantity, the quantity which 0.999r is away from 1.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 10, 2007, 01:10:03 PM
Ok. I failed to get through with the mathematics, now I'll try to reason.
Yes, you failed to get through with the mathematics. I daresay I know a lot more math than you, and as such I have actually seen definitions for the real numbers and for decimal expansions, which you clearly haven't.

Quote
Are you all denying that infinitely small quantities/numbers exist? Do you think that 0.0∞1 = 0?
Yes, I deny that infinitely small quantities exist. I challenge you to produce a quantity, hypothetical or real, in nature, that could sensibly be called "infinitely small". If you want to talk about real numbers, then yes, I again deny the existence of infinitely small real numbers. They don't exist. The string of symbols "0.0∞1" is not a decimal expansion of anything, and is in fact completely meaningless when considered as anything other than an abstract string of symbols. If you want to talk about definitions in mathematics that behave as one might imagine infinitely small quantities would, then such things do exist, but they are completely separate from the real numbers, and in such systems, you still have .999...=1. If you simply want to talk about abstract strings of symbols, then of course .999... and 1 are different strings of symbols, but now you're no longer talking about numbers.

Quote
So, if you go magnify something infinitely (so that you get to an infinitely small quantity), there is nothing?
I have no idea what the "something" is that you are "magnifying infinitely". Do you?

Quote
This would mean that everything is composed of an infinity of nothing.
Nonsense. We are not composed of an infinity of nothing; neither are we composed of an infinity of "0.0∞1".

Quote
If 0.999∞ = 1, we don't exist. I don't know about you guys, but I'm quite sure that I exist.
What are these things which you call "1" and which you call "0.999∞". If you mean decimal representations of real numbers, then you are simply wrong. I suspect you have no idea what they are.
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:11:54 PM
Really, you are all missing it entirely. Your minds are so narrowed upon human mathematics that you don't see actual reality. Mathematics are numbers and paper, that represent reality approximately. We are not infinite creatures, therefore our mathematics are not infinite; though in some equations we use symbols to represent infinite quantities. These infinite quantities are not actually truly usable in exact math, they would result in no solution in any formula that used them. But, to humans, such a difference does not matter. An infinitely small difference is the same as no difference, to us. We have margins which anything that falls into them is "exact" to us, though technically just approximately exact.

In mathematics, the 0.999... is expressed as 1 because the difference makes no difference. 0.0...1 has no effect on humanity, as it is infinitely small and an infinitely small difference always falls within our margin of error. This does not change reality, just our rational human use of it. Reality is infinite, but humans experience it finitely. Our perceptions are not able to directly experience reality, but do so through filters (our minds and sensory organs) that only picks up finite portions of the infinite reality; this means that our subjective reality is finite, but does not change the actualities of reality (they are simply irrelevant to us, part of a different world entirely separate from us). So, to us, 0.0...1 = 0, because we are finite and such a number has no effect on anything human; it is irrelevant; but in reality, 0.0...1 does exist, and we can reason that it does exist (everything is made up of infinite infinitely small quantities). This means that 0.0...1 does not truly equal 0, just that our only use for it is as a 0. The same is for 0.999, because of the same reason; the difference is so small it makes no difference, so we take it as 1 (but in reality it is not 1, just as close as it can possibly get to being 1; which makes no difference to us).

I've beaten this dead horse to bits. It really isn't so complex. Do you understand yet?
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:17:04 PM
Nonsense. We are not composed of an infinity of nothing; neither are we composed of an infinity of "0.0∞1".

So what are you saying? That when I take a particle you are composed of and magnify it infinitely Gandalf will appear and magic me back? Yes, I know it is impossible to magnify something infinitely; for the same reason 0.999... =/= 1. It doesn't matter, through reasoning and common sense it is quite obvious that there are infinitely small quantities. I can think of one that we all experience constantly -- existence. We exist only in a single moment constantly moving up the ladder of time, that moment is an infinitely small quantity of time. Got it? If that quantity of time did not exist, we would not exist. Quite simple.

The rest of what you said basically made no sense (as it revolved around there being no infinitely small quantity), except that you know more about technical mathematics than me; you surely do, but I know more about common sense and philosophy.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 10, 2007, 01:21:28 PM
I understood what you were saying. It's still completely wrong. Mathematics exists independently of the world. Mathematical objects have definitions. Real numbers are mathematical objects. Decimal representations are just that; a way of representing these mathematical objects. It turns out that .999... and 1 represent the same mathematical object, when you use the actual definitions of decimal representation and real number, instead of some philosophical wishiwashy non-rigorous nonsense definition which you seem to be using.

I ask you again: what are the actual objects that you think "1" and ".999..." represent?
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 10, 2007, 01:23:11 PM
So what are you saying? That when I take a particle you are composed of and magnify it infinitely Gandalf will appear and magic me back?
If you were to take a particle I am composed of and magnify in infinitely, the particles would fill the entire universe, implode into a black hole, and you would be very very small. And dead.
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:49:00 PM
Quote
Mathematics exists independently of the world. Mathematical objects have definitions. Real numbers are mathematical objects. Decimal representations are just that; a way of representing these mathematical objects.

According to human mathematic rules, you are correct. Human existence is separate from reality, as is our system of mathematics. Technically this simply means what I said here:

Quote
Really, you are all missing it entirely. Your minds are so narrowed upon human mathematics that you don't see actual reality. Mathematics are numbers and paper, that represent reality approximately. We are not infinite creatures, therefore our mathematics are not infinite; though in some equations we use symbols to represent infinite quantities. These infinite quantities are not actually truly usable in exact math, they would result in no solution in any formula that used them. But, to humans, such a difference does not matter. An infinitely small difference is the same as no difference, to us. We have margins which anything that falls into them is "exact" to us, though technically just approximately exact.

In mathematics, the 0.999... is expressed as 1 because the difference makes no difference. 0.0...1 has no effect on humanity, as it is infinitely small and an infinitely small difference always falls within our margin of error. This does not change reality, just our rational human use of it. Reality is infinite, but humans experience it finitely. Our perceptions are not able to directly experience reality, but do so through filters (our minds and sensory organs) that only picks up finite portions of the infinite reality; this means that our subjective reality is finite, but does not change the actualities of reality (they are simply irrelevant to us, part of a different world entirely separate from us). So, to us, 0.0...1 = 0, because we are finite and such a number has no effect on anything human; it is irrelevant; but in reality, 0.0...1 does exist, and we can reason that it does exist (everything is made up of infinite infinitely small quantities). This means that 0.0...1 does not truly equal 0, just that our only use for it is as a 0. The same is for 0.999, because of the same reason; the difference is so small it makes no difference, so we take it as 1 (but in reality it is not 1, just as close as it can possibly get to being 1; which makes no difference to us).

That reality makes no difference to our reality, so our systems only include that which does. This is thinking contextually, in terms of what matters to us; but not realistically, in terms of what is actually reality. The concept of math transcends the simple rules that humans place upon it; math is not human math, it includes it but is not it. Math is the universal code of representation of reality, and true math is infinite (as reality is infinite in every dimension). Our math is used only to represent what we, as humans, use and perceive; but it just so happens to be able to also (frivolously) represent infinite quantities that mean nothing to us.

What you are doing here is thinking contextually, thinking that these numbers do not matter and do not exist in the context they are used (humanly), therefore translate them into human related concepts (finite numbers). I am thinking in opposition of that, realistically, as the question was in pure form: "Does 1 equal 0.999r?" The pure answer to such a pure question is no, it does not: in reality infinite (and infinitely small) quantities do exist, and what is represented by 0.999r is an infinitely small difference from 1. This answer is the technically correct answer, but not the contextual one. To us, there is no reason to think of 0.999r as anything other than 1; therefore the contextual answer is yes, it does (the reason for this being that it falls within the human margin of "exactness," not because it is reality). The question was pondering upon reality, not human reality.

We are simply showing two sides of the coin, one being human reality (all that matters to us), and the other being actual reality (that which is outside of human scope and context). You are answering the question according to human mathematic rules, I am answering it according to pure math; the math that has no meaning for us, hence is rounded and approximated to form our math. You are absolutely correct in that the human institutions of mathematics accept 1 as the same thing as 0.999r; but if you would care to note, if you asked Einstein if in reality 1 was the same thing as 0.999r, he would answer no (but that it is the same thing to us). The institutions are related only to human dealings, and therefore care not mathematics that do not relate to such.

From another perspective, one that would maybe show some things I have said to be slightly wrong for me to assert (that infinite quantities are not humanly usable) would be that I am speaking from the field of philosophy of mathematics while you are speaking from the field of mathematics. The philosophy of something is the field which expands upon the already accepted institutions, so that it may further encompass actual reality. The philosophy of mathematics would bring more and more of reality under the scope of human reality (though never all of it). So, what I say is partially philosophy, in that it comes out of context of human math and attempts to expand the understanding. Of course what I say is no revolution; all genius mathematicians accept it as fact, while the masses, who don't understand concepts such as infinity, will preach that 1 = 0.999r (and this doesn't matter to the mathematicians, as they also know it makes no difference).

Get me?


It turns out that .999... and 1 represent the same mathematical object, when you use the actual definitions of decimal representation and real number, instead of some philosophical wishiwashy non-rigorous nonsense definition which you seem to be using.

I ask you again: what are the actual objects that you think "1" and ".999..." represent?

I basically went over this above. But, you are basically correct, except that you bring the concept of all mathematics under the banner of human mathematics; mathematics represents reality, human mathematics represents human reality. In human reality (that which contextually matters to humans) 0.999r = 1, as the infinitely small quantity makes no difference in our reality and hence is nothing (equals zero) to us. In actual reality, it is as I argue. I explained this basically and in more depth above.

As for what actual objects, in our mind 1 equals a single unit of something, or the measuring quantity of something equal to 1 unit of the quantity we've assigned to that unit (foot, meter, kilogram, whatever). You know that..

The one thing I can think of that 0.999r represents is a certain amount of time that does not include the present "moment" (an infinitely small quantity of time). Or 1 unit - 0.0r1.
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:50:12 PM
So what are you saying? That when I take a particle you are composed of and magnify it infinitely Gandalf will appear and magic me back?
If you were to take a particle I am composed of and magnify in infinitely, the particles would fill the entire universe, implode into a black hole, and you would be very very small. And dead.

Magnify as in zoom up on, like with a magnifying glass ... not increase in size.
Title: Re: .9999... equals 1?
Post by: Pyrochimp on March 10, 2007, 01:53:43 PM
I love how people write entire essays on this question.  It's not just exclusive to this board, people are really interested in being right about .9r = 1 around the entire internet.

BTW .9r = 1
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 01:57:05 PM
BTW .9r = 1

I love how people pop in, not even reading, and make empty statements against full arguments. Funny.

And no, it doesn't.
Title: Re: .9999... equals 1?
Post by: Pyrochimp on March 10, 2007, 01:59:41 PM
I read the topic, .9r really does = 1 though.

Here, what's 1 - .9r?
Title: Re: .9999... equals 1?
Post by: GeoGuy on March 10, 2007, 02:01:34 PM
Yes it does Erebos.

If, as you claim, .999... does not equal 1, I'm certain a quick Google search should give a multitude of articles proving it.
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 02:02:29 PM
I read the topic, .9r really does = 1 though.

Here, what's 1 - .9r?

Obviously didn't read the topic if you would ask that as a proof against it.

Technically:
1 - .9r ≈ 0
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 02:03:44 PM
Yes it does Erebos.

If, as you claim, .999... does not equal 1, I'm certain a quick Google search should give a multitude of articles proving it.

Nice job using "the google argument."

"This is the way it is, but I'll not prove it; I'll just tell ye to go to google!"
Title: Re: .9999... equals 1?
Post by: Pyrochimp on March 10, 2007, 02:04:31 PM
I read the topic, .9r really does = 1 though.

Here, what's 1 - .9r?

Obviously didn't read the topic if you would ask that as a proof against it.

Technically:
1 - .9r ≈ 0

Why won't you believe that I read the topic :(

Well, I really don't care too deeply if you think 1 != .9r, so you guys have fun.
Title: Re: .9999... equals 1?
Post by: GeoGuy on March 10, 2007, 02:04:51 PM
I didn't say that at all. I want you to run a search for articles proving that .999...?1 so you'll see that there aren't any.
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 10, 2007, 02:06:45 PM
What is represented by 0.999r is an infinitely small difference from 1.

This is where you're becoming confused. .999... is not the closest you can get to 1 without being 1. There is no such number. The symbols 1 and .999... represent the exact same quantity. There is no difference at all, no matter how closely you look at them, between the quantity represented by the symbol 1, and by the quantity represented by the infinite string of symbols .999...
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 02:07:28 PM
I didn't say that at all. I want you to run a search for articles proving that .999...?1 so you'll see that there aren't any.

There aren't any for the simple fact of what I already said above. Read.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 10, 2007, 02:26:46 PM
What a pointless discussion!
Title: Re: .9999... equals 1?
Post by: The Government on March 10, 2007, 02:50:38 PM
yea 1 is acts like a 'symbol' for .99999999...
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 03:35:37 PM
yea 1 is acts like a 'symbol' for .99999999...

lol!
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 10, 2007, 03:39:14 PM
What is 2 x 0.9r? Is it 2?
Title: Re: .9999... equals 1?
Post by: Erebos on March 10, 2007, 03:42:13 PM
What is 2 x 0.9r? Is it 2?

It is 1.9r8, or some weird thing like that.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 10, 2007, 03:48:47 PM
So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?
Title: Re: .9999... equals 1?
Post by: Dioptimus Drime on March 10, 2007, 05:00:36 PM
So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?
.999... = 1. It was concluded that before the thread even started, just ignorant, stubborn people chose to disagree.


~D-Draw
Title: Re: .9999... equals 1?
Post by: beast on March 10, 2007, 05:31:28 PM
Really, you are all missing it entirely. Your minds are so narrowed upon human mathematics that you don't see actual reality. Mathematics are numbers and paper, that represent reality approximately.

This is a mathematical question.  The mathematics shows that 1 is exactly equal to 0.9 recurring.  There are a number of mathematical proofs that have been presented.  A mathematical proof is a fact within the formal system of mathematics.  Your argument about reality is stupid.  How often, in "reality," do you have to deal with equations including the number 0.9 recurring?  The thing about mathematical proofs is that you can't deny them with reason, you have to mathematically prove that they are incorrect.  In this case, that is impossible.

Quote
We are not infinite creatures, therefore our mathematics are not infinite; though in some equations we use symbols to represent infinite quantities. These infinite quantities are not actually truly usable in exact math, they would result in no solution in any formula that used them. But, to humans, such a difference does not matter. An infinitely small difference is the same as no difference, to us. We have margins which anything that falls into them is "exact" to us, though technically just approximately exact.

This is completely false.  Parts of mathematics would not exist without the concept of infinity.  The problem is that you don't understand infinity, or maths.  What is your training in maths?  What level have you studied?  Maths is exact and it's not a case of there being differences so small that the don't matter, rather, if you understand the maths, you'll see that there really is absolutely no difference at all.

Quote
In mathematics, the 0.999... is expressed as 1 because the difference makes no difference. 0.0...1 has no effect on humanity, as it is infinitely small and an infinitely small difference always falls within our margin of error.

What are you talking about?  In mathematics 0.9r is expressed as 1 because there are mathematical proofs that demonstrate that they are the same number.  These are not approximate proofs, and they have nothing to do with the effect on humanity, but rather, are facts within the mathematical formal system.  Nobody decided that because the differences are so small it didn't matter, they demonstrated, clearly, that the numbers are absolutely equal.

Quote
This does not change reality, just our rational human use of it. Reality is infinite, but humans experience it finitely.

You keep saying this, but can you actually provide any evidence that this is actually the case?  I don't think what you're saying is a fact at all, and it is misleading to continually present it as a fact, unless you have some evidence to back it up.  What you're expressing is your evidence, based on the fact that you have had no serious mathematical training.

Quote
Our perceptions are not able to directly experience reality, but do so through filters (our minds and sensory organs) that only picks up finite portions of the infinite reality; this means that our subjective reality is finite, but does not change the actualities of reality (they are simply irrelevant to us, part of a different world entirely separate from us).

If what you're saying is correct, which you have provided no evidence to suggest is the case, how would we see things, hear things, smell things, taste things or touch things differently if we could see reality?  What is your evidence that we do not experience reality?  How is the world, in factual terms, different from how we perceive it?  If you're going to make claims about the world, please put forward evidence to show that you're not just putting forward baseless speculation.


Quote
So, to us, 0.0...1 = 0, because we are finite and such a number has no effect on anything human; it is irrelevant; but in reality, 0.0...1 does exist, and we can reason that it does exist (everything is made up of infinite infinitely small quantities).

Please give an example of 0.0...1 existing in reality.  You claim that it does, so you must have seen a place where it does exist.  Or are you just speculating and presenting your speculation as fact?

Quote
This means that 0.0...1 does not truly equal 0, just that our only use for it is as a 0. The same is for 0.999, because of the same reason; the difference is so small it makes no difference, so we take it as 1 (but in reality it is not 1, just as close as it can possibly get to being 1; which makes no difference to us).

Mathematical proofs don't work like that.  They work by having a given fact, that is already accepted as true, and then using the formal rules of mathematics to show that a new fact has to be true if the previous fact was true. if 1/3 = 0.3 recurring, you have to accept that 0.9 recurring = 1.  Approximations are not made in the various mathematical proofs, they are dealing with facts.

Quote
I've beaten this dead horse to bits. It really isn't so complex. Do you understand yet?

What you've done is present a number of statements as facts, without providing any evidence of that they are actually facts.  You've failed to put forward and rebuttal of the actual mathematical proofs that have been presented, or of the fact that a number of mathematical bodies have demonstrated that they support the mathematical proofs.  You've given no indication of any serious knowledge or training in mathematics and then you've acted like those of us who do understand maths can't understand what you're saying.  We completely understand, it's just that all the actual observational evidence demonstrates conclusively that you're wrong, and there has been no evidence to suggest that you're right.

Rather than putting forward your speculative opinions on a subject that you have not received formal education in, why don't you give up this debate and go and learn about what you're talking about.  Just because you fail to understand something, it doesn't mean that it is wrong.  In fact your understanding of a concept has nothing to do with the factuality of that statement.  This is why I believe that it is crucial that we formulate our opinions based on things that we know are facts, rather than speculating and then behaving as if our speculations are correct.  Dogmatic thinking, as far as I can see, is the source of all the evil in the world.
Title: Re: .9999... equals 1?
Post by: Masterchef on March 10, 2007, 05:36:06 PM
Skeptic, I already conceded.

But I would still like an answer to this:
What is 2 x 0.9r? Is it 2?
Title: Re: .9999... equals 1?
Post by: beast on March 10, 2007, 05:50:39 PM
Yes!  And 54 x 0.9 recurring = 54.

The reason a calculator will say something different is because you're not calculating 0.9r, but actually just 0.999999999 or however many digits you entered.  These are significantly different numbers.

Because we're dealing with an infinite number, work it out starting from the first digit, instead of the convention of the last.

2x 0.9 = 1.8
2x 0.99 = 1.98
2x 0.999= 1.998
...
2x 0.9999999999 = 1.9999999998

So we can see that as 0.9 approaches an infinite number of 9s, the answer also approaches an infinite number of 9s.  While technically our answer is 1.9r8, we have already proven that 0.9r = 1.  Therefore 1.9r must equal 2, regardless of what digits are left at the end.

Likewise, as 0.9 approaches an infinite number of 9s, we can see that that number times 54 approaches 53.9r46.  Since we know that 0.9r = 1, 0.9r46 must also equal 1.  Were we dealing with finite decimals, that would not be the case, but we're not.
Title: Re: .9999... equals 1?
Post by: Masterchef on March 10, 2007, 07:56:09 PM
The reason a calculator will say something different is because you're not calculating 0.9r, but actually just 0.999999999 or however many digits you entered.  These are significantly different numbers.
I just used the windows calculator and divided 1 by 3, which would give me .3r. But when I multiplied the result by 3 again, I got 1. Why? The calculator would not round .9r up to 1, so what happened there? I should have gotten .9r as an answer...
Title: Re: .9999... equals 1?
Post by: RESOCR on March 10, 2007, 09:07:08 PM
well I guess 0.9r=1 as long as 0.0r1=0, I'm just not clear how a positive number alone can equal anything else but a positive number.

But in that case 0.9r(equalling 1)+0.0r1(equalling 0)=1. then 0.9r(meaning1)+0.0r1=1. But since we know that 0.9r+0.0r1 really equals 1, and not 1.0r1, we find a problem, specifically that 1=1.0r1. Which again is all solved if 0.1r1=0, but it would have to extend to every instance of the amount, and not just exactly 0.0r1. This means 1.0r110^infinite would be just 10^infinite. So numbers get dropped somewhere. And I didn't think 1/3=0.3r anyway, isn't it 0.3r+0.3r+0.3r4=1?
Title: Re: .9999... equals 1?
Post by: Pyrochimp on March 10, 2007, 09:19:27 PM
well I guess 0.9r=1 as long as 0.0r1=0, I'm just not clear how a positive number alone can equal anything else but a positive number.

But in that case 0.9r(equalling 1)+0.0r1(equalling 0)=1. then 0.9r(meaning1)+0.0r1=1. But since we know that 0.9r+0.0r1 really equals 1, and not 1.0r1, we find a problem, specifically that 1=1.0r1. Which again is all solved if 0.1r1=0, but it would have to extend to every instance of the amount, and not just exactly 0.0r1. This means 1.0r110^infinite would be just 10^infinite. So numbers get dropped somewhere. And I didn't think 1/3=0.3r anyway, isn't it 0.3r+0.3r+0.3r4=1?

The only problem is that you can't have 1.0r1 in the first place.  Having another number after the repeating number implies that the repeating number comes to an end at some point for that non-repeating number to come in.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 10, 2007, 09:27:59 PM
This is why the question cant be answered. Because 1-0.9r must equal a positive number. But this positive number can never be found.



Because if 0.9r=1, it also means 0.89r=0.9. And 0.79r=0.8. And 0.49r=0.5. So 0.49r2 must equal one, right?

But it doesnt. 0.49r2=0.9r8. Which doesnt equal 1.

so that is saying 0.52=/=1, when it does.

maybe it would be better explained if you divide 1.1r by 1, and you get 1, but if you divide 1.1r by 0.9r, you get something like 1.1r21 I think. It's really late here and I'm pretty tired.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 10, 2007, 09:48:18 PM
double-post, since it really doesnt have much to do with my first, I think were toob usy thinking of the number 1 as definign a point and not defining an infinitely small field. This means 0.9r, 1.0r, and 1.0r1 all equal one, while 0.9r8, 0.9r, and 1.0r all equal 0.9r, 1.0r1, 1.0r, and 1.0r2 are equal, 0.0r, 0.0r1 and -0.0r1 are all equal, etc. Our definition of the poitn of a number is the center of one defining field, which is also where two defining fields overlap.
Title: Re: .9999... equals 1?
Post by: RoundisWrong on March 10, 2007, 11:58:06 PM
If you think that .9r does not equal 1, and you can't come up with a mathematical proof to show it, why do you keep arguing?  There is no such thing as 0r1 in mathematics!  It simply shows that you don't understand the concept of infinity. 
Title: Re: .9999... equals 1?
Post by: texta on March 11, 2007, 04:13:57 AM
If you think that .9r does not equal 1, and you can't come up with a mathematical proof to show it, why do you keep arguing?  There is no such thing as 0r1 in mathematics!  It simply shows that you don't understand the concept of infinity. 

What I love about numbers is that you can make up any number you like and it exists. 0.0r1 certainly could exist. Good luck distinguishing it from 0 though. :D
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 08:58:36 AM
There is no such thing as 0r1

On the theoretical number line which 0.9r is a point, 0.0r1 would be the very next point after zero. Also, on that same theoretical number line, 0.9r would be the number preceding one.
Title: Re: .9999... equals 1?
Post by: RoundisWrong on March 11, 2007, 09:08:23 AM
There is no such thing as 0r1

On the theoretical number line which 0.9r is a point, 0.0r1 would be the very next point after zero. Also, on that same theoretical number line, 0.9r would be the number preceding one.
Even if you don't understand math you should be able to figure it out from basic logic.  0.0r1 cannot exist.  You can't have an infinitely long string of zeros with a 1 at the end because there is no end.  0.0r means that it repeats to infinite which means you can't have a 1 at the end because there is no end.  How can you put at one at the end of something that has no end?
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 09:32:05 AM
Then what comes after 0 on a thoereticalnumber line where 0.9r is displayable?
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 11, 2007, 10:32:00 AM
Then what comes after 0 on a thoereticalnumber line where 0.9r is displayable?

First off, any number line that displays 1 is also displaying .999... Second, I don't think there is a number that "comes after" 1 in the sense that you mean. It's my understanding that in between any two rational numbers are an infinite amount of irrational numbers, therefore for any number you can give me I can give you a smaller one.
Of course, I haven't got the slightest idea what I'm talking about, so we'd better wait for Skeptical to get back on.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 10:53:20 AM
Right. There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 11:28:44 AM
Erebos, if you're talking about "actual reality" outside of human perception/thinking then you should not be talking (literally).  You should not even be thinking.  Thinking about something separate from yourself is impossible because everything you think and perceive is part of yourself.  We cannot think about anything we cannot think about.  We will never understand anything beyond the human capacity to understand. 

You are trying to define something that is impossible for humans to understand (by your very definition) and then calling us stupid for not understanding you.  How are you somehow outside of human perception, human mathematics (as if there is some other kind), and human understanding?  Maybe you think you are enlightened (or something) but regardless, there is no way to "talk" about such things and expect to say anything meaningful at all.

Also, saying mathematics is not useful in talking about maths and numbers is. . . beyond ridiculous.  Let me ask another question:  how do you know there is anything that is infinite? 
Title: Re: .9999... equals 1?
Post by: GaryTheBanana on March 11, 2007, 11:30:27 AM
So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?
.999... = 1. It was concluded that before the thread even started, just ignorant, stubborn people chose to disagree.


~D-Draw
I AGREEEEEEEEEEEEE
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 11:52:32 AM
Why can you have o.9r9 but not 0.0r1?
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 11:57:11 AM
Because .9r9 can reduce to .9r, whereas 0.0r1 is essentially meaningless because with the infinite amount of 0's there will NEVER be an end on which to put the 1. 
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 12:04:42 PM
But there'll never be an end to put an extra 9 on 0.9r9
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 12:14:55 PM
But there'll never be an end to put an extra 9 on 0.9r9
That's true, but that extra 9 is a redundant symbol; it can logically be reduced.  If you want to stick with 0.9r9 then yes, it also is meaningless. 
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 12:20:23 PM
Why can you have o.9r9 but not 0.0r1?
You can't have either, or at least they're not decimal expansions. In theory you could think of them as "sequences" of digits indexed by the ordinal (http://en.wikipedia.org/wiki/Ordinal_number) ω+1 instead of ω, but such things are not decimal expansions, and don't represent real numbers.

Quote
Because .9r9 can reduce to .9r
No, it can't. The expansion .9r9 is a valid decimal expansion, which represents the number 1, while .9r9 represents no number at all.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 12:21:34 PM
0.9r doesn't represent a real number either.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 12:33:16 PM
0.9r doesn't represent a real number either.
Yes, it does.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 12:53:57 PM
What does 0.111... represent?
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 01:03:27 PM
Right. There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.

no, i didnt mean a number smaller than zero, I meant after as in next on the number line, greater than, etc. What is the very first number larger than zero, on a line where 0.9r is a viable, defined point, one step up from however you choose to name the number one step less than it, how I define 0.9r8.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 01:13:37 PM
After reading a bit more I've come to the conclusion that this all boils down to the fact that most people don't know what the definition of a decimal is. I didn't until I read this. To be honest, this debate is not something I've ever come across. If you'd have asked me before I would have said that 0.999r is so close to 1 that it may as well be. Obviously that answer doesn't make any sense now that I've read what a decimal is.

Instead of arguing about it and flexing their epeens, people should just explain the definition of the decimal in a civilised manner and that would end the discussion.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 01:23:26 PM
speaking of decimal points, isn't this like saying that 9r is equal to 10r?
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 01:25:41 PM
speaking of decimal points, isn't this like saying that 9r is equal to 10r?
Maybe in the sense that they are both infinity. 
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 01:28:40 PM
I think the same methods for proving 0.9r = 1.0r can be applied to 9r = 10r.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 01:32:11 PM
So can something be infintely small as long as it's not a decimal number?
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 01:37:19 PM
well by your logic 9r=10r, even though 9r-10r=-1 (no matter how many nines or zeros at the end, as long as it is an equal amount which it should be (infinity=infinity, duh) they would subtract to be 1, or -1) and NOT 0.
Title: Re: .9999... equals 1?
Post by: EnragedPenguin on March 11, 2007, 02:10:42 PM
no, i didnt mean a number smaller than zero, I meant after as in next on the number line, greater than, etc.

Right.There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.

By this, of course, Skeptical meant no smallest number after zero on the number line.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 02:15:25 PM
no, i didnt mean a number smaller than zero, I meant after as in next on the number line, greater than, etc.

Right.There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.

By this, of course, Skeptical meant no smallest number after zero on the number line.

Well whatever he meant, he is right, and it is not what I meant. Maybe it would be better explainable as the smallest non-zero number?

Anyway, ho would you display it? I've been (trying) to display it as 0.0r1, because thats how it makes sense to me.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 02:16:39 PM
You can't have the smallest number after zero because you can always divide it by 2 again to get a smaller number.
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 02:23:42 PM
I was confused with what you were originally saying, Kasroa.  You mean 9.9r = 10.0r and not an infinite amount of nines on the left of the decimal place, right? 

well by your logic 9r=10r, even though 9r-10r=-1 (no matter how many nines or zeros at the end, as long as it is an equal amount which it should be (infinity=infinity, duh) they would subtract to be 1, or -1) and NOT 0.
I don't think that's right, Resocr. 
        9 - 10 = -1, but
   9.9 - 10.0 = -0.1,
9.99 - 10.00 = -0.01, etc. 
So 9.9r 10.0r =/= -1. 
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 02:30:07 PM
I don't know what I'm saying I don't have any further education in Maths :D
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 02:34:39 PM
I was confused with what you were originally saying, Kasroa.  You mean 9.9r = 10.0r and not an infinite amount of nines on the left of the decimal place, right? 

well by your logic 9r=10r, even though 9r-10r=-1 (no matter how many nines or zeros at the end, as long as it is an equal amount which it should be (infinity=infinity, duh) they would subtract to be 1, or -1) and NOT 0.
I don't think that's right, Resocr. 
        9 - 10 = -1, but
   9.9 - 10.0 = -0.1,
9.99 - 10.00 = -0.01, etc. 
So 9.9r 10.0r =/= -1. 

uh, I was more referring to 9-10=-1
99-100=-1
999-1000=-1
etc.

I did not add any decimals. 9r (99999999999999999999....)-10r(100000000000000000000....)=-1.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 02:36:22 PM
I think once you add the "..." it represents something slightly different. Like 0.999... = 1. But 0.999 = 0.999
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 02:44:38 PM
well by your logic 9r=10r, even though 9r-10r=-1 (no matter how many nines or zeros at the end, as long as it is an equal amount which it should be (infinity=infinity, duh) they would subtract to be 1, or -1) and NOT 0.
You can't do this with real numbers, because real numbers have decimal expansions with a leftmost digit, but not necessarily a rightmost so neither 9r nor 10r represent real numbers. Incidentally, p-adic numbers (http://en.wikipedia.org/wiki/P-adic_number) have base p representations with a rightmost, but no leftmost digit. So in the 10-adics, ...999 is an actual number, (although it's important to realize that there is a last 9 but no first, unlike 9r which one would think of as having a first but no last) and it is in fact true, in the 10-adics, that ...999=-1. (If you find this confusing, don't worry about it. The p-adic numbers are a very abstract algebraic concept which only professional mathematicians care about anyways.)

On the subject of the "first number after zero on the number line", as I've said before, there is no such object. The real numbers are dense, which means that between any two real numbers there is another real number. In fact, this is obvious, because if a and b are real numbers, so is their average, (a+b)/2, which lies half way in between them on the number line. As a corollary to this, there is no first number after zero on the number line, because if there were, half of it would lie between it and zero, so it wouldn't actually be the first.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 02:52:46 PM
well by your logic 9r=10r, even though 9r-10r=-1 (no matter how many nines or zeros at the end, as long as it is an equal amount which it should be (infinity=infinity, duh) they would subtract to be 1, or -1) and NOT 0.
You can't do this with real numbers, because real numbers have decimal expansions with a leftmost digit, but not necessarily a rightmost so neither 9r nor 10r represent real numbers. Incidentally, p-adic numbers (http://en.wikipedia.org/wiki/P-adic_number) have base p representations with a rightmost, but no leftmost digit. So in the 10-adics, ...999 is an actual number, (although it's important to realize that there is a last 9 but no first, unlike 9r which one would think of as having a first but no last) and it is in fact true, in the 10-adics, that ...999=-1. (If you find this confusing, don't worry about it. The p-adic numbers are a very abstract algebraic concept which only professional mathematicians care about anyways.)

On the subject of the "first number after zero on the number line", as I've said before, there is no such object. The real numbers are dense, which means that between any two real numbers there is another real number. In fact, this is obvious, because if a and b are real numbers, so is their average, (a+b)/2, which lies half way in between them on the number line. As a corollary to this, there is no first number after zero on the number line, because if there were, half of it would lie between it and zero, so it wouldn't actually be the first.

9r has a leftmost digit, being 9. 10r has a leftmost digit, being 1.

And if you say there is no first number after 0 on a number line, that is saqying no numbers even exist, that 0 is as far forward as you can go.

and if 0.9r is a real number, and 1 is definately a real number, there would be numbers between them.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 02:56:12 PM
Sometimes you just have to accept that things can make sense mathematically but not make sense logically.

Like I was reading the other day a solution to that old paradox where a question such as "Is 'no' the answer to this question?". They managed to find a mathematical "inbetween" answer that can be both yes and no at the same time. A load of rubbish it seems to be, but it makes sense mathematically.
Title: Re: .9999... equals 1?
Post by: EvilToothpaste on March 11, 2007, 03:02:20 PM
9r has a leftmost digit, being 9. 10r has a leftmost digit, being 1.

And if you say there is no first number after 0 on a number line, that is saqying no numbers even exist, that 0 is as far forward as you can go.

and if 0.9r is a real number, and 1 is definately a real number, there would be numbers between them.
Except that 0.9r IS 1. 
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 03:09:10 PM
9r has a leftmost digit, being 9. 10r has a leftmost digit, being 1.
Sorry, I should have said that decimal expansions have a leftmost digit, and finitely many digits before the decimal place, but potentially infinitely many afterwards. (p-adics have finitely many digits after the decimal, and perhaps infinitely many before, and may have no first digit) A decimal expansion is shorthand for the limit of the things you get when you take finite decimal expansions, so 0.999... is the limit of the sequence 0.9, 0.99, 0.999, 0.9999, ..., and the limit of this sequence is 1. p-adic representations are the same way, only the limit is taken in the p-adic numbers instead of the real numbers, so in the p-adics, ...999 is the limit of the sequence 9, 99, 999, 9999, ..., which is -1 (in the p-adic numbers).

Quote
And if you say there is no first number after 0 on a number line, that is saqying no numbers even exist, that 0 is as far forward as you can go.
No, it's not saying this at all. There are lots of real numbers after zero, it's just that none of them is the first. This is no different from the fact that there is no last natural number. Surely you would agree that natural numbers exist, and that there are in fact lots of them, but none of them is the largest? It is equally true that positive real numbers exist, and there are lots of them, but none of them is the smallest.
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 11, 2007, 03:14:19 PM
Sometimes you just have to accept that things can make sense mathematically but not make sense logically.
No. If something doesn't make sense logically, it can't make sense mathematically either. What you mean to say is that something can be counterintuitive, but still make sense mathematically; this is certainly true.

Quote
Like I was reading the other day a solution to that old paradox where a question such as "Is 'no' the answer to this question?". They managed to find a mathematical "inbetween" answer that can be both yes and no at the same time. A load of rubbish it seems to be, but it makes sense mathematically.
It sounds like a load of rubbish to me too, but probably the rubbish was not the mathematics that was being done, but rather the reporter trying to rephrase the mathematics in terms of ordinary language, which can be fantastically wrong even when the mathematics is correct. Ordinary language is sometimes unsuitable for expressing mathematics, and this is very often the case when paradoxes arise. The question, "Is 'no' the answer to this question?" simply has no answer, which is fine when you are talking about language. In order to do mathematics, you must somehow translate the question into a mathematical framework, and so the mathematicians were probably just saying that they had found a neat mathematical framework in which "questions" analogous to that question could be "asked", and would have a well-defined "answer" in their mathematical framework.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 11, 2007, 03:17:10 PM
It was quite scary given that if we are ever invaded by robot aliens we won't be able to confuse them and blow up their heads by shouting paradoxical questions at them.
Title: Re: .9999... equals 1?
Post by: RESOCR on March 11, 2007, 05:58:16 PM
It was quite scary given that if we are ever invaded by robot aliens we won't be able to confuse them and blow up their heads by shouting paradoxical questions at them.

Reminds me of the Dark Tower series.

You just have to ask 'Why did the chicken cross the road?'


back to topic, I think I get what your trying to say, that you progress it enough to the point they difference is indistinguishable, and being no solid difference, they are the same. Maybe? So it would hold true that 0.89r=0.9, 1.09r=1.1, etc?

But I still don't believe there is no real number that would be first, we'd just have to apply a range of 'tolerance', how many decimal places are accounted for. As long as you increase but maintain that there will be a smaller number.

Although someone pointed out a number line plotted on a graph, X axis being the number and -Y axis being the number of decimal places it has, or something to that effect. I liked that.
Title: Re: .9999... equals 1?
Post by: Dioptimus Drime on March 11, 2007, 07:45:13 PM
It was quite scary given that if we are ever invaded by robot aliens we won't be able to confuse them and blow up their heads by shouting paradoxical questions at them.
Like Santa Claus (Futurama)?


~D-Draw
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 12, 2007, 03:02:59 PM
I don't understand some of the terms used in the explanations when they start talking about "limits" etc. They say 0.999... reperesents the limit of "insert lots of fractions being added together here". What does that mean exactly? Can anyone explain the definition of decimals in layman's terms. I understand how 0.999... = 1 but I don't understand why exactly.
Title: Re: .9999... equals 1?
Post by: beast on March 12, 2007, 03:16:57 PM
Limits are just what happens to a function as it gets closer to a set point.

For example if we have f(x)=x+1  and we want to know limit x->3 then we can easily calculate that as limit f(x)=4.  Limits obviously have much more significance in more advanced problems.  For example the other week I was calculating how long it would take my team at work to update the credit card details of 1000 members.  I knew that we make about 20 calls per hour and that we resolve 29% of calls.  However if you graph that equation, you'll quickly see that we would end up calling a small amount of people a large number of times.  For this reason, we imposed the limit of 5 calls maximum per person, and found that we would resolve something like 80% of people within 5 calls.

Often in maths, you'll set the limit at infinity (despite the hilarious claim that maths cannot deal with infinity) and see what happens to a function as it approaches an infinite value for x.

That's about as basic and understandable as I can put it, and hopefully not too incorrect :P
Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 12, 2007, 03:17:44 PM
My explanation from page 1:
Masterchief, are you serious, or are you joking? The identity .99999...=1 is a well known fact. Every so often people doubt it, but such people don't even know the definitions.

First question: what is meant by the representation .98604, or .33333..., or .9999....?

Answer: each of these is shorthand for a sum, where the nth digit is the multiple of 10-n that we are adding. So .98604 is simply shorthand for 9*10-1+8*10-2+6*10-3+0*10-4+4*10-5, and .3333... is shorthand for 3*10-1+3*10-2+3*10-3+3*10-4+....

This immediately raises a second question: what is meant by an infinite sum? You can't add infinitely many things, can you?

Answer: yes, you can, in some situations. You can add 1+0+0+0+0+0+..., because after the first term you aren't changing things at all. You can add 1/2+1/4+1/8+1/16+..., because such a sum can be represented "physically" on the number line as follows: first you go half way from 0 to 1, then you go half way from where you are to 1, then you go half way from where you are now to 1, etc. The sum is 1, because that's exactly where you get when you move half way to one infinitely many times (this is not, of course, a proof, but acts as motivation for the idea of limits.) You cannot, however, take the sum 1-1+1-1+1-..., because that sum alternates between 1 and 0, and never settles on any one number (anyone who tells you that sum exists is either completely nuts or has been staring at the zeta function too long). The definition that makes this work is the definition of limits. A sequence of numbers x_n is just that: a sequence x_1, x_2, x_3, x_4, .... Such a sequence is said to converge to a limit x if for any e>0, there is an N>0 so that the differences between x and x_n are less than e for n>N. An infinite sum is just the limit of the partial sums of the initial terms, provided that limit converges. So the decimal number .999999... really is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc.

They don't teach you all this when they teach you decimals in school, because it's complicated, so they usually just gloss over it, but it is what decimal representations actually are, and so any demonstration that .999...=1 has to use this fact (of course, there are some tricks people sometimes use to "cheat", but those involve taking things on faith, which are true, but which are exactly the things that Erebos is denying above). The lack of such actual definitions is why so many people are confused, or try to deny that .9999...=1. But what's true is that decimal numbers are really just representations of real numbers, and are not always unique. The fact that 1 can also be written as .9999.... is just one example.

So, without further ado-

Proof that 1=.99999...:
We know that .999... is, by definition, the sum 9/10+9/100+9/1000+... which is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc. Simply finding common denominators tells us that 9/10+9/100+9/1000+...+9/10n=99...9/10n=(10n-1)/10n=1-1/10n, and since for any e>0, 1/10n is eventually smaller than e, this limit is 1.

p.s. I decided to change my avatar in honor of this thread.
If there are specific things from this explanation which are unclear, let me know which and I'll explain further.

Beast's post is dealing with the limit of a function at a point (and something about phone banking which doesn't really seem relevent), and he doesn't define it at all. Decimals are defined as the limit of a sequence, which is different (but related). You can see the definition of the limit of a sequence in the post I quoted above.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 12, 2007, 03:28:40 PM
My explanation from page 1:
Masterchief, are you serious, or are you joking? The identity .99999...=1 is a well known fact. Every so often people doubt it, but such people don't even know the definitions.

First question: what is meant by the representation .98604, or .33333..., or .9999....?

Answer: each of these is shorthand for a sum, where the nth digit is the multiple of 10-n that we are adding. So .98604 is simply shorthand for 9*10-1+8*10-2+6*10-3+0*10-4+4*10-5, and .3333... is shorthand for 3*10-1+3*10-2+3*10-3+3*10-4+....

This immediately raises a second question: what is meant by an infinite sum? You can't add infinitely many things, can you?

Answer: yes, you can, in some situations. You can add 1+0+0+0+0+0+..., because after the first term you aren't changing things at all. You can add 1/2+1/4+1/8+1/16+..., because such a sum can be represented "physically" on the number line as follows: first you go half way from 0 to 1, then you go half way from where you are to 1, then you go half way from where you are now to 1, etc. The sum is 1, because that's exactly where you get when you move half way to one infinitely many times (this is not, of course, a proof, but acts as motivation for the idea of limits.) You cannot, however, take the sum 1-1+1-1+1-..., because that sum alternates between 1 and 0, and never settles on any one number (anyone who tells you that sum exists is either completely nuts or has been staring at the zeta function too long). The definition that makes this work is the definition of limits. A sequence of numbers x_n is just that: a sequence x_1, x_2, x_3, x_4, .... Such a sequence is said to converge to a limit x if for any e>0, there is an N>0 so that the differences between x and x_n are less than e for n>N. An infinite sum is just the limit of the partial sums of the initial terms, provided that limit converges. So the decimal number .999999... really is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc.

They don't teach you all this when they teach you decimals in school, because it's complicated, so they usually just gloss over it, but it is what decimal representations actually are, and so any demonstration that .999...=1 has to use this fact (of course, there are some tricks people sometimes use to "cheat", but those involve taking things on faith, which are true, but which are exactly the things that Erebos is denying above). The lack of such actual definitions is why so many people are confused, or try to deny that .9999...=1. But what's true is that decimal numbers are really just representations of real numbers, and are not always unique. The fact that 1 can also be written as .9999.... is just one example.

So, without further ado-

Proof that 1=.99999...:
We know that .999... is, by definition, the sum 9/10+9/100+9/1000+... which is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc. Simply finding common denominators tells us that 9/10+9/100+9/1000+...+9/10n=99...9/10n=(10n-1)/10n=1-1/10n, and since for any e>0, 1/10n is eventually smaller than e, this limit is 1.

p.s. I decided to change my avatar in honor of this thread.

I bolded the parts that I don't understand.


Title: Re: .9999... equals 1?
Post by: skeptical scientist on March 12, 2007, 03:54:37 PM
Okay, the parts you bolded were the definition of a sequence, the definition of a limit, and the proof that the limit of the sequence of partial sums of the series represented by .999... is 1 (don't worry if you didn't understand that last sentence right now.)

First, what is a sequence of real numbers?
A sequence is a list of numbers indexed by the natural numbers (1, 2, 3, 4, 5, etc.), or equivalently it can be thought of as a function from natural numbers to real numbers. Often a sequence is written x1, x2, x3, x4, ... meaning that the first number in the sequence is x1, the second is x2, etc. The numbers in the sequence are called the terms of the sequence; for instance, the nth term of the sequence x1, x2, x3, ... is the number xn. (You can also talk about sequences of things other than real numbers, such as sequences of complex numbers, or sequences of points in the plane, etc., but for our purposes all we need are sequences of real numbers.)

Some example sequences:
The constant sequence zero: 0, 0, 0, 0, ...
The sequence 1/n: 1, 1/2, 1/3, 1/4, 1/5, ...
The sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
The sequence of powers of 1/10: 1/10, 1/100, 1/1000, ...
etc.

Second, what is a convergent sequence, and what is a limit?
A sequence of real numbers x1, x2, x3, ... is said to converge to a limit l if for every real number ε greater than zero, there is a natural number N such that for every n>N, the distance |xn-l| between the nth term of the sequence and the limit is smaller than ε. Roughly speaking, a sequence converges to l if the points on the number line defined by the terms of the sequence eventually get as close to l as you want.

Examples:
The constant sequence zero: 0, 0, 0, 0, ... converges to 0, because it is always as close as you could possibly want to zero.
The sequence 1/n: 1, 1/2, 1/3, 1/4, 1/5, ... converges to 0, because given ε>0, we can consider the real number 1/ε. There is some natural number N larger than 1/ε (this is sometimes called the Archimedian property of the real numbers). Then for n>N, 1/n<1/N<ε, so |1/n-0|<ε. This is exactly what it means for the sequence to converge to 0.

Third, what is an infinite sum?
An infinite sum, usually called an infinite series, is simply the sum of an infinite number of terms arranged in a sequence, and is defined by means of limits. If the terms of the series are y1, y2, y3, ..., then we define the nth partial sum sn of the series as the finite sum y1+y2+y3+...+yn. The series is said to converge to a limit s if the sequence of partial sums s1, s2, s3, ... converges to s (as defined above for a sequence), in which case we write y1+y2+y3+...=s.

Now does the proof make some sense? (By the way, can you see this symbol: ε? I want to make sure it displays properly on your browser. It's the lower case Greek letter epsilon, and it is almost always used as I have used it here in the context of limits in mathematics.)

p.s. The contents of this post constitute material which is not generally taught except in college math courses, and some high school calculus classes.

Edit: I decided to define infinite series in more detail as well.

You can also check out the Wikipedia pages on limits (http://en.wikipedia.org/wiki/Limit_of_a_sequence) (of sequences) and series (http://en.wikipedia.org/wiki/Series_%28mathematics%29).
Title: Re: .9999... equals 1?
Post by: cmdshft on March 13, 2007, 05:11:40 PM
tl;dr

Use limits. It's pre-calc High School math.
Title: Re: .9999... equals 1?
Post by: Kasroa Is Gone on March 13, 2007, 05:16:28 PM
I lose it when all the maths lingo pops up.
Title: Re: .9999... equals 1?
Post by: Wendy on March 19, 2007, 01:36:17 AM
A simple way of explaining it is that there is no number in between 0.9 repeat and 1, and because of that they are the same number. And to whoever said that 0.3 repeat isn't exactly 1/3, you're wrong. There is no exact number for a third, if you don't take the concept of infinity into account.
Title: Re: .9999... equals 1?
Post by: SPrinkZ on March 20, 2007, 12:00:16 AM
Shit, I can't believe Masterchief thought that .9 (repeating) didn't equal 1. I knew that when I was like...crap, I don't even recall how young I was.
Title: Re: .9999... equals 1?
Post by: Dioptimus Drime on March 20, 2007, 12:19:09 AM
Shit, I can't believe Masterchief thought that .9 (repeating) didn't equal 1. I knew that when I was like...crap, I don't even recall how young I was.
I bet your mother thought you were cool.

~D-Draw
Title: Re: .9999... equals 1?
Post by: SPrinkZ on March 20, 2007, 12:33:23 AM
Shit, I can't believe Masterchief thought that .9 (repeating) didn't equal 1. I knew that when I was like...crap, I don't even recall how young I was.
I bet your mother thought you were cool.

~D-Draw

No, she thought I was fucking ICE COLD, NIKKUH.

Like this ice cold: (http://static.flickr.com/20/71166193_46ec43c2e2_m.jpg)
Title: Re: .9999... equals 1?
Post by: Franc T., Planar on March 20, 2007, 01:01:32 AM
There are a few idiots on a different forum trying to claim that .9999... is actually equal to 1, which anyone with half a brain knows is false. There is no way that any number can equal a different number. Please, can anyone who isn't a fucking moron back me up on this?

http://www.kirupa.com/forum/showthread.php?p=2080661&posted=1#post2080661

You do not understand what infinity means.

You fail at life.

MASTERCHIEF2219 FTL
Title: Re: .9999... equals 1?
Post by: BOGWarrior89 on March 20, 2007, 12:03:58 PM
Boy, this looks familiar.
Title: Re: .9999... equals 1?
Post by: Midnight on March 20, 2007, 12:05:16 PM
Thread truncated at first binary zero.