.9999... equals 1?

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Re: .9999... equals 1?
« Reply #30 on: March 10, 2007, 02:33:23 AM »
Erebos--

you lose.

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skeptical scientist

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Re: .9999... equals 1?
« Reply #31 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?
-David
E pur si muove!

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EnragedPenguin

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Re: .9999... equals 1?
« Reply #32 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.
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Masterchef

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Re: .9999... equals 1?
« Reply #33 on: March 10, 2007, 11:10:19 AM »
I give up. :D

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skeptical scientist

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Re: .9999... equals 1?
« Reply #34 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.
-David
E pur si muove!

Re: .9999... equals 1?
« Reply #35 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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EnragedPenguin

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Re: .9999... equals 1?
« Reply #36 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.

« Last Edit: March 10, 2007, 12:57:10 PM by EnragedPenguin »
A different world cannot be built by indifferent people.

Re: .9999... equals 1?
« Reply #37 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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skeptical scientist

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Re: .9999... equals 1?
« Reply #38 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.

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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.

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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?

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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".

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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.
-David
E pur si muove!

Re: .9999... equals 1?
« Reply #39 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?
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #40 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.
« Last Edit: March 10, 2007, 01:19:04 PM by Erebos »
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Re: .9999... equals 1?
« Reply #41 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?
-David
E pur si muove!

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skeptical scientist

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Re: .9999... equals 1?
« Reply #42 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.
-David
E pur si muove!

Re: .9999... equals 1?
« Reply #43 on: March 10, 2007, 01:49:00 PM »
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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:

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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.
« Last Edit: March 10, 2007, 01:50:42 PM by Erebos »
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #44 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Pyrochimp

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Re: .9999... equals 1?
« Reply #45 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
Some people are ****ing stupid! ~ George Carlin

Mathematical proof of the flat Earth:
[{(Diameter of Earth)*(tan[distance from Earth to sun/distance from North pole to equator])}2]/0

Re: .9999... equals 1?
« Reply #46 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Pyrochimp

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Re: .9999... equals 1?
« Reply #47 on: March 10, 2007, 01:59:41 PM »
I read the topic, .9r really does = 1 though.

Here, what's 1 - .9r?
Some people are ****ing stupid! ~ George Carlin

Mathematical proof of the flat Earth:
[{(Diameter of Earth)*(tan[distance from Earth to sun/distance from North pole to equator])}2]/0

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GeoGuy

Re: .9999... equals 1?
« Reply #48 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.

Re: .9999... equals 1?
« Reply #49 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
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #50 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!"
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Pyrochimp

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Re: .9999... equals 1?
« Reply #51 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.
Some people are ****ing stupid! ~ George Carlin

Mathematical proof of the flat Earth:
[{(Diameter of Earth)*(tan[distance from Earth to sun/distance from North pole to equator])}2]/0

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GeoGuy

Re: .9999... equals 1?
« Reply #52 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.

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EnragedPenguin

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Re: .9999... equals 1?
« Reply #53 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...
A different world cannot be built by indifferent people.

Re: .9999... equals 1?
« Reply #54 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #55 on: March 10, 2007, 02:26:46 PM »
What a pointless discussion!

Re: .9999... equals 1?
« Reply #56 on: March 10, 2007, 02:50:38 PM »
yea 1 is acts like a 'symbol' for .99999999...

Re: .9999... equals 1?
« Reply #57 on: March 10, 2007, 03:35:37 PM »
yea 1 is acts like a 'symbol' for .99999999...

lol!
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #58 on: March 10, 2007, 03:39:14 PM »
What is 2 x 0.9r? Is it 2?

Re: .9999... equals 1?
« Reply #59 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.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.