Is 0.9 recurring equal to 1?

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pauxlo
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Re: Is 0.9 recurring equal to 1?

Sgt. Groovy wrote:
Zarel wrote:Assume that ö is the smallest possible number greater than 0.
ö > ö/2 > 0, therefore ö/2 is the smallest possible number greater than 0, and ö is not the smallest possible number greater than 0.
If you stick to real numbers, you're right. But then again, so were everyone before Bombelli who said that sqrt(-1) doesn't exist. What if ö is like i, that it doesn't make sense in the realm of real numbers, but if you ignore this seeming contradiction and just go ahead as if did exist, you may break into a new realm of numbers that nobody could even dream about before. That's how Nobel prizes are won!
(There is no Nobel prize for mathematics.)

If you extend the number space, you have to extend the <-relation too, and the operators.

I assume your extension Ö of R also extends the <-relation, since you want 0 < ö < (anything real bigger than 0). If it also extends the multiplication operation, you have a number 1/2 * ö = ö/2. So, if you say that ö is the smallest possible positive number, than we have either ö/2 < 0 < ö, or 0 < ö < ö/2, or 0 < ö = ö/2. The last one means that Ö is not a field (since in any field x = x/2 can only occur for x = 0). The other two cases mean that the <-operation on Ö is not consistent with the addition/multiplication operations.

There are extensions of R which extend +, * and < consistently (for example the hyperreal numbers), but these have still 0 < ö/2 < ö for 0 < ö.

Paŭlo
Midnight_Carnival
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Re: Is 0.9 recurring equal to 1?

Darkmage wrote:
Midnight_Carnival wrote:"correct" and "incorrect" are similarly abstract judgements/concepts,etc... which can be arbitrarily applied to anything. "misuse" is use which the people around you don't agree with.

"1+1=2" is as meaningful as saying " black cats are evil" or "4 is a lucky number".
No offense, but, regardless of the need to pretend we can have absolute clearly defined values interacting in a structured way in order to bypass a lot of pointless speculation, certain claims are best left to the highly religious, and mathematics makes a <urine>-poor religion.

Well, 1st of all, correct and incorrect are not that "abstract", it is "abstract", though, if an action is morally correct or incorrect, but if we use a method we can use that method correctly or not.[Those are two different meanings.]

2nd, about 1+1=2 having "no meaning" I got to disagree, maths are a method, we stablish that a unity is equal to 1 and the double of that is 2 and so on with the rest of the numbers, also with the operations, as + is the add, so just adding 1 to 1 is the double, so is 2, empirically, just take a pen, define a pen as 1, then, take a couple of pens and define it as 2, you wil notice that 1 + 1 equals 2, that is true. On the other hand you are telling math can become a religion, I guess you meant that it can become an unfounded belief, mut maths are not that, you can be given some formulas, but those formulas are the shortened path of more complex operations that you can ask and be answered, you only got to take some basis, but again is not nonsense basis(no offend), you just set a symbol for an idea, for the sake of communication, we do not tell you believe in 1=0'9~ because I say so, and then, go everynight an pray to the numbres and all that "religious" stuff, we just tell:

Do this operations: 1/3 = 0'3~ (that's right, right?)

Now try this: 3*0'3~ = 0,9~ (Is this ok?)

Then we will try something different: (1/3)*3 = 3/3 (I guess nothing's wrong here.)

So, we will make final deduction, 3/3 = 1, then 0.9~ = 1 We did the same operation in different ways.

So, the only option left is to say maths do not work properly, they are highly susceptible to be wrong at their claims of exactitude, is a science that fails at his purpose, it is just useless and we should prevent it's propagation. (Irony)

Edit: Also what you say about real/unreal, it does not mean directly for which you can or can't see, you can see many things that are not real but you think they are, for example, a delusion; also numbers are not the real thing you can see, they are concepts. If you state that non-visible things do not exist, tell me how do you see love, satisfaction, and so on..
Ok, last thing's first:
love: a horrific afliction I am most fortunate to be allmost imune to
satisfation: I'll tell you when I find out
so on: pretty much the same, and you can see it, it looks like this thread, and the other 0.9999(rec) thread we had a while back.

About the different meanings of the terms "correct vs incorrect": What I think you mean is that there is a difference between "correct" as in socially agreed uppon and "correct" as in corresponding to reality: I'd counter this as follows: 1) reality is a social construction which has a coincidental relationship with things people experience. 2) even if we could talk about an objective realtiy, I think numbers and mathematics would not form any part of this becasue they exist in people's heads only.

Take the word "horse", if I used that to designate the way I felt towards someone on the forum, it would most likey be judged as "incorrect". Yoou can't "horse something". But words change their meaning all the time, the change in meaning affects the concepts to which they refer and the way in which the concepts interact. Mathematics was designed so that this change would be limited for various reasons... Mathematics can be seen as a language in many respects. It serves its purpose, and should be propogated, but it should never become the basis for any form of ideology .

(sorry, I don't have any nice equations for you)
...apparenly we can't go with it or something.
Sapient
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Re: Is 0.9 recurring equal to 1?

Midnight - stop repeating yourself... the "horse", etc. analogies have nothing to do with this discussion. Even if it was relevant, you've already made the point multiple times. Either stay on the topic or, if that's too difficult, just don't post.

Really I think this question has been answered and then beaten to death.
If anyone has a reason this thread should not be locked, I'd be glad to hear it.
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HomerJ
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Re: Is 0.9 recurring equal to 1?

OP, my immediate answer to questions like this will always be: "That is a philosophical question..." and I think 5 pages to answer a single-line question qualifies the problem for this answer.

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Sgt. Groovy
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Re: Is 0.9 recurring equal to 1?

There is no Nobel prize for mathematics.
But there is one for economics.

Just think of the possible applications of ö in economics. In retail, the prices are always set a little bit under integer figures, so that they appear lower to the buyer. That is, €4.99 instead of €5.00, so that the price seems to be "four euros and change" instead of "five euros." But you still have to substract one whole cent for this trick to work. That's one cent less profit for every item sold. But if you would substract ö instead, you could make it work without earning any less, because the difference would be incomputable. In the scale of global economics, this would mean an annual difference on billions!
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Re: Is 0.9 recurring equal to 1?

Sgt. Groovy wrote:But there is one for economics.
Not exactly
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Re: Is 0.9 recurring equal to 1?

Not exactly
Can you provide a mathematical proof for that?
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Re: Is 0.9 recurring equal to 1?

Wikipedia says so!!!
Zarel
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Re: Is 0.9 recurring equal to 1?

Sgt. Groovy wrote:What if ö is like i, that it doesn't make sense in the realm of real numbers, but if you ignore this seeming contradiction and just go ahead as if did exist, you may break into a new realm of numbers that nobody could even dream about before.
Four points:

1. Even in number systems with infinitesimals, 1 - ε ≠ 0.9~ = 1.

2. You can adjoin infinitesimals to a number system, but they usually still satisfy the property ε > ε/2 > 0.

3. It is possible to artificially construct a number system in which ε is the smallest number greater than 0 (for instance, the integers, where ε = 1), however, even then, 1 - ε ≠ 0.9~ = 1.

4. When people ask "What is 0.9~?" they usually mean "What do mathematicians usually mean when they say '0.9~'?" not "Please construct an elaborate fantasy number system that no one actually uses in which 0.9~ is not 1."
Last edited by Zarel on July 7th, 2010, 7:08 am, edited 3 times in total.
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PeterPorty
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Re: Is 0.9 recurring equal to 1?

Zarel wrote:
Sgt. Groovy wrote:If you stick to real numbers, you're right. But then again, so were everyone before Bombelli who said that sqrt(-1) doesn't exist. What if ö is like i, that it doesn't make sense in the realm of real numbers, but if you ignore this seeming contradiction and just go ahead as if did exist, you may break into a new realm of numbers that nobody could even dream about before. That's how Nobel prizes are won!
Four points:

1. Even in number systems with infinitesimals, 1 - ε ≠ 0.9~ = 1.

2. You can adjoin infinitesimals to a number system, but they usually still satisfy the property ε > ε/2 > 0.

3. It is possible to artificially construct a number system in which ε is the smallest number greater than 0 (for instance, the integers, where ε = 1), however, even then, 1 - ε ≠ 0.9~ = 1.

4. When people ask "What is 0.9~?" they usually mean "What do mathematicians usually mean when they say '0.9~'?" not "Please construct an elaborate fantasy number system that no one actually uses in which 0.9~ is not 1."
I'm actually for Sgt. Groovy in this one... although I am pretty sure the guy who creatd this thread just wanted help with his math homework... I'n that case, 0.9999.... = 1, but otoh, it IS possible that it's not.
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Zarel
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Re: Is 0.9 recurring equal to 1?

pauxlo wrote:(There is no Nobel prize for mathematics.)
The closest analogous prize would be the Fields Medal.
Sgt. Groovy wrote:If you stick to real numbers, you're right. But then again, so were everyone before Bombelli who said that sqrt(-1) doesn't exist.
The difference is that sqrt(-1) doesn't exist in the real numbers. Constructing a number system in which it does exist leads to many practical applications.

0.9~ does exist, and it's equal to 1. Redefining it breaks a number of fundamental axioms that must now be redefined.

When we constructed the complex numbers, we didn't have this problem, since sqrt(-1) was never defined in the first place. We just defined it in a way that all of our previous axioms still worked.

By redefining what 0.9~ means, you'll have to redefine addition, subtraction, multiplication, division, exponentiation, equality, ordering. Do we still satisfy the three properties of equality? Do the distributive laws still apply? Do we satisfy the axioms of group theory? Ring theory? Field theory?

No, math does not work by saying "I wish these two things that were previously equal no longer are." You create new areas of mathematics by changing axioms like the Parallel Postulate, not results like 0.9~ = 1. Changing 0.9~ = 1 is like changing your website by applying White-Out to your computer screen - it doesn't work.
PeterPorty wrote:although I am pretty sure the guy who creatd this thread just wanted help with his math homework...
I sincerely doubt that. No one would ever have a 0.9~ = 1 question on their homework. Recurring decimals are completely irrelevant to any sort of mathematical study - in lower math, approximations are good enough, and in higher math, when approximations aren't good enough, people don't use decimals, they use fractions or other exact expressions.
PeterPorty wrote:I'n that case, 0.9999.... = 1, but otoh, it IS possible that it's not.
I don't think you know what the word "possible" means.

Again, refer to what I said earlier:

You can change what the symbols "1" and "+" mean, but you can't change the meaning nor the facts behind "1+1=2". You can only change how you write it.
Sapient wrote:If anyone has a reason this thread should not be locked, I'd be glad to hear it.
Not all of the more recent discussions have been resolved. If you think they're off topic, split the thread. If you think they're on topic, let us continue posting.
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Tonepoet
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Re: Is 0.9 recurring equal to 1?

I'd submit that such factors are largely unimportant, as for most practical intents and purposes whenever .9~ occurs its so impractically close to 1 that it may as well be rounded up. Even if it was a huge problem, you said it yourself, we cannot simply make wishes of the math and make it so.

What you guys have told me so far is that the transcendence of infinity is sequentially impossible. If that's the case, after reexamining the evidence, how do we come to the conclusion that .9~ x 10 = 9.9~? I believe the general method of multiplying by 10 to displace decimals usually relies upon moving the last digit and replacing it with a zero. This makes the proof unreliable because either A) There is no operating on the transcendence of infinity so this procedure simply cannot be done or B) Perhaps the transcendence of infinity can be done and the numerals are displaced after the infinitesimally small difference we're trying to define because 0 gets placed at the very end of our sequence, making .9~x10 indeterminate as infinity itself.

And we do know that there is a point at which 1 stops being 1 because of the existence of unequivalency. 1 isn't 2, 3 15, -2, 500,000 or .1, therefore it has boundaries and limitations. Determining exactly what the topography of 1 is, is important in order to determine what is what isn't equivalent to 1. I do not have a problem with 1 being represented in other ways so long as it's systematically congruent with logic of math. That 1/3 x 3 is 1 makes sense because 1/3 is defined as 1 split into 3 unilateral pieces. The value of 1/3rd is intrinsically affixed to the definition of 1. In fact all numbers are intrinsically related to the value of 1 as a frame of reference but we must be especially prudent in knowing in what way this is so.

Also, I know as a rule of thumb that usually moving the decimal point 1 over is usually a safe way of determining what x10 is but these rules of thumbs can be misleading. In the sequence of multiplication by 11 for example, it would seem safe to assume that merely writing the amount you're multiplying it by twice. 11x1 is 11, 11x2 is 12, 11,x3 is 33 and so on, however once we reach the double digits, this stops being the case and we have to do the operation normally because we've run out of unique integers to multiply by 10 and add itself to. So since this case may have special properties sensitive to similar numeric artifacts, I must be insistent as to following the protocol of proper procedure before admitting .9~ x 10 = 9.9~ as evidence.
Zarel answered this quite beautifully in mathematical terms. To put it another way, there is no number that is as close as possible to 1 without being 1 because you can always get closer.

Again we run into infinity. Whatever number you pick, no matter how close to 1, will have an infinite amount of numbers that are closer to 1. Therefore, to say XXX is as close to 1 as you can get without being 1 is false, no matter what XXX is.
Normally this would be true but I don't see why running into infinity should be a problem with .9~ because .9~ is infinite in enough ways to represent such a theoretical value. If it does represent this theoretical value, it cannot be 1 because 1 has a value that is inherently separate from the value just before 1. If I'm granted the above and can eliminate the process by which .9~ x 10 = 9.9~ I believe the only evidence we have left as to what .9~ can equate to are infinite sequences that attempt to eliminate a remainder that cannot be eliminated.

Shortly put, that the value of .9~ can only be this sequence would make very much sense, since .9~ is a number with a beginning and no end, just like the sequence itself.
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Re: Is 0.9 recurring equal to 1?

4. When people ask "What is 0.9~?" they usually mean "What do mathematicians usually mean when they say '0.9~'?" not "Please construct an elaborate fantasy number system that no one actually uses in which 0.9~ is not 1."
Don't put words in my mouth, I never said anything about 0.999..., my ramblings were an answer to a totally different question.
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Re: Is 0.9 recurring equal to 1?

Sgt. Groovy wrote:Don't put words in my mouth, I never said anything about 0.999..., my ramblings were an answer to a totally different question.
Fair enough. But my other post still applies. And there are still problems with that approach:

1. Nearly every known infinitesimal number system still satisfies ε > ε/2 > 0, so the infinitesimal is not the smallest possible value.

2. Number systems in which a minimal element greater than another element are necessarily quantized, and thus incompatible with being adjoined to the real numbers unlike sqrt(–1).

3. Number systems in which a minimal element greater than another element are usually not infinitesimal at all (I mean, quanta necessarily can't be infinitesimal since they have a rather obvious size). For instance: The integers. 1 is the smallest integer larger than 0, but 1 is not an infinitesimal by any definition.

4. Number systems aren't created for fun, they're created because they're meaningful. Complex numbers have applications in electricity and magnetism, and surreal numbers can model games and economics. I do not know of any meaningful application of a number system with a minimal infinitesimal element greater than zero.

5. I'm not even sure if it's possible for an infinitesimal quantized number system to exist - "infinitesimal" and "quantized" seem mutually contradictory to me.
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Re: Is 0.9 recurring equal to 1?

"infinitesimal" and "quantized" seem mutually contradictory to me.
Like wave and particle?
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