Alexander Gieg says:
May 11, 2011 at 1:02 pm
What can be summed up by saying that in the decimal system rational numbers that aren’t usually “shown” in their repeating decimal form do in fact have a repeating decimal representation:
1 / 1 = 0.9999… = 1
1 / 2 = 0.4999… = 0.5
1 / 3 = 0.3333… = (no short representation)
1 / 4 = 0.2499… = 0.25
1 / 5 = 0.1999… = 0.2
1 / 6 = 0.1666… = (no short representation)
…
2 / 1 = 1.9999… = 2
2 / 2 = 0.9999… = 1
2 / 3 = 0.6666… = (no short representation)
…
And so on and so forth. In fact, it wouldn’t surprise me to find someone out there arguing the repeating decimal representation to be the most appropriate decimal version.
By the way, are there practical circumstances (let’s say, some interesting algorithm) in which this representation, rather than the usual one, would be preferable or even easier to use?
Uli says:
May 11, 2011 at 2:32 pm
I always liked this explanation:
1/3 = 0.33333…
2/3 = 0.66666…
3/3 = 0.999999… = 1
Xamuel says:
May 11, 2011 at 3:33 pm
I hate to do it, but I’m going to have disagree with the explanation, though I’ll grant it’s a very common mis-explanation. I addressed the question here: http://www.xamuel.com/why-is-0point999-1/ To summarize it here: the question is not so much why 0.999… equals 1, but rather, what ARE real numbers in the first place? One naive approach is to define real numbers to BE their decimal representations, with arithmetic DEFINED by the algorithms, and this is what anyone would implicitly think after K-12 math (since we never tell them otherwise). According to this naive definition, 0.999… is NOT 1, as these decimal representations are certainly different. But what the explanations in your post demonstrate is that this naive definition is a bad one, because it causes algebra to break (for example, using the naive definition, the distributive law breaks and a sequence becomes able to converge to two different limits simultaneously). Since we can’t stomach having such badly behaved arithmetic, we basically *define* 0.9999…=1 (and 1.999…=2, and so on) using equivalence classes. TL;DR: 0.999…=1 by definition.
Paul says:
May 11, 2011 at 5:00 pm
Xamuel, may I ask which of the two explanations you are disagreeing with, and on what level (as an persuasion to the layman, as a proof to blahblah, or otherwise)? And why do you think most people implicitly take the decimal expansion as their definition of the real numbers, is it what you gathered from your non-math friends?
Xamuel says:
May 12, 2011 at 12:27 am
Paul: Fact is, all the standard “proofs” of 0.999…=1 go about things the wrong way. The general pattern is, they present an argument which uses some Fact X, which fact is some well-behavedness fact about the reals. They then conclude that if 0.999… is not 1, that Fact is wrong, which would make the reals badly behaved, so 0.999…=1. But this is backwards. However much we want Fact X to hold, it depends on WHAT the reals actually are.
(.999…)=9*(.999…)/9=(10-1)*(.999…)/9: We have used the distributive law. This law is not true of formal decimals under the arithmetic of the addition and multiplication algorithms.
Limit argument: We use the fact that a sequence can converge to at most one limit (formalized with epsilons and deltas): This fact is not true of formal decimals under the arithmetic of the addition and multiplication algorithms.
3/3=3*(1/3)=3*(.333…)=.999…: We have used some facts about division and multiplication which aren’t true of formal decimals under the algorithmic addition and multiplication.
Correct Conclusion: We should not use formal decimals as our definition of the reals. We should use Dedekind cuts or Cauchy sequences. Or we can use Stevin’s construction (google it), where we do use formal decimals, but DECLARE that 0.999…=1. By definition. In ORDER to make other desired facts work.
Incorrect Conclusion: “So 0.999…=1. Because reals gotta be well-behaved. Don’t ask what reals are! Nothing to see here!”
Xamuel says:
May 12, 2011 at 12:34 am
I admit I’ve never actually polled random people about what numbers are. But in my experience, elementary math is most often taught as though reals are formal strings of decimals, with arithmetic defined by algorithms. You’re right, this might be something worth investigating.
Xamuel says:
May 12, 2011 at 12:36 am
And now that I check, Google does not correctly find Stevin’s construction
Here’s a link: http://en.wikipedia.org/wiki/Construction_...7s_constructionThe Mathematician says:
May 12, 2011 at 10:35 pm
Hey Xamuel,
Thanks for your comment. Yes, I am implicitly assuming that 0.9999… is a real number (or, at least, has certain properties of real numbers). The proofs I used work just fine, under this assumption. That does not imply this is a mis-explanation, only that it requires belief that 0.9999… has the properties of other numbers. The people I have showed these simple proofs to seem satisfied (you excluded), implying that they believe that 0.9999… has the properties that other numbers have. In light of that, I do not know of evidence for your claim that the wrong question is being answered or the wrong assumptions are being made.
Em says:
May 13, 2011 at 12:20 am
When you round it is. 0.99999999999999999 is 0.00000000000000001 away from one. So, technically you could count it as one unless yo are working on a rocket ship… Being exact is important at that point.
David Schreier says:
May 13, 2011 at 6:11 am
Yeah, Xamuel has got it, – definition of the Reals needs a bit of sprucing up. You can make the case for the non-existence of any Real Number that is NOT of the form .9999999…, because if the number does not includes an unknown element, then it is “imaginary”. This would butt heads with what we commonly think of as an imaginary number. Anyway, the whole question goes nowhere until there is a change in Western philosophical direction concerning ontology, and that is not happening for a while.
christopher says:
May 13, 2011 at 4:15 pm
if .999999… is equal to one, does that mean math in incapable of describing a number which infinitely approaches, but is not quite equal to one? what if i take one, and then subtract the smallest possible amount from it, it should be almost but not quite one. and i would figure it would be represented by .999999…. forever.
The Physicist says:
May 13, 2011 at 4:20 pm
Unfortunately, there’s no “smallest amount”. Necessarily, if you take away any actual amount away from one you’ll end up with something smaller than 0.999…
christopher says:
May 13, 2011 at 5:59 pm
but what if i took that amount away, giving me something smaller then .9999…. then kept adding a little more, continuously over an infinite amount of time? the number infinitely approaches one but never hits it. and if i wanted to describe what this ever changing number approaching one without ever realizing one is, how would i do so if i cannot use .99999…. is there some other notation for it?
Neal says:
May 16, 2011 at 1:16 pm
You know, there’s another number system that might be useful for analogies here. We all learn very early the difference between these “pseudo-numbers” and the actual “numbers” themselves.
I’m talking about , of course. and so forth. These are all different symbols, different pseudo-rational numbers, but they’re all representatives of the same rational number. How do we know two formal fractions are equal? Precisely when they cross-multiply to equal numbers.
As we know, something similar happens with real numbers (I’m taking Xemuel’s definition of equivalence classes of pseudo-reals, BTW, because I like it — it’s just the Cauchy sequence def, but slightly pared down, I think), except in this case two decimals are equal precisely when their difference goes to zero. So since ,
I’ve never seen anybody compare directly to before, but it seems to me to make good sense.
(I wrote up something a little more extensive here.)
(PS I am hoping that this will take tex?)
The Physicist says:
May 16, 2011 at 1:54 pm
We got a LaTex plug-in for this site. To write something in LaTex write”$latex” then your code, and end with “$”.
(I added in the “latex” for the last comment already)
The Physicist says:
May 16, 2011 at 3:28 pm
@christopher “1″? You’re exactly describing a “limiting sequence” or more specifically a “Cauchy sequence”. It turns out that one of the properties of the number line is that a limiting sequence only approaches one point.
In this case you can definitely construct something that approaches 0.9999…, but at the same time that sequence will also approach 1.
They’re the same after all!
Hamlin says:
June 2, 2011 at 3:02 am
i say 1=0.99999….. cause when u subtract 0.99999… from 1 the number that comes out will be 0.00000…. and there cant be a number at the end of those stream of 0 cause if there was then the 0.9999… wouldnt really be going on forever so making the number added with 0.9999 to make 1 will be 0 and 1-0 can only be 1 making 1-0 and 0.99999….+0 equal or 0.9999999…..=1
Hi says:
June 3, 2011 at 11:38 pm
1/3 != .3333333333
1/3 = .3333333333…. + 1/ (3 *10 ^infinity)
2/3 = .666666666….. + 2/(3*10^ infinity)
3/3 = .99999999….+ 3/(3*10^infinity) = .999999999…. + .000001 = 1
o.O
.9999999999….. != 1
0.9999… = (9*0.9999…) / 9 = ((10-1) 0.9999…) / 9
= (10*0.9999… – 0.9999…) / 9
= (9.9999…99990 – 0.9999…) / 9
= (9 + 0.9999…99990 – 0.9999…99999) / 9 = (9-.000…0001) / 9 = 8.9999999999/9 != 1 o.o
John says:
June 7, 2011 at 6:01 pm
In another thread http://www.askamathematician.com/?p=6992 you said that because two different numbers equaled each other, the original hypothesis could not be true. Why doesn’t this apply here? (Rather, wouldn’t it prove that 0.99… isn’t a real number [a non-real hypothesis?])
The Physicist says:
June 7, 2011 at 6:53 pm
That was me (a Physicist!).
In that case, the fact that 0 and 1 are definitively different numbers was used to prove another point. If, however, the fact that 0 isn’t 1 was in doubt, then that fact (0 isn’t 1) couldn’t be used to prove anything else.
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human mathematics says:
September 5, 2011 at 1:41 pm
It depends on the number system. If you use hyperreal numbers, they are not the same.
STEVE A JEFFREY says:
Your comment is awaiting moderation.
September 30, 2011 at 5:53 am
AND ALL THE WORLD WONDERED AFTER THE BEAST.
In the example you gave the left side makes some sense; you can talk about apples and oranges, and fractions of them. But on the right side you’ve got an “apple/orange”, which isn’t a real thing. The same sort of problem crops up when one starts talking about physics.
Every equation describes something. They’re like very succinct sentences. Perhaps you could “add them in thirds”, but what you get out is very unlikely to make any sense.
For example, here are three sentences I found by googling random words (specifically: wildebeest, summer camp, and water).
-”Gnus belong to the family Bovidea, which includes antelopes, cattle, goats, and other even-toed horned ungulates.”
-”And for this round, we’ve shaken things up a bit, and opened up the promotion to a mix of bands on the 2011 Summer Camp Line Up.”
-”We offer information on many aspects of water, along with pictures, data, maps, and an interactive center where you can give opinions and test your water knowledge.”
Each says something, and each kinda needs to be used in the right context to make any sense. Combining in thirds you just scramble them even worse:
-”Gnus belong to the family Bovidea and for this round, we’ve shaken things up a we offer information on many aspects of water, along”
You could wander the world looking for a situation where this sentence makes sense, but there’s no reason to expect you’d find it. Moreover, the new sentence itself doesn’t makes sense. Despite the fact that all three of the original sentences were grammatically correct (except for starting with “and”), the new sentence has several errors.
The same is true of the equations of QM and GR. They’re talking about wildly different stuff.
Let me know if that makes sense or clears anything up, and thanks for asking!
-Physicist
Ok you said to let you know if this clears anything up.
Actually you can get a GNU by adding DNA 1/3 COw+ 1/3 goat+ 1/3 Antelope= 1 GNU (wilderbeast)..
These three familes make up the GNU which is probably related to the original ancestoral species………..
So in the same way three equations out of millions of equations may have made sense together in the big bang.
They may have originally been the same species like the GNU.
So you can add QM and GR in 1/3s and the right equation will be a contradiction between the two like the GNU.
Does this make sense to you.?Sorry about the delay.
A computer might be able to accidentally generate a theory of everything, but how would you be able to tell when you’ve got the right set of equations? Moreover, if the TOE involves math that hasn’t been imagined yet, then there are no equations that could be combined to get it.
I can’t say with absolute certainty that your idea won’t work, but I can say that it would work about as well as any method one might use to generate random equations.
If you wanted to write a play you wouldn’t program a computer to combine words at random, you’d just sit down and right it.
-PhysicistWell this is one of the most interesting presentation of question I have seen.
1st part of your question asks “can we estimate the probability that we can come up with a sentence with grammar and syntax by adding random sentences in thirds.”
To this i can say there is a lot of work going on this field. 1st step to this is for a computer to be able to comprehend the question as humans. There have been a lot of work in this field especially using in the field of machine learning and artificial Intelligence. Although the technique will take a lot of time to master since the number of permutation are in billions and thus needs an very effective algorithm and computer very very high computing power.
2nd part of question asks.
And if we can get one in a billion sentences to make sense does this mean we can extropolate to adding random equations in thirds and getting a thoery of everything that works for QM and GR.?
Well getting one out of billion sentence and combining the equation of physics to get Unified Field theory or theory of everything are equation in the same genre still with a lot of differences and complexity. The complexity manly arises due to mismatching of how different forces change there nature and the cases where singularity comes to picture. There is no problem in solving the 1st part that comes close to encounter singularity. The particle Changing there behaviour at speed very close to light and very low mass are also few problems. I won’t deny that solving the 1st problem will help us solving the 2nd but that wont solve the problems completely.
One part of your question asks if we will ever find out whether the equation computed is correct or not. Well this is the most famous problem in theritical computer science. This is the problem of Verification and actually Finding the answer. Finding A answer in lot of cases is very complex and is not computable in polynomial time but verification of the same problems is lot less complex. So if we can find the answer to the problem than verifiaction to the probelm is going to easier. But From a random set of hypothesis verification of each of them to find is going to lot more complex.
This is just a disucssion type answer. Understanding the theoritical answer of your question completely will requrie a very using advanced knowledge of the laws of computation, Thoeritcal computer science , Complex laws of physics, Np completenss, Alogirthm of randomness and probablity and statistics and the answer can ammount to a size of a book.
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To build a Cosmos Mockup computer using visual 180 computers and C.
You import random equations QM and GR with Maths type 6.
And combine them in a physics spreadsheet 1/3 APPLE+ 1/3 ORANGE+ 1/3 ORANGE= 1 APPLE/ORANGE.And print out the results with a dot matrix printer.
And analyse the output on the bakckboard looking for equations that makes sense.
As it is a mock up of Cosmos it will produce similar resultsv to Cosmos when it si working on 10^500 different sets of rules for the multiverse.
You can build a Cosmos mockup on an IBM computer running C for under a thousand dollars…………………
Compare this with the cost of CERN and you will get similar results.