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This is the talk page for discussing changes to the 0.999... article itself. Please place discussions on the underlying mathematical issues on the Arguments page. If you just have a question, try Misplaced Pages:Reference desk/Mathematics instead.

view · edit Frequently asked questions Q: Are you positive that 0.999... equals 1 exactly, not approximately? A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Misplaced Pages article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so. Q: Can't "1 - 0.999..." be expressed as "0.000...1"? A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10 toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of ${\displaystyle {\frac {0.000\dots 1}{10}}}$ would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0. Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1? A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit ${\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}$ is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1. Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1. A: No. By this logic, 0.9 < 0.999...; 0.99 < 0.999... and so forth. Therefore 0.999... < 0.999..., which is absurd. Something that holds for various values need not hold for the limit of those values. For example, f (x)=x/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that ${\displaystyle 0.x_{1}x_{2}x_{3}...}$ must be less than ${\displaystyle 1.y_{1}y_{2}y_{3}...}$ for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation. Q: 0.999... is written differently from 1, so it can't be equal. A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers. Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount? A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0, which is not a number, but rather part of the expression ${\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}$, the right limit of x (which can also be expressed without the "+" as ${\displaystyle \lim _{x\downarrow 0}f(x)}$); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application. Q: Are you sure 0.999... equals 1 in hyperreals? A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, ${\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}$, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly? A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system. Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right. A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway. Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article? A: The criteria for inclusion in Misplaced Pages is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Misplaced Pages policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Misplaced Pages policy on original research.

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One important skepticism reason is missing...

Talk:0.999.../Arguments

My Insane Rantings

So 1/3 does equal .3..., so you could say the continuous threes are basically and infinite about of 3s. However infinity is a concept. Since .3...... involves a concept you cannot mulitply is by 3 and justify it equaling 1. Techiqually you cant multiply by a concept, so when you mulitply .3... by 3, youre actually multiplying 1/3, a non-concept number, by 3. However since a calculator can't understand concepts it gives you .9..., which is another concept.

SO taking all this in account, you can't actually multiply .3... by 3 because it is a concept. When you do that you are actually multiplying 1/3 by 3, which is not a concept, and gives you 1. —Preceding unsigned comment added by 75.189.196.97 (talk) 20:18, 21 October 2009 (UTC)

Ho ho! This person forgot to make his own section, instead inserting it into an old discussion where it didn't belong! Oh well, just gives me the right to title his post myself. Hope you like it, anon.

Anyways, here's your answer, anon (unfortunately, I could not find a font big enough):

.

NUMBERS

.

ARE

.

CONCEPTS!

.

HTH. --COVIZAPIBETEFOKY (talk) 21:37, 21 October 2009 (UTC)

How?

How can one number be the same as another? I don't doubt that 0.999999999 etc is close to 1 but it is not one. If that was the case you can say subtracting 1 from 0.999999999 would give you zero, but that's not true. Also is 1.9999999999999999 the same as 2? etc etc?

It is sufficient to say that it is close enough to, to say it equals 1, but technically it is not 1. 203.134.124.36 (talk) 03:07, 23 October 2009 (UTC)

I'll assume that you meant to add ellipses to the end of each of your trailing nines, to indicate infinitely many nines, rather than some large finite number of nines.

Can you, perchance, tell me exactly what number you believe 0.999... - 1 is, if not 0? Also, what is 1 - 0.999...?

In answer to each of your questions:

• How can one number be the same as another?

It is important to make the distinction between a number and its representation. 0.999... and 1 are different representations for the same number.

• Also is 1.9999999999999999 the same as 2? etc etc?

Yes, 1.999... = 2, as well as 99.999... = 100, 0.0999... = 0.1, 3.4999... = 3.5, 4.3879999999... = 4.388, etc.

• It is sufficient to say that it is close enough to, to say it equals 1, but technically it is not 1.

No, it is not sufficient to say that 0.999... is 'close enough to 1' to say it equals 1. When we write that equality, we mean exactly. To take an analogy that was mentioned in the arguments page a little further, the difference between 1 and 0.999... can perhaps be thought of as being the difference between a whole pie, and a pie in which I've made a single cut from the center. I haven't actually removed anything, so there's the same amount of pie.

Hope that helps. --COVIZAPIBETEFOKY (talk) 04:44, 23 October 2009 (UTC)

It's important to remember that we're dealing with the decimal number system for representing numbers. we're so used to it that we often skip certain steps in our head without realizing what we're doing. For example, .1 is not really a number, it's just a decimal representation of a number. It says in the tenths value we have 1 quanity. Or to put it simply: 1/10. Or, if we have 1 pizza, it has been cut into 10 equal slices and 9 of those slices are missing, leaving us 1 slice.

When you look at a number like .999... and think think it's less than 1, your brain is skipping a step. It sees that .9 and doesn't really care about the rest because it knows no matter what - this number is less than 1. But that's because you're so used to working with the decinal number system that you jumped over actually looking at what .999... really represents. What it really means is every place value from the tenths, to the hundredths, on down, has the quantity 9 in it. Then you have to look at what that means.

Let's go back to our pizza. What does the number .938 represent? 9 tenths. 3 hundredths. 8 thousandths. On our pizza that means we cut it into 10 slices, save 9 of those 10. Then with the remaining slice we cut that into 10 equal slices as well, we save 3 of those slices, then we take one of the remaining 7 slices from that cut and cut that into 10 slices as well, and save 8 of those. We are left with the following: 9 slices 1/10th sized. 3 slices 1/100th sized, and 8 slices 1/1000th sized. As well as 6 slices 1/100th size and 2 slices 1/1000th sized we didnt use and will throw away. also notice if you add up .9 + .03 + .008 and + .06 + .002 (the leftover slices we didnt use) youll get 1.

This may seem trivial but there's something important to note here, what's the maximum amount of slices of pizza you can ever have in a given place value? The answer is 9. You may be tempted to say 10, but the 10th slice is used for the next place value. Remember how our 1/100th sized slices in the previous example were actually part of the 10th slice of 1/10th sized slices? (note if you actually do have 10 slices then you really still just have 1 slice of the previous place value (ie 10/10 = 1)

So .999... means every place value is maxed out. We have 9 slices, the maximum, for every single division on down, forever. So what does this mean?

Well let's take a look at our pizza and start at the top. We see we have 9 slices of 1/10th size. 9 slices of the 1/100th size. 9 slices of the 1/1000th size...etc You'll quickly realize that you're not actually missing any pizza here. The "left over" pizza (remember how we had .06 and .002 left over when we cut our pizza into .938?) after your divisions is actually being used to divide further. The "missing" 1/10 slice from our first 9 slices of 1/10th size that would give us 10/10 slices or 1 whole pizza, is actually there, it's just being used to hold all 9 slices of the hundreds. And the "missing" 1/100 slice from our 9 slices of the 1/100th size is being used to hold all 9 slices of the 1/1000s and so on. So in actuality, all "missing" slices are in use for dividing the pizza down to every place. Every place has its maximum 9 slices, and every places "missing" 10th slize that would "complete" the pizza is still there, it's just being used to divide down to the next lowest place value. There is no missing slices and we have a full pizza. .999... = 1.

If you want to prove this yourself. Start from the top down. Take an imaginary pizza and cut it into 10 slices for your first place value. Then cut one of those slice into 10 slices for your 1/100ths. Then cut one of those slices into 10 for your 1/1000ths. Then cut one of those slices into 10 for your 1/10,000ths. Now take a break and look at the pattern you have.

9 slices 1/10th size 9 slices 1/100th size 9 slices 1/1000th size 10 slices 1/10,000th size

Looking familiar? .9 + .09 + .009 +...But wait a minute you say, I thought we couldn't have 10 slices? That's right. So those 10 slices of 1/10,000th become 1 slice of 1/100th which add our total of 1/100th slices to 10 which becomes 1 slice of 1/10th and that means we now have 10 slices of 1/10th or 10/10 or 1.

But say we didn't take that break, we just kept cutting and took one of those 10 1/10,000th sized slices and cut it into 10 for 1/100,000th and cut one of those and so on forever....

I'm sure you now see that if you keep going forever what you really have here is just every possible division of the pizza by 1/10 (which is infinite) and what this leaves you with is 1 whole pizza cut in a way that every possible decimal slice is available to you, and no slices are missing.

You have 1 pizza divided into every possible division of 10. This just happens to give you 9 slices of each division, which just happens to be the same thing as .999. repeating decimal.

You could also have 1 pizza divided into every possible division of 2, which would give you only 1 slice of each division. This gives you 1/2 + 1/4 + 1/8 + 1/16...(forever) = 1, but there's no provocative way to write that so most people don't have a problem with it. Whereas writing out 9/10 + 9/100 + 9/1000...(forever) = 1 as .999... = 1 is provocative and grinds some gears, but now you know what it really means and perhaps it isn't so scary :) PS If this was too long winded for this discussion page i apologize, dont mind if it's deleted. 76.103.47.66 (talk) 00:28, 25 October 2009 (UTC)

One number may be the same as another number if they represent the same thing. "4/4" and "1" look differnent on the page, but they are simply different ways of representing the same thing, they are both equal to one. Similarly, 0.9999999... is just another way of representing the number 1.--RLent (talk) 21:16, 31 December 2009 (UTC)

first paragraph should

mention that, like all fractions, 0.999... is a sum (9/10 + 9/100 + 9/1000), and, in fact, a sum of infinite addends, these adding up to "1". 0.999... is a sum just as 1.000... is a sum. (1 + 0/10 + 0/100 + 0/1000). Once the reader is reminded that decimals -- whether terminating or not -- are sums, it will no longer be surprising in the least that two of them (1 and 0.999...) can be the same, any more than it is surprising that 1, 1.0, 1.00, 1.000, and 1.0000 are the same sum.

By contrast with the above, correct initial reminder, the current phrasing is misleading. 1.0 and 1.00 don't "represent" the same number: they are sums whose value is the same number. The first "represents" 1 + 0/10 and the second "represents" 1 + 0/10 + 0/100. Likewise, 1 and 0.999... don't "represent" the same real number (as is currently stated). Rather, they represent the sum 1/10^0 (single addend) on the one hand, and 9/10^(-1) + 9/10^(-2) etc (infinite addends) on the other. As soon as the reader is reminded of this, 99% of the rest of the article is unnecessary. Decimals are, very simply, shortcuts for sums of powers of ten.

"Sum of infinite addends" usually isn't defined at all. It's a series (or, for full precision, the sum of a series), which is a special case of limit of a sequence. There is a significant difference between how finite and infinite decimals are constructed (although, of course, the finite decimals can be seen as a special case of the infinite ones, corresponding to the sequences which become constant - but then even the finite decimals cease to be seen as "sums"), and I don't think we should try to hide that difference. Huon (talk) 21:57, 2 November 2009 (UTC)

If 0.9999... equals 1, I cannot understand why a monkey typing for an infinite amount of time cannot re-write everything written by Shakespeare (see the 'infinite monkey' article). There seems to be a contradiction here.86.153.54.29 (talk) 21:12, 25 November 2009 (UTC)

A monkey typing for an infinite amount of time can re-write all of Shakespeare... (they aren't guaranteed to do so, just almost sure to, but that difference is a very subtle one). --Tango (talk) 21:25, 25 November 2009 (UTC)

Given enough time and a key typing process that is truly random, they will. (This assumes that the typing process is random enough to guarantee that all possible subsequences of keystrokes are equally likely.) — Loadmaster (talk) 23:50, 2 December 2009 (UTC)

Sources

The "Introduction" and "proofs" sections are entirely unsourced. I'm no math whiz, but is this normal for WP math articles since it's certain math processes fall under WP:FACTS? Ten Pound Hammer, his otters and a clue-bat • 01:49, 7 November 2009 (UTC)

Many of the mathematical articles do not have extensive references. See Rational number and Logarithm, for example. That being said, WP:MOS(math) and WP:SCG recommend including a useful amount of quality references. — Loadmaster (talk) 02:09, 7 November 2009 (UTC)

Yes, but is this because of WP:FACTS or because of lazy writers? Ten Pound Hammer, his otters and a clue-bat • 02:09, 7 November 2009 (UTC)

I think it's probably because mathematicians don't think it terms on "this is true because X says so", we think in terms of "this is true because I can prove it". Yes, we could provide references but they wouldn't add anything to the article from the point of view of a mathematician. (Proper maths papers do have references but only to give credit where it is due. They arguments given in this article are so trivial we don't need to give credit to anyone for them.) --Tango (talk) 02:20, 7 November 2009 (UTC)

Okay, so it is WP:FACTS then. Just wanted to make sure. Ten Pound Hammer, his otters and a clue-bat • 02:22, 7 November 2009 (UTC)

I wouldn't say it was FACTS. FACTS is about things that everyone knows are true, this is about things we can prove are true in the article. There is a difference. --Tango (talk) 02:28, 7 November 2009 (UTC)

Same general idea. It's just that rare breed of information that doesn't need a secondary source to back itself up. Ten Pound Hammer, his otters and a clue-bat • 02:29, 7 November 2009 (UTC)

Wouldn't it be original research then? :) --Sandman888 (talk) 00:37, 15 November 2009 (UTC)

Yes, but if that research is completely reproduced in the article and is an indisputable logical argument, then is there are harm in it? --Tango (talk) 00:42, 15 November 2009 (UTC)

I see a bunch of sources inside the "proofs" section. — Carl (CBM · talk) 03:21, 15 November 2009 (UTC)

Moreover, see the "Routine calculations" section of Misplaced Pages:No original research. Nyttend (talk) 17:28, 7 December 2009 (UTC)

Really, guys

I've learnt this in highschool. And I mean learnt by being proven logically to me, using Aristotelian logic, common sense and mathematic rigour, not taught by professoral authority. Maybe here in Europe we have a higher level of highschool teaching than in other places (no offence) but we're giving Misplaced Pages such a bad name by arguing about such a common knowledge and common sense thing.

I know that Wikipedians must be in consensus when something is "disputed" and that this usually sounds like "this is true because all the authorities say it's true" but this isn't so. Actually no one with an inch of reason and sense who has actually given some thought to this problem can say that 0.(9) does not equal 1. All those who have other (academic) preocuopation and first see this due to reason well explained by cognitive psychology feel that this isn't right. Some react emotionally and write down their opinions on Misplaced Pages.

Again, I know that we should be in consensus in case of "disputed" points of view, but really is this actually a "dispute" or a "point of view" or is it common sense and evident due to the flawless logical demonstration?

Would we for example take seriously anyone arguing on this pages that Washington DC is located in China or that London is the capital of Russia? Do we need to proove here that grass is a plant and not an animal? Do we need consensus that the city of Paris is not located on Mars but actually in France? We don't. If anyone would assert those kind of things on these talk pages they would be at best blatantly ignored. SO why not use the same policy here? —Preceding unsigned comment added by 82.77.239.121 (talk) 12:55, 30 December 2009 (UTC)

There is no dispute among mathematicians, European or otherwise, and I see no substantive contribution to this discussion in your comment above (notably, no reference to a reliable source). --macrakis (talk) 14:49, 30 December 2009 (UTC)

A bit of a dispute is evident in archive 14. Tkuvho (talk) 08:52, 1 January 2010 (UTC)

Well, indeed, but I don't think there is any dispute here - it's long been established that we can present this as factual in the article, and any dispute gets farmed off the talk page to Talk:0.999.../Arguments (I guess there's the debate about whether we should simply delete those comments instead, but either way, it doesn't clutter up this page). Regarding things that seem obviously true, such as Paris being in France, see Misplaced Pages:When_to_cite. Whilst common knowledge doesn't need a source (the article gives Paris as an example), it's not clear to me that this is common knowledge. But still, we have references anyway. Mdwh (talk) 14:59, 1 January 2010 (UTC)

I'm just asserting, Kyrie Makraki, that the fact that 0.(9)=1 is subject-specific common-knowledge per Misplaced Pages:When_to_cite. We appear laughable if we allow it to be disputed, even on archived talk pages. I never said it's disputed by mathematicians, I said that it's disputed by some on these talk pages and that diminishes Misplaced Pages's credibility. Personally I was abhorred that some good-faith knowledgeable mathematics contributors on these pages were taking the dispute seriously and were trying to prove the thing, not only as an exercise of mathematical rigour, but actually as support for a presumably unusual statement, in an almost journalistic way. just check out the talk pages if you don't believe me. —Preceding unsigned comment added by 82.77.239.121 (talk) 03:25, 8 January 2010 (UTC)

Are you referring to formulas of the sort ${\displaystyle .{\underset {H}{\underbrace {999\ldots } }}\;=1\;-\;{\frac {1}{10^{H}}}}$ with infinite hypernatural H? Tkuvho (talk) 09:08, 8 January 2010 (UTC)

To me it sounds as if 82.77.239.121 is advocating wholesale removal of any discussion about the merits of 0.999...=1 on the talk page. I disagree. The widespread confusion about the equation is is part of what makes 0.999... notable in the first place, and it's no surprise that some of those who doubt the equaity make their way to the talk page. Of course I prefer them going to the arguments page and we should move such discussions there, but more importantly I prefer them discussing the merits of the equality anywhere but in the article proper. If we tried to stifle the debate on talk pages, it would probably lead to increased good-faith wrong edits on the article. And who knows, maybe some of those who doubt the equality can even be convinced by those discussions (though others probably are here just to troll). Huon (talk) 19:47, 8 January 2010 (UTC)

Sure

You have to be sure though when claiming that it is one, to point out that it is really (1 - .0000...1)
While the difference between .999... and 1 is infinitely nothing, it cannot be dismissed because it is everywhere.
.000...1 is the first thing there is greater than zero, and it's between every change between every two numbers all the way up to one. It's everywhere, but it's nothing. I believe that it is the graviton number, but I cannot prove it. --Neptunerover (talk) 16:31, 21 January 2010 (UTC)

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