*(note: this post has been closed to comments; comments about it on other pages will be deleted!)*

Okay, so I'm still on vacation, but it's clearly time for me to step back into this. Thanks to the people at scienceblogs, namely Goodmath/Badmath and the EvolutionBlog for taking up my cause. And thanks very much to the people in the comments of my other posts for doing their best to explain. Some of my explanations here use some of their ideas.

Let's do this by classifying the problems people have with my posts, from the most ridiculous to the least (roughly):

You must be a public school teacher, and I fear for your students. You don't know enough math to teach it. Stop filling their heads with nonsense.

Wow. I actually teach at a private school. Some of my former students have gone on to get perfect scores on SAT's, study advanced math at top-level universities, and place well at the national level in math competitions. I have B.A. in math from a top-20 school. All the information online (you can start here and here) agrees with me. In addition, while I know that there are public high school teachers who are not very good with math, all of the ones I know who teach at, say, the pre-calculus level or higher also agree with me. Please don't insult the entire profession just to discredit me. If you're that worried, go to graduate school and study math, and teach it yourself.

.999... is clearly less than 1, but mathematics isn't advanced enough to handle infinity, so you can't prove it. My intuition isn't flawed, math is.

This is another one that's so wrong that I barely know where to start. Historically speaking, this debate is quite old. In the 19th century, this apparent paradox (.999... is clearly less than 1 vs. .999... equals 1) was addressed by some great mathematicians (Cauchy and Dedekind, among others). In particular, they formalized the notion that the real numbers are infinitely finely divisible, and in their formulations, all arithmetic operations worked the way they seemed like they should. Using their formulations, all proofs result in .999...=1. Those formulations are discussed more at the wikipedia site linked above. Mathematics has been handling infinity well (using definitions that don't even require the use of the word 'infinity') for at least 100 years now. Until you've studied that work, you might want to be careful about saying that math can't handle infinity.

Variations on: 1 - .666... = .444..., but .999... - .666... = .333..., so 1 and .999... can't be equal.

You'd think that people trying to argue with mathematicians would at least check their work:

1 - .666... = .444...? If that's true then to check it, I should be able to add .666... and .444... and get 1. But .6 + .4 is already 1. So the sum of two larger numbers can't also equal 1. The check fails.

Variations on:

10x = 9.5

- x = .5

9x = 9So x = 1 and also x = .5. SEE!! I can use your stupid method to prove that 1 = .5!

If x is .5, then 10x is 5, not 9.5. Your equations are unrelated, so *of course* you can prove something false.

(A lot of birds with one stone):

10 * .999... isn't 9.999..., it's 9.999...0

1 - .999... is .000...1

2 * .999... is 1.999...8

This is a common mistake among my students. You're mistaking notation for mathematics. Notation is *not* mathematics. Mathematics is the study of ideas about patterns and numbers, and we have to invent notations to communicate those ideas. Just because you *can write* something that looks mathematical, that doesn't imply that what you wrote *has meaning*. The number written as .999... has meaning:

9*(1/10) + 9*(1/100) + 9*(1/1000) + ...

Every 9 in the list means 9*(1/*something*). But .000...1, for example, is an abuse of notation. It doesn't correspond with any meaning, so it doesn't communicate anything. If you think it means something (and putting 1 at the END of an ENDLESS list of zeros shouldn't mean anything), you're going to have to tell me what's in the denominator of the fraction represented by that decimal place. If you can't tell me that denominator, you're not using the notation right. If you tell me the denominator is 'infinity', then see the next entry.

1 - .999... is 1/infinity, which is a number bigger than 0.

Mathematicians don't really use the noun 'infinity' very much, and when they do, it's usually as a shorthand for an idea that is relatively easy to define without the concept of infinity. They do use the adjective 'infinite' to describe lists and sets, and the adverb 'infinitely' to describe how the real numbers are divisible. While some intuitive ideas can be captured by using the idea of 'infinity' as a kind of number, you have to be very careful with it, and standard arithmetic doesn't usually work. But as an intuitive idea, anything that might be written as 1/infinity never behaves differently from the number 0. I can't prove that, however, since 1/infinity doesn't really mean anything. Using infinity as a number creates fallacies that even the doubters of .999... = 1 would disagree with.

.999...

effectivelyequals 1, but it doesn'tactuallyequal 1. It rounds off to 1. You can never really .999... going on forever because you can't live long enough to write it.

These are all really arguments that claim that .999... isn't really a number, and that you therefore have to stop writing it or round it off at some point. Look, either you allow the possibility that there could be infinitely many (note the use as an adverb!) decimal places or you don't. If you don't allow it, you'll have a lot of trouble with proofs that pi or square-root-of-2 can't be written using a number that has finitely many decimal places. If you do allow it, you have to be prepared to discuss what happens if they all equal 9 at the same time, and you have to discuss it without rounding or talking about when they end.

.999... = 1 if you allow limits, but not if you're just talking about numbers. The limit of the series isn't the same as adding up the numbers.

The evolution blog linked at the beginning of this post has an excellent discussion about this. The upshot is that once you admit that there's an infinite geometric series here (which you have admitted as soon as you merely *write* .999...), there is no difference between the limit and what the thing actually equals. They have to be the same by any defintion that is internally consistent.

Your fraction argument only works if I admit that 1/3 equals .333.... I don't think it does, so I don't think your arguement works. 1/3 can't be precisely expressed with decimal numbers.

Well, at least the people who argue this are not abusing notation, and they're not attacking me personally, and they understand that the assumptions have to be correct in order for the argument to be correct. So I'm giving some credit to this one. But unfortunately, .333... really does equal 1/3. If you think 1/3 equals some other decimal, you're going to have to tell me what it is. If you think that you can't express it with decimals, then remember that the very word 'decimal' itself comes from our base 10 number system, and that's a biological coincidence due to our 10 fingers. In a different base, 1/3 might be no problem, but 1/2 might be. Any problem results from notation, not from the concept of 1/3. Remember, notation is not math, notation just communicates math.

Okay, I really have to go now...I'M ON VACATION, PEOPLE!!!

"You even had to tell us that we had to forget EVERYTHING that we know about math (including simple long division, multiplication, addition, and even fractions!). Are you seriously saying that this is LOGICAL? We can't even start from basic facts and work up to a proof of some sort?"

Ok, I'll admit that was a little out there. I meant to say "forget everything you know of advanced mathematics". Essentially, take a look at this as though you're in junior high school, or fresh out of high school.

"Your way just goes, "0.99|9 does not look the same to me as 1, therefore 0.99|9 != 1." This is what is called "circular logic." I don't think that's the kind of logic you want to be toting around."

Ok, let me address this. When you first heard about this whole thing, whenever that was in your life, I am willing to bet you looked at the two numbers proposed as being equal and thought "What? Those aren't equal..."

Now, why is that two things that are "equal" would need so many things to prove that they are "equal"? Shouldn't they be incredibly easily and beyond a doubt (i.e. we shouldn't even have to be here) proven equal?

Next, tell me how this "circular logic" is wrong. What I see here is two different numbers. To use an analogy, I see an apple and an orange. To me, you are claiming that the apple is an orange. And you happen to have many people believing it.

Again, I ask you, why is it so hard to grasp that when two numbers look different, it's quite possible that they ARE different. If you're saying that type of logic is wrong, then have fun going around thinking that all oranges are apples, and that apples don't exist. Because that's exactly what you're doing here.

"You seem to agree that the process is endless, yet you still insist on having this "end" (the 7) show up. I wasn't "trying" to take the 7 out. It just isn't ever displayed in the answer (the 2.99|9 part) because it never becomes part of it, as you certainly must understand, since you stated so.

Just like all the other 7s in the 10th, 100th, 1000th, etc. decimal places that have been turned into 9s, the 7 is not noted in the answer.

To state my point more clearly, for the equation:

0.9999 x 3 = 2.9997

the 7 at the 10th decimal place is being ignored, as well as the ones in the 100th and 1000th places. According to you, we should make sure that these 7s be noted within the finite answer of 2.9997."

Ok, let me once again rephrase this: The 7 in the answer does exist. However, when you look at the "end" (which we both agree doesn't exist) we see a 9. However, by doing the math in a finite setting, we end up with a 7 at the "end". So, if we take the 7 from the infinite finite multiplications to get the answer (not sure how to phrase this part... but I mean every time we increase this:

0.99 x 3 = 2.97

0.999 x 3 = 2.997

0.9999 x 3 = 2.9997), and take the 9 that is seen when we look at it infinitely, then both must exist simultaneously.

I don't mean to propose any specific "notation" of how this should be written, but I merely wish to show how I arrive at these numbers. However, if you were to write it, I think I'd have it as 2.99|9...7 or possibly 2.99|9|7.

"What is really lazy is to just end the 0.3333... with a 1/3 tacked at the end (actually, it isn't really "0.33333333333...(finite number of 3s) + 1/3" but "0.3333333333...(finite number of 3s) + 1/300000000...(however many zeros is needed so the sum is 1/3).) If that's your answer, then why bother at all with dividing 1 by 3? Leave it at 1/3 in the first place!"

True, mine is also lazy. However, it takes in a number more representative of the answer, and when you use it, you get the answers that you should get.

However, I'd agree that we should just leave it as 1/3. It's simpler, anyone can understand it, and we don't end up with things like this.

"1/2 = 0.5 is not to be trusted. Sure, it seems so simple and obvious, but the process used to get that answer is the same one that failed for something as mundane as 1/3, so LOGICALLY we can't be sure that it's correct, can we? I'd like to see you try to logically state that 1/2 = 0.5 while at the same time insisting that 1/3 != 0.33|3. Or do you actually agree that 1/2 may not equal 0.5?"

Actually, no. 1/2 doesn't do quite the same thing as 1/3. 2 is an even number, while 3 is odd. All of the fractions that is 2 to any power (1/2, 1/4, 1/8, 1/16, etc) give you nice, easy decimal values. It's because it's cut in half each time. It operates on a fairly different system than 1/3.

It's easy to explain, really. when I cut something in half, I get half of it. Makes sense, doesn't it? But a third is a third, and it doesn't work quite so well, for some reason.

“WOW...it just gets better. If your special "numbers" don't exist then my point is proven exactly! And by the way, those numbers do not exist.

ok...so I showed that those number are indeed possible? how exactly? Because you just can't be wrong. Ok, think about it...you're trying to prove to me that .999... != 1. So to prove that you insist that there are numbers between .9|9 and 1. And those numbers exist why? because .9|9 != 1?? you're coming to a conclusion assuming the point your trying to prove to be true. It makes no sense. You can't say that A is true because B implies A, therefore A implies B to prove that B is true. So please give me some sort of proof that the number .9|91 crap exists.”

You showed that the numbers are possible assuming we write them in the form I’ve shown. To be fair, let’s look at it this way and assume that .99|9 is NOT the same as 1 (I know, it’s hard to grasp). I’m merely using this because in my mind it’s the next number down the number line, in a sense. If you want to show me another number down the number line to use, which is definitely not equal to 1 but it is fairly close to it, fine.

Assuming that they are not equal, then there would have to be infinite numbers between them, correct? At least, that’s what you’ve told me. So, I assume that it would be easiest to write them as .99|9…1. This 1 would represent the next chain of infinite numbers. So then you could have things like .99|9…2, or .99|9…3. The 1, 2, or 3 is separate from the recurring decimals, as it’s part of that “in-between” area.

This can go on infinitely. Please, if you want to propose that .99|9 is equal to 1 in order to prove me wrong in these regards, then show me a different number to use. After all, with so many numbers, there must be one that is only slightly farther down the line.

Actually, andy hit it on the nose. I think that about solves part of the problem, here.

“And honestly, I'm not sure why I'm arguing with you unless you have a masters or PHD in math. I have a B.S. in math so I've been well educated in the field. I don't go on "gut feeling" anymore when it comes to mathematics.”

Ah, pulling the “I have a degree so I’m smarter than you” card? Well, in that case, you’re not allowed to use a computer until you get a degree in computer science. You’re not educated enough, and go on too many “gut instincts” with them. You’re a danger to computer society.

Please, buddy, don’t act like a degree makes you “god of math knowledge”. I am just as entitled to my right to have my opinion and share it as you are. Sorry if having a lesser person debate with you is too much, since obviously degrees mean everything. That’s why Bill Gates is the richest man on earth. Oh wait, that’s right, he dropped out. Since he didn’t get his degree, he obviously wasn’t smart enough to use computers, so he shouldn’t have been able to make Microsoft, and earn nearly 1 million dollars per year.

If you’re going to use that on me, then I suggest you shut up and go away, because I have better things to deal with than arrogant math freaks who think that just because they’ve gone through college they’re better than me.

“this problem basically boils down to what's the next number after 0? or what's the next number after any number. most would say 0.000....1, whatever that means, but if you say 0.0000...1 = 0; then it makes perfect sense for 0.99999.. = 1. They both exploit the same problem, what's the next number after any number? is there a branch of mathetmatics that studies this?”

I think he’s hit this on the nose better than I have. You have to admit that there is some number that comes after zero, if not perfectly after it. And, going on down, we eventually need to get other numbers. If you deny that, then we have no numbers.

Posted by: Catalyst | June 28, 2006 at 09:07 AM

@ Lzygenius,

ummmm...no I meant 10^-(infinity+1). I was trying to show a number between 10^-x and 0. So what's a number less than 10^-x? 10^-(x+1). For example, 10^-2 < 10^-1. So no, I didn't mean 10^-(infinity-1). Let's work on our basic algebra skills here before devulging into our beliefs on how the mathematics at infinity works.

Anyway, I feel that I'm beating a dead horse here. For those that don't believe it still, then just look at the multitude of websites available in this subject offered by trustworthy mathematicians. No, it's not a conspiracy and no it's not a flaw in mathematics. If you don't like repeating decimals then blame our base 10 number system, not math.

Posted by: Chip | June 28, 2006 at 09:07 AM

@ Catalyst,

haha, I actually do have a B.S. in computer engineering too. But this is getting very retarded. I'm not trying to start name calling stuff nor do I think I'm "holier than thou" because I have a math degree. I'm simply saying that I've studied mathematics to a great degree. If you study a bit more then you too would start to get the point. So I apologize if I insulted you. Any discussion that leads to name calling is pretty pointless. So let me get back to the point...

There is no "next number" on the real number line. The real numbers are dense. Suppose we assert that a is "right next" to b. That is, you can't "fit" any real number between a and b. However, if we take the average of a and b (a+b)/2 then we should get a real number. As I have said, since the real numbers are a field then valid binary operations on 2 different real numbers will result in another real number. So we have now just found another real number between a and b, which contradicts our statement that there can't be any number between a and b. So can you see how it is flawed to believe that there is a real number "right next" to every real number? There are many areas of mathematics that deal with this concept. Number theory and topology are just a couple of fields that deal with this.

Again, I apologize for seeming to make you feel "inferior" to me.

Posted by: Chip | June 28, 2006 at 09:35 AM

The reason why .999...1 cannot be a real number is this: Every decimal place represents a term in a series, whether it be finite or infinite series. For example, .225 = 2/10 + 2/100 + 5/1000. .999... = 9/10 + 9/100 + 9/1000 + 9/10000 + ... So the .x notation just represents a base 10 digit divided by some power of 10. so the "number" .999...1 = 9/10 + 9/100 + 9/1000 + ... + 1/10^??? We know that 1/10^(-infinity) is not a real number (if you want a proof then I can give one to you). So what exactly is that real number which divides 1 after that never-ending string of 9's? Think about it...infinity is not a real number so you can't divide a number by infinity. This is why a 1 does not exist after a never-ending string of 9's.

Posted by: Chip | June 28, 2006 at 10:14 AM

Allow me to summarize:

Premise: Real numbers are dense. This means that between any two real numbers, there are an infinite number of real numbers between them.

Premise: There are no numbers between .999... and 1

Possible conclusions:

1) .999... = 1

2) .999... < 1, but it is not a real number.

3) .999... < 1, and it is a real number, and so are an infinite number of numbers such as .000...1, .000...2 and so on.

So, if you want to argue that .999... < 1, you better start arguing that it isn't a real number.

Posted by: mark | June 28, 2006 at 10:27 AM

[Ok, let me address this. When you first heard about this whole thing, whenever that was in your life, I am willing to bet you looked at the two numbers proposed as being equal and thought "What? Those aren't equal..."]

I will say that, yes, I did initially balk at the idea that 0.99|9 = 1. But then I looked at the proofs showing that this was indeed true and was convinced because I could not find any contradiction in the steps taken (and believe me, I tried). It especially was convincing since the steps were based on easy-to-understand (for me at least) highschool math. It now especially makes sense to me once I uncoiled my thinking from assuming that 0.99|9 has an "end" somewhere in there, thus making it less than 1. This insight is hard to explain in words, but the fact that I can't plot 0.99|9 anywhere (except on top of 1) on a number line without resorting to assuming an "end" to the line of 9s now convinces me on even an intuitive level.

You say that something like this should be apparent to anyone without showing mathematical proofs. Well, it sure is "apparent" to me that the Earth is flat (shouldn't I be able to notice the curvature?), the Sun revolves around the Earth (if Earth is the one moving, why don't I feel it?), and of the three lines shown in this "optical illusion":

http://en.wikipedia.org/wiki/Muller-Lyer_distortion_illusion

the middle one is obviously longer than the other two.

See how this form of "logic" doesn't really work? It may seem obvious at first that 0.99|9 != 1, but when you actually go and check it out (using basic equations), you come back with the fact that this is not so.

[Ok, let me once again rephrase this: The 7 in the answer does exist. However, when you look at the "end" (which we both agree doesn't exist) we see a 9. However, by doing the math in a finite setting, we end up with a 7 at the "end".]

I'm sorry, but you're still assuming that there is an "end" onto which the 7 can be placed. You keep saying that you agree with me that there is no end, but then say,

[I don't mean to propose any specific "notation" of how this should be written, but I merely wish to show how I arrive at these numbers. However, if you were to write it, I think I'd have it as 2.99|9...7 or possibly 2.99|9|7.]

Where are you putting that 7? At the end of the 9s? Actually, the better question is, why should the 7 even be noted at all? As I said before, we don't take note of all those other 7s that came before-hand. Stop saying that you agree that there is no end and then follow up with statements that assume that there is an end (whether you're imagining a moving end or a dangling end is unclear).

[It's easy to explain, really. when I cut something in half, I get half of it. Makes sense, doesn't it? But a third is a third, and it doesn't work quite so well, for some reason.]

..."for some reason"? This argument from ignorance is the best you can come up with? I highly recommend that you go and look up what bases are in mathematics. The reason 1/2 yields neatly to 0.5 while 1/3 yields a repeating decimal 0.33|3 is simply because we are doing the math in base-10. Such numbers as 2, 4, 5, 8, etc. divide neatly into non-repeating decimals solely because their prime factors consist ONLY of the prime numbers 1, 2 and/or 5 (which, surprise!, are the factors for 10). Other numbers that include prime factors other than 1, 2 or 5 in them end up with repeating decimals when they are used to divide 1 in base-10.

And are you seriously suggesting that you can divide an object (say, pizza) in halves, but can't physically divide it into thirds? Seven equal pieces? 13 equal pieces? Hmm? They physically do exist in reality, it's just the way we arbritrarily group things in tens that makes things difficult.

Oh, and here's a little something extra to mull over:

Since I stated that 3 x 0.99|9 = 2.99|9, then it must be true that 2.99|9 divided by 3 is 0.99|9. Try to do it in long division (imagine that the dashes and slash are connnected, and add in the lines that are missing above the 29s):

0.999999...

------------

3/ 2.999999...

-27

29

-27

29

-27

2(and so on)

Posted by: Monimonika | June 28, 2006 at 01:30 PM

Argh!! Ignore the stars (for space-taking purposes) in the final equation:

***0.999999...

**------------

3/ 2.999999...

***-27

*****29

****-27

******29

*****-27

*******2(and so on)

Posted by: Monimonika | June 28, 2006 at 01:37 PM

0.9999999999999... = 1

If you disagree, you're a fucking retard.

Posted by: pEnG | June 28, 2006 at 01:46 PM

http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

If you don't believe it yet, then I suggest you write to wikipedia and tell them that they are publishing fraudulent material. I'm sure they'll listen to you.

Posted by: Chip | June 28, 2006 at 03:00 PM

Yes Catalyst, I admit I did succumb to the temptation to flame... please forgive me a harmless little jest. I hope you can see through your understandable (but hopefully not deep) anger towards me and consider a few more ideas. And I will admit you are correct; I indeed know very little about math. But I do know certain proofs when I see 'em.

1. Many posters have provided numerous excellent (and correct) proofs that 0.999... = 1. You have not offered a rigorous counterexample to any. Since you say you have a math degree, I am sure that you took a semester in proofs, or number theory, or formal logic, or some other class with a similar name and purpose. There you learned to write formal proofs or present counterexamples, using strict terminology, did you not? So far your arguments rely on your intuition and rhetoric, not rigorous formality, and rigorous formality is of course the foundation of mathematics.

2. In math, there are numerous examples of things that are equal, but do not appear so at first. In fact, you might even say that most of meaningful mathematics deals with things that are equal but do not appear to be so....:

Quote: "Now, why is that two things that are "equal" would need so many things to prove that they are "equal"? Shouldn't they be incredibly easily and beyond a doubt (i.e. we shouldn't even have to be here) proven equal?"

Take pythagorean theorem for example. (I'll assume you're familiar with it, having a math degree and all). It's indisputably true, isn't it? But there are numerous proofs to prove it is equal. And as for myself, not having a math degree (unless you count a minor), those proofs are neither easy nor intuitive to me! But upon inspection I am compelled to admit their truth. Perhaps I've taken your quote a bit out of context, and if so I apologize and admit laziness, but the quote reflects a common lay-attitude towards many of the more technical aspects of math, which I though I'd point out. To digress a bit... some mathematicians have become very famous for proving things that defy all common sense.

3. This whole silly 0.999... = 1 debate is not even REMOTELY the most mind-bending or counterintuitive truth of formal math. If you disagree with this equality... what do you have to say about the cardinality of infinite sets? Are there more real numbers than integers? They're both infinite, right? Or are they? Is it possible for one kind of infinity to be bigger than another? Hmm. What about a set of all sets? Can you define one for me?

There are other confusing topics out there too. I wish you an opportunity to enjoy exploring them.

If I've flamed you, or pissed you off, I only hope it serves to encourage you to pursue your interest in math further and deeper. As your posting passion demonstrates, you obviously are interested in math topics, and that alone is highly commendable in my book. I hope you don't stop pursuing this particular topic until the proverbial "Aha!" catches you!

Posted by: Lurker | June 28, 2006 at 09:24 PM

When Catalyst says this:

'Ok, I'll admit that was a little out there. I meant to say "forget everything you know of advanced mathematics". Essentially, take a look at this as though you're in junior high school, or fresh out of high school.'

I think he makes it clear he's yanking your chains.

Posted by: adamsj | June 28, 2006 at 09:57 PM

If the whole argument is that it doesn't make sense on a high school level...that isn't a disproof of this result. :)

Since not everyone is on the same playing field let's keep it real simple;

Find a number between 0.99|9 and 1.00|0 and we'll all accept that these two numbers are different.

Posted by: Ranbir | June 29, 2006 at 09:35 AM

pi is exactly 3!!!

Posted by: Chip | June 29, 2006 at 03:10 PM

Chip - I can define a number system where pi does equal three... but it makes counting very, very difficult!

jbs

Posted by: John B. Sandlin | June 29, 2006 at 10:25 PM

The real hang up I see in most of the posts revolves around understanding a never ending series. If I define a number, 0.2|2 for example, and at some point - time, the universe, and everything ends - my 2's still continue. Hmm. I should have made that 0.42|42.

HA! I've found the formula for immortality!

jbs

Posted by: John B. Sandlin | June 29, 2006 at 10:29 PM

Good call on this being your last post on this tired topic. Let's have some *actually challening* maths!

Posted by: Paul "the tree" Carpenter | June 30, 2006 at 03:56 AM

Your methods are all correct!! I stand by your argument, you are right!!

Posted by: SunXia | July 02, 2006 at 08:41 PM

I'm not a mathematician, so I would not dare to enter the actual debate. I did want to say, though I greatly admire you for standing up for math teachers who teach the "higher" levels of math. I agree with you that no one should be discrediting the whole profession just because they disagree with a particular post.

Posted by: thebizofknowledge | September 25, 2006 at 09:57 AM

Enjoy and have fun with your vacation....Keep up the good work..

Posted by: Juno888 | May 15, 2007 at 03:44 AM

This debate is a wonderful demonstration of the difference between truth and truthiness, or intuitive acceptability. It is a truth that 0.999... is equal to 1, but it just isn't truthy enough for most people to really believe, even if they can regurgitate it on command.

I posted (but did not write) an anonymous fable at http://quomodocumque.wordpress.com/2007/09/30/they-had-not-the-habit-of-definition/#comment-686

that bears on this question.

Posted by: John Cowan | October 01, 2007 at 08:48 AM

I wanted to expand on what Monimonika said above.

Some people have trouble understanding what an infinite number of digits after the decima means. It means that if x = 0.999... (an infinite number of 9s after the decimal), then 10x = 9.999... (still the same number of 9s after the decimal). That's why 10x - x = 9. The infinite number of 9s after the decimal cancel out when subtracting.

Posted by: Dave | December 01, 2007 at 10:00 PM

When you are talking about how .6 + .4 = 1

But .666~ + .444~ =/= 1

Because you said that .666~ is a bigger number. However, .66<.6

Isn't the farther you go down in decimal places, the smaller the number gets?

Posted by: Tony | February 03, 2008 at 11:25 AM

apparently infinity is every single number to ever exist, so when ever your writing a answer to your maths problem, or typing it, your writing/typing infinity, so infinity has already been proved millions of years before now, because when math was first invented they used numbers.

Posted by: some_genius | August 31, 2008 at 05:13 AM

Catalyst said that 1/3 != 0.333... because they don't look the same on paper. Others don't believe that numerals of different bases still represent the same number. These people need to understand that this is just the NOTATION. That is, a VISUAL REPRESENTATION of the number. There are many visual representations of the same number. The same way as there are many ways I can say a word: by using DIFFERENT LANGUAGES.

If I want to represent the number NINE, I can do so in many ways:

I can draw nine dots, for example, or I could use one of these:

nine

9 (decimal)

9/9

27/3

1001 (binary)

10 (octal)

102 (trinary)

and many many more. They are different representations of the concept we know as "the number nine", but they are still the same number!

So, if we want to represent what we call "the number one", we can do that in many ways too:

. (One single dot)

one

1

1/1

2/2

0.999...

0.999|9

and many more.

They are just representations. We (humans) defined what these representations mean so that we can communicate concepts to each other in an unambiguous way. We defined that 0.999... = 1 just like we defined 0.333... = 1/3

Why did we do this? Because decimals aren't capable of representing certain numbers. This doesn't mean that number doesn't exist, just that the DECIMAL REPRESENTATION or LANGUAGE doesn't have a word for it. So we created one.

Another example: the number PI is an infinitely long sequence of non-repeating decimal digits. Because this is hard to write, we often use the PI symbol to represent it. The symbol is a representation of PI equal to its numeric value. It doesn't look the same on paper, but we defined it to mean the same thing. Its all NOTATION.

Posted by: dan | December 22, 2008 at 03:09 PM

The only hard thing to explain, at least in my mind is this part of the proof:

.999~ = 1

The simple equality relates a whole number, aka an integer, with a never-ending decimal, which has been taught to me, to be a non-integer. The gut feeling, and what number theory seems to suggest is that an integer should not be equal to this never-ending decimal, even if the proof is correct.

Posted by: Joker | January 26, 2009 at 06:25 PM