0.999... = 1, proofs etc

[adam_grif]

Alright, I'm sure most of you are aware that 0.999... is an integer exactly equal to 1. But of course, when you tell people they generally refuse to believe it, insisting there's something wrong with your proof. What do you people do to convince them otherwise?

The standard proof I start with is:

1/3 = 0.333...
2/3 = 0.666...

.'. 1/3 + 2/3 = 0.999... = 3/3 = 1

Then if they insist on there being something wrong, usually relating to xeno's paradox and how it "never really becomes 1", I'll explain to them about how it only has infinite digits because you're expressing it in Base 10. If we convert into base 12, it becomes:

1/3 = 0.4
2/3 = 0.8

0.4 + 0.8 = 1

Is this a solid proof? I saw the first one online, but not the second, although it seems intuitive and I'm sure it's been raised before.

[Hawkwings]

Here's a lengthy post about it:

http://polymathematics.typepad.com/poly ... nally.html

[Surlethe]

Well, 1 - 0.99999... < e for every small number e. Since this is true for every small number e, 0.999... = 1.

[mr friendly guy]

Ok, I get the argument, ie you can't really find a number between 0.999... and 1 and how it only looks different from 1 because you are expressing it as decimals (using base 10).

However my question is, besides being useful as an abstract intellectual exercise, does this have any practical applications? Or perhaps put in the way a layman might ask, why should we care?

[adam_grif]

Ok, I get the argument, ie you can't really find a number between 0.999... and 1 and how it only looks different from 1 because you are expressing it as decimals (using base 10).

However my question is, besides being useful as an abstract intellectual exercise, does this have any practical applications? Or perhaps put in the way a layman might ask, why should we care?

None I can see for the lay person.

http://en.wikipedia.org/wiki/0.999#Applications

All applications seem confusing and/or barely relevant to little old me.

[Khaat]

Or perhaps put in the way a layman might ask, why should we care?

Merchandising is the only application I can think of where the preference of ignorance of a layperson would have any value, and only to the seller.

"Only $1999.99!"
"That's two thousand dollars."
"No, it's under two thousand dollars!"

[Terralthra]

Or perhaps put in the way a layman might ask, why should we care?

Merchandising is the only application I can think of where the preference of ignorance of a layperson would have any value, and only to the seller.

"Only $1999.99!"
"That's two thousand dollars."
"No, it's under two thousand dollars!"

Since it's not $1999.99..., it is under two thousand dollars. Your understanding seems a bit off.

[Wyrm]

When it comes down to it, the infinite repeating decimal 0.999... invokes many mathematical concepts that the layman has difficulty grasping, including (but not limited to) "infinite", "repeating" and "decimal". To help someone accept that 0.999... = 1, you have to find out which of these concepts they have difficulty with.

Let's start with the most innocuous-seeming concept of them all, "decimal". People think they understand decimals, but in general they don't. They merely know how to use them, and for the purposes they use them for, that is all that is needed. However, they don't actually understand what a decimal is, because they think that a decimal is a number. Decimals are not numbers, they are a way to construct numbers. Decimal notation allows you to sift through all the anonymous numbers on the number line, pick out the one number you want, name it, and allow you to talk about it. Arithmetic is a way to manipulate these constructions such that they mimic the operations on numbers (addition, subtraction, multiplication and division).

An important point to remember about a construction schema: unless the schema is very carefully arranged, the schema is not necessarily injective — distinct representations do not necessarily refer to distinct numbers. You can underline this point by pointing out that nobody has a problem with sin(π/2) = 1, cos(0) = 1, or 7⁰ = 1. These are perfectly legitimate ways to construct the value 1. The claim 0.999... = 1 is the same; "0.999..." constructs a number that happens to equal the number constructed by "1".

On to the concept of "repeating" when joined to "decimal". People who claim that ".9 repeating doesn't equal 1, it gets closer and closer to 1," are misunderstanding what the "repeating" refers to. They think that the 0.999... is spitting out a bunch of numbers that get closer to 1. Of course, it isn't. What they refer to bears far more resemblance to a sequence than a number. A sequence is a different beast altogether from a number, and is not equal to any number. While people are hard-pressed to say what 0.999..., they do claim it is a number.

Remember that decimal notation is a way of constructing a number — one number. Repeating decimals are an extension of that concept to enrich the numbers availible to the notation, but the purpose is the same: to construct a single number. Only a single value is produced, no matter how much the decimal is repeated. The object produced is not a sequence but a single number that has a definite, fixed value. However, a sequence is implicit in the definition — that's what's repeating in repeated decimals, and helps us to find the number that we are referring to in the construction. In particular, "0.999..." implicitly invokes the sequence (0.9, 0.99, 0.999, ...), and the number that "0.999..." is the single, unique number that the sequence (0.9, 0.99, 0.999, ...) 'gets close to' as we run through the sequence. If you have someone who claims that "0.999... only gets closer and closer to 1," you can rephrase that claim the following way:

"So you admit that the terms in the sequence (0.9, 0.99, 0.999, ...) get closer and closer to 1, and not any other number? Well, the value of 0.999..., according to the rules of repeating decimal notation, is precisely that number that the terms in the sequence you just named gets closer and closer to, which is 1."

But how do we know that a single number is named by this sequence, rather than two or even a group of numbers? That brings us to the last and most mind-bending concept we invoke here: infinity.

Perhaps people look at the two and think, since 0.999... is infinite and 1 isn't, the two cannot possibly the same number. How can something be infinite and not infinite at the same time? Well, if your victim opponent has been paying attention, he would realize that the only infinity here is in the construction, not in the number itself. Of course, he might insist that, because 0.999... requires an infinite sequence of calculations, you cannot find its value exactly (in contrast to 1 which yields is value quite readily).

Well, yeah, if you do it the stupid way.

Of course we cannot carry out an infinite number of calculations in a finite time. However, this does not mean we cannot say some things about the infinite sequence we have specified above. We note, for instance, that each term in the sequence is a finite decimal, which has a finite construction, no matter how many 9's there are. Furthermore, it is quite easy to demonstrate that each term in the sequence is 1-10^{-n}, where n is the place of the term in the sequence. With this handy expression for each term, we can do an end-run around the infinite number of calculations by invoking the limit of the sequence. We can find limits of sequences without going through an infinite number of steps, all that is required is that we can deduce general properties of the sequence.

Limits of sequences, if they exist, only refer to unique numbers, because it can be shown that a sequence that does not converge to a unique limit breaks the definition of a limit.

Once your target has wrapped their head around the complete mathematical foundation of 0.999..., getting him to admit the equality is duck soup.

Convincing a 0.999... = 1 skeptic requires you to guide them through the problems in his understanding, so there's no one-size-fits-all proof that will convince everyone; if everyone intimately understood all the concepts involved in the proofs of 0.999... = 1, they'd already be convinced of its truths.

If what you have is a 0.999... = 1 denier, then no amount of handholding will convince them.

[Surlethe]

A sequence is a different beast altogether from a number, and is not equal to any number.

Technical quibble: if you construct the reals with Cauchy sequences, you consider all Cauchy sequences of fractions, mod out by the obvious equivalence relation, then call the resulting set of classes "\mathbb{R}". So sequences can be 'equal' to numbers if you consider sequences as representative of the number equivalence classes. (Of course, if our victim skeptic is sophisticated enough to employ this nitpick, he's not a victim skeptic in the first place.)

[Ryan Thunder]

(0.999... + 0.999... = 1.999...8 =/= 2 and 1 + 1 = 2) => 0.999... =/= 1

I win?

(I'm kidding)

[Hawkwings]

the general response to that is that 1.999...8 is not an acceptable way to denote a number.

[Surlethe]

No, it is an acceptable way to denote a number. However, 1.999...8 is a terminating decimal, whereas 0.999... is not. The problem here is that 0.999... + 0.999... is not a terminating decimal -- there's no "end" where the carryover can demand its pound of flesh and turn a 9 into an 8. It's a fallacy of the sort similar to dividing by zero in those faux proofs where 1 = 2.

Slightly more formally, note that 0.999 ... is the limit of the sequence {0.9, 0.99, 0.999, ...}; it's a fact that if you termwise add two such sequences, the limit of the sum is the sum of the limits, so 0.999... + 0.999... is the limit of {0.9+0.9, 0.99+0.99, ...} -- i.e., it's the limit of {1.8, 1.98, 1.998, ...}. Is this different from 1.999...? Well, 1.999... is the limit of {1.9, 1.99, 1.999, ...}; employing the same logic, let's look at the sequence of differences: {0.1, 0.01, 0.001, 0.0001, ...}. This obviously goes to zero, so 0.999... + 0.999... = 1.999... (= 2). Informally, the 8 at the end just gets pushed off to infinity; in the limit, it doesn't exist.

[Ryan Thunder]

Informally, the 8 at the end just gets pushed off to infinity; in the limit, it doesn't exist.

Just for the record, I suspected as much, but I wasn't sure.

[Zadius]

How about this:

(1) x = .999...
(2) 10x = 9.999...
(3) 10x-9 = .999...

Now, since the right-hand sides of both (1) and (3) are equal, then the left-hand sides of (1) and (3) must also be equal:

(4) 10x-9 = x
(5) 9x = 9
(6) x = 1

[Axiomatic]

Okay, stardestroyer.net, I am amazed. You have done something I have never before seen achieved on the net.

Someone brought out the old 1 = 0.999... and it DIDN'T lead to an incredibly long and tiresome discussion where some people could not be convinced that this was the case.

You have actually created a place on the internet where everyone knows that 1= 0.999...

[Marcus Aurelius]

Okay, stardestroyer.net, I am amazed. You have done something I have never before seen achieved on the net.

Someone brought out the old 1 = 0.999... and it DIDN'T lead to an incredibly long and tiresome discussion where some people could not be convinced that this was the case.

You have actually created a place on the internet where everyone knows that 1= 0.999...

That's because in SDN and SLAM in particular your argument actually needs to have some merit. You can't argue just based on vague feelings or intuition. Mathematics is an area where the merits of an argument or lack thereof are usually easier to see than say politics or art1, so slam dunk cases like this usually do not generate a very lengthy discussion. Most other sites follow the Miss Manners principle, which means that you can argue almost anything as long as you don't resort to ad hominem attacks or foul language. The latter of course is not even forbidden here...

1I realize that this does not apply to very advanced math problems, but I was referring to cases like the one in question.

[Surlethe]

Actually, advanced mathematics is written precisely so that the merits of an argument are easy to see - as a professor of mine put it, "Proofs are not written to be intuitive, they are written so they are easily checked for logical correctness." It's just that advanced math is so difficult to intuit and requires so much background in jargon, it may as well be gibberish to a layperson.

[Hawkwings]

Have you seen the advanced math discussions here? The reason nobody comments on them is because only Surlethe and Kuroneko understand even the notation itself, let alone what the actual argument is.

[Darth Wong]

Have you seen the advanced math discussions here? The reason nobody comments on them is because only Surlethe and Kuroneko understand even the notation itself, let alone what the actual argument is.

Kuroneko is quite professorial, and Surlethe is young. To be honest, most working engineers don't use 90% of the math they learned in school. Very few problems presented to a typical working engineer (before he enters management, which apparently happens to something like 3/4 of engineers) are best solved through manual use of complex math; they are usually solved through numerical iterative methods in a computer.

Mind you, that doesn't mean they should stop teaching engineers complex math; it just means that most engineers tend to allow their higher math skills to atrophy somewhat over time, although they can still easily do what they view as simple math, which is still far beyond what the average person considers to be simple math.

[Darth Wong]

Ok, I get the argument, ie you can't really find a number between 0.999... and 1 and how it only looks different from 1 because you are expressing it as decimals (using base 10).

However my question is, besides being useful as an abstract intellectual exercise, does this have any practical applications? Or perhaps put in the way a layman might ask, why should we care?

Well, I suppose one might argue that you should care because the concept of infinity is important to this discussion, and the concept of infinity is important to the concept of limits, which is in turn important to the concept of analytical calculus, which is in turn necessary for Newtonian physics, which is in turn the foundation of almost all modern mechanical and civil engineering, without which you would be living in a straw hut and raising chickens in your backyard.

Admittedly, that doesn't mean the layperson really needs to know it himself. It just means that it's important for someone to know these things. I suppose you could say the same thing about all science. There are people in this world who quite seriously denigrate the entire concept of math education for the masses based on the same "I don't see why the average person needs it" reasoning. Rosie O'Donnell did this quite prominently.

[Surlethe]

Have you seen the advanced math discussions here? The reason nobody comments on them is because only Surlethe and Kuroneko understand even the notation itself, let alone what the actual argument is.

Kuroneko is quite professorial, and Surlethe is young.

It's interesting that Kuroneko and I are cited as being the only ones to engage in complex math discussions, since off the top of my head, Wyrm, Steel, Zadius, and Grog all participate as well. I'm sure there are others, too, whom I've forgotten.

To be honest, most working engineers don't use 90% of the math they learned in school. Very few problems presented to a typical working engineer (before he enters management, which apparently happens to something like 3/4 of engineers) are best solved through manual use of complex math; they are usually solved through numerical iterative methods in a computer.

Mind you, that doesn't mean they should stop teaching engineers complex math; it just means that most engineers tend to allow their higher math skills to atrophy somewhat over time, although they can still easily do what they view as simple math, which is still far beyond what the average person considers to be simple math.

Old fart offering up excuses, eh? Seriously, it makes perfect sense for engineers to permit higher math skills to atrophy: unless you're in the business of writing computer models, you don't need them. Even then, the higher math needs of an engineer who does need them are going to be much different from the higher math needs of, say, a mathematical physicist; one will need to be able to model immediately relevant real-world situations, while the other has to describe universal patterns with general mathematical language - the difference between needing an accurate way of numerically describing a stress tensor field in a material and needing a logically rigorous way of handling a particular smooth class of liftings from a manifold into a tensor bundle.

You can say the same for physicists, and there draw a distinction between theoretical and experimental fields. One prof I've studied under does experimental particle physics and he has only a vague ideas of what theories in particle physics describe. Even mathematicians won't be well-versed in areas beyond their fields of specialty - I know a Lie theorist who hasn't ever taken anything beyond an introductory course in PDEs. Hell, if in ten years you ask me about number theory, I probably won't be able to give more than a vague, handwaving description.

Ultimately, you really only have enough time to do your job, and if the job doesn't involve higher math, you're going to let your ability to do it wear away.

[Xess]

I didn't get this till I took Calculus in university. Then I was introduced to the concept of limits and it made perfect sense. The whole limit of of 0.999... = 1 made the whole thing make sense. Even if it isn't a proper mathematical proof.

[Steel]

My first proof was the one x=0.9... So 10x-x= 9x =9 thus x=1, however I now find the one where the limit of the sequence 0.9, 0.99, 0.999, ... Tends to 1 as 10^-n more compelling.

An interesing fact with limits in general is that until you first prove the limit exists (and is finite) you cannot manipulate the object you are finding the limit of, otherwise you can get any answer you want, any answer at all.

To illustrate, lets try and work out what 0/0 is (yep, thats zero divided by zero)

Well first we could set it up as

lim x->0 (the limit as x goes to zero)of x/x, which we can 'obviously' see is lim x->0 of 1, which is one. So zero over zero IS one?

Well we know that lim x->0 2x = 0 as well, so what happens if we approach 0/0 with (2x)/x, that becomes 0/0 as well, but we can see that zero over zero is apparently 2... 2 doesn't equal 1, so something went wrong, and that is the fact that "0/0" does not exist.

More values:
x^2 also tends to zero as x->0, so whats 0/0 if we go via (x^2)/x, well then its zero! How about x/(x^2)? Then its infinity.

There are much more subtle ways things can go wrong, but this is one of the key places you need analysis, otherwise you can run in to all sorts of problems if you start to manipulate things that don't exist.

A (slightly) less artificial example is what is the integral from -1 to 1 of 1/x?

You can split the up into two integrals, the first from -1 to a (a<0) and the second from b(>0) to 1, then take limit as a and b go to 0 and it matches the original... Hopefully.

Turns out if you have a=-b, you'll get the answer is 0, but you could choose a=-t and b=t^2 and have t->0 and get that its -infinity and vice versa for +infinity or anything you like in between.

An even more odd scenario that can crop up is where you have a limit that in general doesn't exist, but you want a specific case of it where it does, and then you're even more constrained in what you can do to it.

Knowledge atrophy isnt a good thing ever, but in the case of engineers, having at some point known how to do something makes a difference i think. Even if you dont directly use it later i think those kinds of skills help form a basis for intuitions and also the action of learning the principles and applying them is going to improve your abilities in other fields.

Also, I'm classed as part of the boards mathematical elite eh? Guess that masters was worth it then Back to phd work...