 Robert makes a good point that deserves attention.

But really isn't this just the fault of our number line? If you make the argument that you can't find a number between .999... and 1, well your number line isn't good enough. If a number line is only integers, there is nothing between 1 and 2. Doesn't mean they're equal. But that's how the number line is, and there's nothing in between them. A "number line" is just an abstract concept we as humans have made up to organize quantities onto something more tangeable. But in this argument it serves no purpose.

The reason why you can't find the number between 1 and 0.999-> is because there isn't one, no matter what "line" you do or don't use. There isn't such a number because 1 - 0.999-> = 0.

But let's say for the sake of argument that 1 - 0.999-> is not 0. What would it be? Well if it's not 0 then it's not an infinite series of 0s after the decimal. So there'd have to be a 1 in there somewhere. An infinite series of 0s ending in a 1? But if it "ends" in a 1, then it's not infinite is it? And subtracting that finite value from 1 would produce a finite series of 9s, rather than an infinite series of 9s. 1 - .999... = 1/oo

.999... + 1/oo = 1

No? I am surprised by all the people who are still trusting their intuition. One of the first things I learned as a math student is that intuition will often betray you.

For those who have not looked yet, see my proof at http://adrake.blogdns.com or if you wish you can download the pdf directly from the following link

If you have any questions, comments, refutations, etc. please voice them through whatever medium you prefer.

Best Regards, I somewhat see this, but what is
.999...
X .999...
-----------
?

and is .999... / 1 = .999... or 1? Sorry bunk, 1 - .6|6 != .4|4

Try it on a calculator...

1 - .6 = .4
1 - .66 = .34
1 - .6666 = .3334
1 - .6666666666 = .3333333334

If you drag it out you get infinite sixes, infinite threes. How can you have a four _after_ an infinite number of threes? By definition, _infinite_ never ends. You can't put something after the end of something that never ends.

The problem is that the decimal representation system is not equipped to represent things like this. Whenever the words "infinite", "infinitesimal", or "irrational" come into play, the decimal system is inherently inadequate.

This is not a limitation of math, or numbers, or anything of the sort. It's a limitation solely of the decimal system we use to represent these quantities visually. Representation is separate from concept. Accept, and understand, this and you free yourself from a great many false limitations, not just in math. This has been the stumbling block of several commenters both on this post and its predecessor.

There are a great many directions this discussion can take. But at the heart of it is the fact that mathematics is built not on fundamental truths, but on postulates and working assumptions. Such things are necessary in order to have progress. Changing the postulates and assumptions yields work that is just as valid and valuable. But one must recognize that the foundations are different, and so results can't be transferred directly.

We can certainly argue over whether a given set of postulates matches physical reality (i.e., can you really subdivide a unit of space infinitely? Max Planck would say no, that infinitesimals are physically unreal). But this does not change the fact that mathematically and conceptually, these things can still be useful. Or at the very least, both interesting and correct within their own context. Hi,
I would like to thank all convinced "0.999.. != 1" people for a great amusement they gave me this evening (while reading their comments). Internet fatally lacks more of this kind of fun :)

As a university student of mathematics, I of course know that 0.999.. is equal to 1, which can be proven in million different ways, but non-mathematical people won't get any of these. Sorry, better education needed here.

Blog author: thans for a good topic & I wish many successes when dealing with non-math-positive people! Ok, not an uber-math guru here, I'm thinking from more of a logic perspective, but I think Robert's right. Here goes:

I think the problem here is that we are accepting approximations as exact numbers. The problem with thinking about infinity, is people still like to think it ends somewhere. How many times have you heard infinity plus one? But if it never ends, how are you ever going to add to it? The same goes for an infinite series. .999 repeating is a decimal point followed by an infinite string of 9's. Which makes it an ever-closer approximation of 1, but never 1 exactly.

People are taking it for a given that the decimal representation of 1/3 is .333 repeating. While that is the closest approximation we can make algebraicly(did i spell that right?), it is only an approximation, it is not 1/3. The same goes for all fractions whose decimal representation involves an infinitely repeating series of digits. All the proofs I've seen thus far that .999 repeating equals 1 seem to accept that these approximations are exact.

If you consider .999 repeating as a function, you would accept the limit as being 1. But any limit defined by x to infinity, will never be an exact result of the function. It would just get closer.

I don't know how cogent that all was, but it made sense to me :) Robert:

Alright, let's turn this into algebra. So you've got 99/100, 999/1000, 9999/10000...can you think of an equation to formalize that? It could be:

(x - 1) / x

So, if we're moving into infinity then x approaches infinity. Let's set x equal to "infinity" (which I'll represent as oo):

(oo - 1) / oo

By your own definitions this is equal to .999..., correct? Correct.

So, what is infinity minus 1? Still infinity, by definition (Note: infinity is not just a "really big number", it goes on FOREVER - take one star out of *infinite* stars, and how many stars are left? Still an infinite number). So we have:

.999... = (oo - 1) / oo = oo / oo = 1

Therefore,

.999... = 1

QED.

This is not the best proof in the world, but it's easily derivable from Robert's starting point. Because he couldn't formalize it, he didn't understand that it led right to the solution...if he had followed all the steps.

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

A big thanks to polymath for fighting the good fight against the ignorant hordes, and thanks also to polymath's trolls for all the entertainment they have provided! Congratulations, you folks fail at understanding formal mathematical abstractions, but you'll be DAMNED if you'll let actual mathematicians teach you otherwise! Reminds me of the Flat Earth Society. "A "number line" is just an abstract concept we as humans have made up to organize quantities onto something more tangeable. But in this argument it serves no purpose.

The reason why you can't find the number between 1 and 0.999-> is because there isn't one, no matter what "line" you do or don't use. There isn't such a number because 1 - 0.999-> = 0."

Then any argument to prove that .999...=1 shouldnt use the concept of a number line. Sums and lists are number lines with arbitrary components. And yeah its abstract, but so is an infinitely repeating series. There is a number between .9999... and 1. It just can't be represented with that particular number line. Sorta like the shortest distance between 2 points is a straight line - except when its not. This is why I never want a trial by jury.

I especially like the people who say that a repeating decimal is just an approximation. Remember long division? It's that thing you should have learned in 3rd grade. Eric, this is the problem - 1/3 = .333... by definition. It's not an approximation, it's a NOTATIONAL representation of the exact number you get when dividing one by three. Does this number actually exist, in reality? Could you divide a pear into three perfectly equal parts? No...because pears are made up of non-continuous components (subatomic particles) which can not be infinitely divided (to our knowledge). But that presents no problem for formal mathematics, because formal mathematics deals entirely with Platonic concepts - the world of the perfect. You can conceptualize of a purple flying goat; we could even draw one, have a biologist and a physicist work out the dynamics of its physiology, how it flies, etc. Does that mean it exists? No - it just means we can describe it. Repeating decimal representations are notation for a mathematical concept, not an approximation of some real-life quantity. It is exact, by definition.

Also by definition, if there is no number between two real numbers, then the two numbers are identical. There is no number between .999... and 1, therefore they are equal BY DEFINITION. Arguing otherwise requires a complete misunderstanding of mathematical notation. One might as well argue that false can be true, or that if a = b and b = c then it's possible that a != c. They're logical impossibilities. The only people who would make those arguments would be people who do not actually understand the notation involved. Math grad student and avowed math geek here, and I think a lot of us are pulling out tufts of hair trying to present this in easy terms.

You really need a course in advanced calculus / basic analysis before having the tools to handle this rigorously.

Based on Peano axioms, you can construct the positive integers, then build up from there to the rationals. Typical homework assignments at this stage might be something like:

for a,b,c rationals, prove (a+b)+c = a+(b+c)

Then using Cauchy sequences (or Dedekind cuts if you prefer) you define real numbers.

At that point, you still haven't even considered decimal expansions. Every real number is simply defined as a Cauchy sequence. Addition, multiplication, etc, are all operations on Cauchy sequences, not on decimal expansions. You have to prove such mundane things as a*b=b*a

At that point all the heavy lifting is done. You can show the correspondence between decimal representations and the Cauchy sequences you are used to working with, and things like 0.999... = 1 start making perfect sense.

Being fairly sloppy here, but you get the idea. Basic analysis was a long time ago for me.. Garthnak has the concepts right, though a "real" mathematician would likely cringe at seeing ".999... = (oo - 1) / oo = oo / oo = 1"

The quantity "infinity", represented here as "oo", is not compatible with arithmetic operators, unless we apply limits, which by definition changes the rules a little.

But even so, oo / oo is "undefined". In calculus terms, it "diverges", or is "unbounded". It has no real number solution. Even under a limit.

So a "real" mathematician would be more likely to formulate this in terms of inifnitesimals, as opposed to ratios of differences. Because at least when you divide by inifinity, you can take a limit and get a hard number: zero. "Mathematics" is just an abstract concept that we humans have invented as well, but one that starts from axioms and proves everything else starting from those axioms. You can argue that it's all bullshit, but then you can't say 1+1=2 and you're in the realm of grade-school phenomenology. There are lots of fine mathematical philosophers out there, and you're welcome to go get a PhD with any of them, but if you come in arguing that 1+1 is not 2 because of your great intuition, you'd better be prepared to feel really, really stupid. WaterBreath, I kind of knew that and it's why I said it's not the best proof in the world :-P I'm not a formal mathematician. However, I think I at least related the right concepts involved.

I also realized that a better representation for his series would be (10^x - 1) / (10^x). Not that it really matters that much, but I should point it out. I agree with ondra. You cannot make a base 10 representation of 1/3. In base 3 it is .1 and 2/3 is .2 . So .1+.2=1 in base 3. .99999.... is a concept that is not in the bounds of reality. It is equal to .22222.... in base 3 and is not a number, but a representation of the closest number to 1 without being one. However, because of the limitations of base 10 in numbers divisible by 3, .99999... does equal 1 in base 10. both sides of the argument are flawed in that each is attempting to describe continuous concepts in discrete terms.

...but for the sake of experimentation:
1 != 0.999|9
1 > 0.999|9

(the > in this example is infinitely small) scott, if 1 > 0.999..., then show me the number that you would add to 0.999... to make it equal to 1. Here's someone that explains my thoughts earlier a lot better. i was just trying to tie the idea of infinite nines into the [i]comparison[/i] of the two representations. i as i said before, i do not believe it would be prudent to try and describe a discrete value where the concept in question is continuous. For those insisting that the algebraic argument is flawed because there is one less nine to the right of the decimal place in 10x = 9.9999... as in x = .9999..., consider:

How many nines are there after the decimal point in x = .9999...? Obviously, since it's the definition of what we're talking about, there's a countably infinite number of nines there. Aleph-nought, to those in the know, simply "infinity" to the uninitiated. So to advance the "one less" argument, how many nines are there after the decimal point in 10x = 9.9999...? Presumably, it's one less than countably infinite. The only way to differentiate, then, would be to say that this number, one less than countably infinite, is itself finite. Let's say that it is. Let's call it y.

Add that 1 back to y. y+1 is still finite, no?

Do y'all see? There's no other way to interpret infinity, even at its lowest cardinality, than that subtracting any finite number from it, no matter how big, just yields the same infinity as you started with. The arguement it seems is does infinity exist. Is matter infinitly small? Does the universe expand into infinity? Can you really get infinitly close to something without touching it? You really can't do math to a number that keeps moving! "The arguement it seems is does infinity exist."

I disagree, Adolfo. Whether something "exists" has little bearing on what is possible in math. In math _we_ make the rules. If it can be imagined by our mind, we can also imagine the rules that govern its manipulation.

Whether a particular concept "really" exists, as a physical law or phenomenon, is an entirely different question. And the only significance of its answer is to determine whether math using infinities or infinitesimals may be used to accurately model the physical universe. As I said in my first post, this has nothing to do with the validity of the math itself. Only it's application in the given domain.

Math is useful in modelling physical reality. But it can and does exist separate from that. 1^oo = 1
(0.999...)^oo = 0

(where oo symbolizes infinity)

therefore, 0.999... cannot equal 1

As a non-terminating sequence, 0.999..., despite its infinite number of significant figures, will always be less than 1. Any number less than 1 when raised to the power of infinity will approach zero.

The only pathway in which this will fail, is if you come into the proof assuming that 0.999... is the same as 1.

I will use the same 'disprove-me' logic that Polymathematics uses to declare his truth:

Show me a zero, a decimal point, and any finite series of 9's that add up to a number that is not less than one.

0.9 < 1
0.99 < 1
0.999 < 1
0.9999 < 1
0.99999 < 1

As the number of 9's approach infinity, can we not assume that each number will continue to be less than 1? And thus 1^oo can never be the same as (0.999...)^oo ? And thus 1 can never be the same as 0.999...

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