.999 = 1?

Factual Questions
41

While everything you say is true, I don’t think the OP is making that point. That is a very advanced point to make. If somebody said “well, it’s trivial to define a space where these numbers are distinct” most people who have ever done higher level math will probably nod and say “ayup.”

I don’t think it’s fair to disparage the OP, but we HAVE been using the aforementioned rigorously defined real numbers as the de facto standard for a while now. I mean, I guess it’s possible he’s making the very advanced mistake of misapplying late 18th century mathematics to modern number systems, but I think it’s more likely that he’s just misunderstanding the modern common real number system with no middleman involved.

42

Many thanks to Indistinguishable and Senegoid for well thought out responses.

Re: “Similarly, when a mathematician says “0.9999…”, what they mean, by that same definition, is “The number which is >= 0, and also >= 0.9, and also >= 0.99, and also >= 0.999, and so on, AND also <= 1, and also <= 1.0, and also <= 1.00, and also <= 1.000, and so on.” What number satisfies all these properties? 1 satisfies all these properties. Thus, when a mathematician says “0.9999…”, what they mean, by definition, is 1”

Pretty stupid notation, why not just say “1”. :wink: Seriously is the some Mathematical Authority which has proclaimed this definition? What if I just want to talk about the geometric series 9/(10^x ) for x = 1 to Infinity and I do not want any assumptions that say well intuitively if we get arbitrarily close to some number we’ll just call it that number. I’m not to say that this is not reasonable to make such assumptions, but they are assumptions not proofs.

What is the smallest number less than 1? If I say it is .999… as defined by the geometric series geometric series 9/(10^x ) for x = 1, what proof can be offered I am wrong with out the assumptions made by a limit process?

I don’t know why any one would find that mathematical trickery like saying 10 x .999… is 9.999… where one must add this mysterious 9/infinity to the end is comforting. I would argue it is the opposite, a trick is something to fool you into believing something other than the actual reality.

I will repeat once more that when you use the Limit process you are making an assumption that x is close enough to n, that we will just call it n. Perhaps it is n, but it is not a proof is it?

I don’t need a proof that 1 -1 = 0. I can accept that a given. But to me it seems many have taken the result of a limits as a mathematical practicality to the conclusion that they are in fact some undeniable truth.

Okay so you can say by the axioms of the real number system, .999… must be 1. Fine but the real number system is not perfect is it? Does it describe (-1)^(1/2)? Does it describe 1/infinity?

I can accept that by the rules proclaimed by the axiom of Real Numbers or whatever .999… must be 1. But can some one tell me what 1/infinity is or why it is not logical to assume 1 - .999… would be 1/infinity?

43

The real number system IS perfect. In fact, so is this number system I just made up, comprised of the set {Dog,Cat,Fred} where Dog+Cat=Fred and Fred-Cat=Fred are the only valid forms of manipulating the set.

We get results that work in the real world with the Real Number system. Trying to philosophically suss out why 1-.999… =/= 1/infinity or what the square root of -1 is in a system that doesn’t deal with it is madness.

There’s a rather recent famous proof that you can take a sphere and, doing a bunch of slices and rotations, clone it into five spheres, completely identical (even in size and volume!) to the first. It’s not useful, but it’s kind of cool. It’s the same with trying to do what you’re doing.

If you WANT to define a number system where 1-.999…=Fish, by all means. If you can do math that makes science or real world applications easier, then definitely go for it! The mistake you’re making is that you’re asking a question that basically has no meaning. It’s the math version of “Does a dog have a Buddha nature?” No, in the real numbers sqrt(-1) doesn’t mean anything, that doesn’t mean it’s imperfect, it just means that we don’t particularly care about the result. There’s no point trying to criticize it.

The point is, yes, mathematical systems are rather arbitrarily defined to have certain properties. Yes, some of them lead to counter-intuitive results like .999… = 1 or don’t cover all cases like deciding sqrt(-1) doesn’t make sense. However, despite their arbitrariness we get RESULTS with them. Like cell phones. And the internet. If you want to invent a system that deals with all the crap that doesn’t make intuitive sense, and can still help us make cell phones, great! If not, then you have to accept that given standard conventions, 1/infinity has no meaning and it’s wrong to say 1 - .999… = 1/infinity because the standards and conventions of the number system say so. And that same convention invented cell phones! And the internet! Even if you find it hard to swallow, you can’t really argue with what it produces.

44

Your proof of the convergence uses limits does it not? Does it not at its foundation say well “it gets as close as we want it to get and therefore we will assume by time we get to infinity that the difference is 1/infinity and let’s just call that difference zero?” Does it not do that? If except limits only as a practicality for say calculating the amount of fuel needed for a rocket is only going to be off by 1/infinity, good enough. But if your looking to prove the universe has no dirt under it’s fingernails… maybe you need to do better?

45

There is no numerical error because by definition there is no remaining error. See my above post. By definition in the real number system there is no such thing as 1/infinity, nor is there any “leftovers” of some tiny value that we’re “ignoring for convenience.” See my last post.

Again, if you want to define a number system that makes us cell phones that makes all the weird counter-intuitive stuff disappear, I’m sure we’ll all thank you. But pretending the real number system should be defined as something other than it is, or insisting it’s “hiding” properties it doesn’t have is a bit fruitless.

46

You could try to explain it. But your explanation would carry the flaw which is at the core of the discussion here. You need limits, or something equivalent, to adequately discuss how repeating decimals work.

Why say 4/6 when we can say 2/3? Numbers can have lots of different names.

Assuming you meant “the largest number less than 1”, the answer is that there isn’t one. That is a fundamental difference between the reals and the integers.

47

Earlier, I gave meaning to .9999… by defining it as a limit, and suggested that it’s best if you can prove your alleged limit – But I didn’t define what a limit is or how you compute the actual value of one, or prove anything about. I’m not sure if anyone else in this thread has, either.

If any of us actually felt it necessary to do that, we’d just by typing our Calculus I textbooks into this thread. Can we assume that all the parties to this discussion at least know those basics of Calculus?

I’m not getting the impression that erik150x does. And I’m not seeing that anyone else has tried much to explain that, as that is exactly what I would mean by typing in our Calculus I textbooks. And no, I’m not going to do that here. I’ll just point out the problem – more clearly, I hope, than has been done so far.

Forget all that horsemanure about 1/infinity and stuff like that. At the end of that path lies madness, not meaningful results. And skip all the nonsense about finding the [del]smallest[/del] largest number less than 1 that isn’t 1. There isn’t one.

From the days that Newton and Leibniz developed Calculus until sometime in the 19th century, nobody knew what limits were or how they worked, and mathematicians couldn’t do much formal proofs or logic dealing with them. They just had a bunch of formulas for derivatives and integrals that seemed to work, and they built up Calculus from there.

A formal development of Calculus was not possible until somebody (sorry, I’m not enough into math history to know the names) came up with the epsilon-delta definition of a limit, and the corresponding epsilon-delta method of proving theorems about them.

In order for this entire thread to make any sense, everybody needs to wrap their minds around the epsilon-delta definition of a limit. There’s no infinities involved. There’s no 1/infinities involved. There’s nothing about finding just the right number that’s infinitesimally close to another number. Before you can deal with any abstractions like those, you have to know how to work with limits first.

48

You can reprise another thread that did the issue to death An Infinite Question - Cecil's Columns/Staff Reports - Straight Dope Message Board

My posting was this: I’ll repeat (with a few edits to improve it) it since I think it is hard to dispute. :smiley:

Another way of looking at the issue, is to go back to the question: why do these infinite length representations exist anyway?

The answer lies in the unique prime factorisation theorem. Sometimes known as the fundamental theory of arithmetic. Simply put, for any natural number you can select, there is only one set of factors of that number that are all prime. So 20 = 225 42 = 337 and most importantly for this discussion, 10 = 2*5 and 3 = 3.

It is never possible to come up with an alternate set of prime factors for any natural number.

If you want to express a fraction (or rational number) in decimal form one of two cases are possible. Either the denominator will have prime factors which are only 2s and/or 5s, or there will be some other prime in the denominator’s factors. If the only prime factors are 2 and 5, you can represent the number as a terminating sequence in decimal notation. Any other number than 2 and 5 and the representation is not terminating. Any.

So, 1/4 = 1/(22) = 0.25 Note that this is exact.
6/32 = (33)/(22222) = 0.1875 and again, this is exact.

But, say, 4/7 = 2*2/7 = 0.571428571428571428571428571428…
and note the recurring sequence 571428. Another thread pointed out the notation 0.(571428) to denote this. (When I was at school we used a superscript dot.) The parentheses representation is much easier to read, so I’ll stick with that.

Now, note that 2 and 5 are important only because they are the factors of the base we are using for the representation (decimal). We could easily represent the number in a different base. In base 2, aka binary notation, we only have one prime factor - 2. The only fractions that can be represented in a terminating form are those where the denominator is a power of 2.

So, in base 2:
1/4[sub]10[/sub] = 1/(22)[sub]10[/sub] = 0.01[sub]2[/sub] Exactly.
1/8[sub]10[/sub] = 1/(222)[sub]10[/sub] = 0.001[sub]2[/sub], exactly and
3/8[sub]10[/sub] = 3/(22*2)[sub]10[/sub] = 0.011[sub]2[/sub]. also exactly.

But 1/5[sub]10[/sub] is 0.001100110011001100110011001100110011001100110011001101… or 0.(0011) in base 2. Which is only exact if you sum the infinite series: 1/8 + 1/16 + 1/128 + 1/256 + 1/2048 + 1/4096 …

Yet in base 10, 1/5 was terminating and thus exact without recourse to a series.

How about trying base 3 then?

1/3[sub]10[/sub] = 0.1[sub]3[/sub] Exactly.
2/3[sub]10[/sub] = 0.2[sub]3[/sub] Also exactly.
1/3[sub]10[/sub] + 2/3[sub]10[/sub] = 0.1[sub]3[/sub] + 0.2[sub]3[/sub] = 1 Exactly.

1/2[sub]10[/sub] in base 3 doesn’t terminate - since 3 has only one prime factor: 3.
So, 1/2[sub]10[/sub] in base 3 is 0.111111111111111111111111111111111… = 0.(1)

Now how about that!!! Guess what is going to happen?

0.111111111111111111111111111111111…[sub]3[/sub] + 0.111111111111111111111111111111111…[sub]3[/sub]
= 0.(1)[sub]3[/sub] + 0.(1)[sub]3[/sub] = 0.(2)[sub]3[/sub]
= 1/2 + 1/2 = 0.2222222222222222222222222…[sub]3[/sub] = 1
in base 3. Look familiar?

It is also worth noting that in base 2:
0.111111111… = 0.(1) = 1. Which is identical to the expression for the sum of the infinite series 1/2 + 1/4 + 1/8 …

In base 10
1/3 = 0.33333… = 0.(3)[sub]10[/sub]
2/3 = 0.66666… = 0.(6)[sub]10[/sub]
1/3 + 2/3 = 0.(3)[sub]10[/sub] + 0.(6)[sub]10[/sub] = 0.(9)[sub]10[/sub]

So 1/3 + 2/3 = What?

Which brings us full circle. 0.9999999… = 0.(9) is not a process, it does not exist in any definition that uses a finite number of terms. It is a notation for a sum of an infinite series. No different to any of the other infinite series above. If you mess about changing bases you can make any fraction drop in and out of having a terminating representation. You didn’t change a notation into a process or back again. (Not unless you define 1.(0) as a process too. Which means you define all numbers as an infinite process. Which we don’t. Or to put in the obverse: 0.999999… is just as much a process as 1.00000… - neither are.)

49

I really don’t get what you mean with this. The integers give us perfectly good results when we’re counting cocoa beans, the reals give us good results when we measure their lengths, the complex numbers give us good results when we spin cocoa beans around on graph paper or broadcast pictures of cocoa beans, and the quaternions give us good results when we spin cocoa beans around in three dimensions.

(Numbers don’t stop at quaternions, necessarily, but after that I find mathematicians tend to refer to them more as matrices or vectors than as ‘numbers’ as such.)

There are even numbers that represent concepts you can’t really apply to cocoa beans directly, and while most people think those ideas don’t amount to a hill of beans, they can still be quite interesting.

My point is, one, ignore the posts I made in this thread back in 2000, and, two, the concept of ‘number’ is complex enough to extend in many dimensions well beyond the reals you’re used to. (Also, the naming scheme we have is idiotic. The ‘reals’ are no more real than any rotation matrix, the ‘imaginary’ numbers no more imaginary than any integer, and the ‘complex’ numbers no more complicated than watching a compass spin around.)

50

I know that there are complexes, quaternions, etc etc. My point was that disparaging a number system that have provably given awesome technology to us because it does not account for your pet philosophical peeve (real or imagined) isn’t really a winning strategy. Yes, complex numbers are very useful, and with them we can do things that we couldn’t with reals (and so on and so on). That’s why I said if he can invent a system that does what it is he’s doing, and still do awesome things like make Cell Phones, go for it.

51

Risking repetition, I’d like to clarify my previous posts. Various approaches are proposed, perhaps none more or less “correct” than another. But some of these end up with “… by definition”; others end up making an explicit statement that dividing by infinity is illegal. Either can be unsettling for someone seeking a simple answer.

I still think best is to simply note that 0.999… = 1 follows directly from the Axiom of Archimedes.

That axiom
[ul][li] is very much common-sense;[/li][li] requires no mention of infinity or infinitesimals(*);[/li][li] is a prerequisite for some of the compass-and-straightedge constructions of ancient Greece![/li][/ul]

What’s not to like? :smiley:

(* - indeed the Axiom can be viewed most simply as outlawing infinities and infinitesimals.)

52

Ahh… well okay you’re saying that 1/infinity just doesn’t plain exist? Is that correct? Not just undefined.

For the record I never specified the “real number system” , I guess that maybe is an assumed aspect of the question. I am just talking about the intuitive notion of a number which is as closest as you can possibly get to 1 with out being 1?

But this leads into the paradoxical world of the smallest number possible or the largest number possible for that matter. How does one make that leap from just a very very big number to actual infinity? Or vice verse from a very very small number to an infinitesimal?

Real numbers don’t allow for infinitesimals, that’s the deal right? Why not?

Imaginary numbers don’t exist on the real number line and irrationals don’t exist on the rational number line, so what.

The set of all real numbers is uncountable, right? In a sense even bigger than the infinity of the countable rationals? Right, What does it mean to say one infinity is bigger than another, here we have these sets of numbers which don’t even have number to describe their quantity? Poor things. So when i talk about 1 not being equal to .999… maybe I am not thinking about just real numbers, and why should I? Imaginary numbers can be quite real. In fact I think there are many fields that couldn’t do with out them.

Let’s not confuse “real” numbers with reality. In reality I say 1 is not equal to .999… , but perhaps in reality .999… doesn’t even exist or having any meaning.

Just because 1/infinity doesn’t exist in the real number system doesn’t mean it doesn’t exist. Isn’t that why its always described as “undefined”?

Alas maybe it is a question of .999… = 1 or not requires further clarification and context. But just from an intuitive perspective, even if you want to think about the sum of an infinite geometric series, or the point at which Zeno’s arrow finally touches the target there is bit of mystery about that point just before we finished the geometric series or the point just before the arrow hits the target… there is this air of a quantum leap to get to the final point. I don’t know, but i like to think of that point as .999…

Thanks to all for your comments, I concede by the axioms of the “real” number system and the powers invested in The Limit process that .999… = 1.

I swear though that just after I saw the exit sign marked 9.999… to Albuquerque and just before I got to the exit marked 10 San Bernardino, there some other exit. Can’t remember for the life of me what it was. Well maybe it was just a dirt access road and not a real exit.

53

If you want to talk about different infinities, study Cantor, he did it all.

To reprise another point.

By definition 0.999999… is
9/10 + 9/100 + 9/1000 + 9/10000 …
= Sumn = 1, n = infinite

There is no scope for intuition in this - this is what the notation means. Note that we are not talking about real numbers either, this is a sum of rational numbers. There is no need to bring reals, or worse, algebraic numbers into the question. Thus no need to consider numbers bigger than Aleph Null.

This brings to mind the fallacy of Zeno’s pardox. It is rooted in the notion that there is somehow a different (and bigger) infinitude of geometrically decreasing points between the arrow and the target than the time steps needed to count those points. There isn’t. That is key. Indeed Cantor effectivlely showed this too. You can show that there are Aleph Null of both, and they can always be put in a one to one relationship. This means that the idea that there is a jump at the end that can’t be bridged in Aleph Null (usually called “infinite”) steps is false.

Cantor spend most of his life torturing himself in order to get these results. It isn’t as if they are easy to work out a-priori.

54

Well, what does x/y equal generally? We say that z =x/y when x = y times z. For example, 2=6/3 because 6 is 3 times 2.

So what would it mean for 0 to equal 1/infinity? It would mean that 1 = 0 times infinity. Does that make any kind of sense? Generally mathematicians think not, because if you assume that you get led into contradictions, e.g., 2 = (0+0) times infinity, so 2 = 1.

55

I don’t encounter this notation much. What I’m gleaning from Wolfram is that Aleph-null is what’s usually refered to as “countably infinite” (cardinality of integers), and the continuum is “uncountably infinite” (cardinality of the set of reals, equal to N to the aleph-null where N in an integer), or am I wrong on that?

And what’s this about there being an infinity bigger than continuum? I thought it stopped at uncountably infinite, but they list the charming property c[sup]c[/sup]=F. What set has that cardinality?

56

I have studied calculus, and more. The Limit process is an assertion and a very useful and valid assertion I think to accept. If your trying to calculate the instantaneous velocity of a something with changing acceleration, and endless other practical problems as well as theoretical problems. Newtons Laws of motion are also very useful and practical, yet in the end not exactly correct as we know from Einsteins Theories of Relativity.

You may say… “Forget all that horsemanure about 1/infinity and stuff like that. At the end of that path lies madness, not meaningful results.”

Well, sir you are entitled to your opinion, and you may even be right. But my assertion remains true that when you use the limit process you are making assertions which are not technically proven, however arbitrarily close they me be to being correct. They are not proofs. I still stand by the idea that using the limit process to prove 1 =.999… and not something arbitrarily close to 1 on the basis of a convergent geometric series which proves the existence of a limit… an answer which is as close to correct as we want it to be, but never quite gets you there. And when talking about the difference between .999… and 1 that distinction to me would be huge.

57

I like that in 2000, Derleth believed there were “obscenely old” threads on the SDMB. :smiley:

58

Every set S has a power set P(S) = {t | t ⊆ S} and it can be proven that |S| < |P(S)|
Therefore larger and larger infinities can be found:
aleph-null < |C| < |P(C)| < |P(P(C))| < |P(P(P(C)))| < …

P(C) has “natural meaning,” e.g. as the number of curves in a plane.

IIRC, Paul Cohen proved that the question of whether every possible infinite size occurs in the above list is undecidable.

59

Now that you say it it’s so simple. I didn’t think of using power sets (though if I did I admittedly probably wouldn’t have assumed that the |A| < |P(A)| rule necessarily worked on infinities).

60

To put this more succinctly…

Saying that the sum geometric series .999… converges to 1, is to say that there is a limit for that sum which is 1. Saying that there is a limit for that sum is saying that it can be proven that .999… is as close to 1 as I wish to prove. Yet you cannot prove it IS one. Right?

Well, heck, I could have told you .999… is a close to 1 as you can get without any calculus. Well that’s not counting the new number I came up with, described as:

(1 - .999…)/2 + 1 - .999… :wink:

which is halfway between .999… and 1.