Is There a Rigorous Proof Of 1 = 0.999…?
Yes.
First, we have not addressed what 0.999… actually means. So it’s best first to describe what on earth the notation [tex]b_0.b_1b_2b_3…[/tex] means. The way mathematicians define this thing is
[tex]b_0.b_1b_2b_3…=\sum_{n=0}^{+\infty}{\frac{b_n}{10^n}}[/tex]
So, in particular, we have that
[tex]0.999…=\sum_{n=1}^{+\infty}{\frac{9}{10^n}}[/tex]
But all of this doesn’t really make any sense until we define what the right-hand side means. There is an infinite sum there, but what does that mean? Well, we put
[tex]S_k=\sum_{n=1}^{k}{\frac{9}{10^n}} \ ,[/tex]
then we have a finite sum. So, for example
[tex]S_1=0.9, \ ~S_2=0.99, \ ~S_3=0.999, \ etc.[/tex]
So, in some way, we want to take the limit of this sequence.
Let’s consider a particularly simple sequence to illustrate the idea behind the definition of a limit of a sequence: 1/2, 1/3, 1/4,… The terms in this sequence get smaller and smaller. You might think that it’s obvious that it goes to 0, or that it’s obvious that a smart mathematician can prove that it goes to 0, but it’s not. It’s impossible to even attempt a proof until we have defined what it means for something to go to 0. So we have to define what the statement “1/2, 1/3, 1/4,… goes to 0” means before we can attempt to prove that it’s true.
This is the standard definition: “1/n goes to 0” means that “for every positive real number [itex]\epsilon[/itex], there’s a positive integer N, such that for all integers n such that [itex]n\geq N[/itex], we have [itex]|1/n| < \epsilon[/itex]”. With this definition in place, it’s quite easy to prove that “1/n goes to 0” is a true statement. What I want you to see here, is that we chose this definition to make sure that this statement would be true. The first mathematicians who thought about how to define the limit of a sequence might have briefly considered definitions that make the statement “1/n goes to 0” false, but they would have dismissed those definitions as irrelevant because they fail to capture the idea of a limit that they already understood on an intuitive level.
So the real reason why 1/n goes to 0 is that we wanted it to! Similar comments hold for the sequence of partial sums that define 0.999… It goes to 1, because we have defined the concepts “0.999….”, “sum of infinitely many terms”, and “limit of a sequence” in ways that make 0.999…=1. Can we define number systems such that 1=0.999… does not hold? Of course! But these number systems are not as useful, because they don’t conform to our intuition about limits and numbers.
Now that we know what a limit and an infinite sum is, let me give a fully rigorous proof to the equality 1=0.999… This proof is due to Euler and it appears in the 1770’s edition of “Elements of algebra”.
We know that
[tex]0.999…=\sum_{n=1}^{+\infty}{ \frac{9}{10^n} } = \frac{9}{10} + \frac{9}{10^2} + \frac{9}{10^3} +…[/tex]
This sum is a special kind of sum, namely, it’s a geometric sum. For (infinite) geometric sums, we can find its limit easily:
Let
[tex]x=\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+…[/tex]
Then
[tex]9x=0.999…[/tex]
But, we also have [itex]10x=1+\frac{1}{10}+\frac{1}{10^2}+…[/itex], so [itex]10x-x=1[/itex].
This implies that [itex]x=\frac{1}{9}[/itex].
Hence,
[tex]0.999…=9x=1[/tex]
Does this proof look familiar? It should! It is essentially the same as Proof #2 in the previous post. The only difference is that every step is now justified by operations with limits.
See this supportive article: https://www.physicsforums.com/insights/why-do-people-say-that-1-and-999-are-equal/
The following forum members have contributed to this FAQ:
AlephZero
Fredrik
micromass
tiny-tim
vela
Advanced education and experience with mathematics
[QUOTE=”JonnyG, post: 5216818, member: 547631″]Once I saw the thread title and saw it went to 5 pages, I knew there would be at least one crack pot…*sigh*[/QUOTE]
[MEDIA=youtube]kIbEj1CIpuU[/MEDIA]
Once I saw the thread title and saw it went to 5 pages, I knew there would be at least one crack pot…*sigh*
It’s not ridiculous since saying “there is a contradiction in math” implies DIRECTLY that “1+1=3 can be proven”. It is called the rule of explosion or “ex falso sequitur quodlibet” [URL]https://en.wikipedia.org/wiki/Principle_of_explosion[/URL]
Maybe to clarify some expressions:
1=0.99999…. where the expression on the right is seen as an infinite sum. This actually, technically , means that the sum on the right _converges_ to 1. The sum on the right is a (monotone) Cauchy sequence (of partial sums) bounded above so that, within the theory of Real numbers, it has an upper bound. This upper bound is 1. This is how we define equality in this respect. In this case of standard Reals, the sum is over a countable index (there are issues of convergence otherwise). There are other “models” of the Reals, meaning copies of the Reals where only first-order properties (though not second-order properties) are satisfied, where one may define uncountable sums.
I can also say ##1+1=0## :D in ##mathbb{Z}_2## that is true, but people conviniently leave it unmentioned when they make this claim, for all of us are used to operating in ##mathbb{R}##
[QUOTE=”DMartin, post: 5216732, member: 569662″]I prefer to keep it in degrees throughout, and although you can subtract the 89, I won’t.
To explain it, it’s another series of numbers that suggest something, but somewhat different from the other. It suggests that for any number like 89.99999, there’s another number between it and 90, arrived at via x/(sine x). You can then add more 9s, and it still applies. However many 9s you add, this rule applies, so it seems very reasonable to say that it applies to 89.99999…….. as well. That means there’s at least one number between 89.9999….. and 90.
[/QUOTE]
OK, which number?
[QUOTE]
I think there can be contradictions in mathematics,[/QUOTE]
So you think ##1+1=3## is valid and can be proven? Because that is equivalent to what you just said.
[QUOTE=”William White, post: 5215774, member: 562357″]they are mutually contradictory statments.
You have a number.
Its exactly 1.
If the line is continuous there is a number greater than 1 that is smaller than every other number greater than 1.
If this is not the case, then all adjacent numbers are equal, so the line is not continuous.[/QUOTE].[/QUOTE]
[QUOTE=”DMartin, post: 5215860, member: 569662″]Well, it’s a matter of taste to some extent. You say you can prove that within a particular artificial system, a number 0.99999999…. exists, but 1 minus that number, or 0.00000….0001 doesn’t exist.
1-minus that number has an infinite number of zeros. Again, equality here means convergence in terms of a sum, and sum is not done on infinitely-many terms, it is done as a limit.
And a challenge to you: if you believe Mathematic sis a “particular artificial system” , provide a consistent one that generates more answers and is somehow more useful.
[QUOTE=”DMartin, post: 5216547, member: 569662″]Incidentally, and I’m not going to get into discussion about this, if one should show that there’s always a number between 0.9999…. and 1, that might suggest that they’re different. Well I can show that there’s always a number in between 89.9999…… and 90. It’s the expression x/(sine x) again, it always gives a number nearer 90, however many 9s there are.[/QUOTE]
Since you claim that you can show a number between 89.999… and 90, [B]what is that number[/B]? Just as 1 and .999… are the same number (proved in the first post in this thread), so too are 2 and 1.999…, 2.5 and 2.4999…, .1 and .0999…, 90 and 89.999… and many, many more. Every one of these pairs of numbers represents exactly the same number. If you subtract one number in a given pair from the other, you get 0.000… — IOW, zero. There is no difference (hence no distance) between the numbers in any given pair, other than how they appear.
As for ##frac x {sin(x)}##, if x is the [B]real number[/B] 90 (i.e., in radians), ##frac x {sin(x)} approx 99.5568##. If x = 90° (in degrees), ##frac x {sin(x)}## is [U]exactly[/U] [U]equal[/U] to 90. If x is merely close to 90°, then ##frac x {sin(x)}## will be close to 90, but so what?
As I said before, mathematicians almost never use degrees in trig calculations.
[QUOTE=”DMartin, post: 5216638, member: 569662″]I do believe in mathematics, but I just take a different view of it than you do. Gödel’s work meant that many others saw the edges of it, from the 1930s on, so my view isn’t a controversial one.[/QUOTE]
I don’t see why you’re going on about Godel. Godel never showed mathematics to be inconsistent. It’s a theorem about provability in first-order logic, so I don’t see how this is relevant to ##1=0.999…##.
Your view definitely is a controversial one.
[QUOTE=”DMartin, post: 5216556, member: 569662″]I never said it wasn’t useful! It’s very useful. And it doesn’t matter that it’s a bit rough round the edges either.[/QUOTE]
It’s not rough round the edges. I have no idea where you get that from. Just because you don’t believe in mathematics, doesn’t mean that it’s rough around the edges.
[QUOTE=”DMartin, post: 5216547, member: 569662″] I think mathematics is not a consistent system[/QUOTE]
OK then. I take it that you don’t have a proof of this at all.
If you think mathematics is inconsistent, then perhaps you shouldn’t fly an airplane either since that uses mathematics waaaaaaay more advanced than ##1=0.999…##.
[QUOTE=”DMartin, post: 5216540, member: 569662″]You seem to think we should define things first, and then do our exploring. But sometimes the exploring helps to inform the definitions, and this is how mathematics has actually developed. This means one can point out a mathematical structure without necessarily having a rigorous definition. But a definition I’ve given for the relevant number, whether a loose one, and whether or not it fits with your definitions, is an angle x between 0 and 90 degrees such that x/(sin x) = exactly 180/π.[/QUOTE]
Things like ##0.1## are just a notation. They mean something. And we define what they mean. What they mean might have an independent meaning, but the notation itself is just a notation. Same thing when I write ##sin(pi) =0##, then I must first define what I mean with the notation ##pi##. Sure, I can have different notations, but you need to be clear first. So when you’re going to talk about ##0.000…001##, then that is fine, but you are using a notation and I would like to know what that notation means.
[QUOTE]
and this is how mathematics has actually developed.
[/QUOTE]
Actually it isn’t. You should read up on the history of mathematics. Mathematicians have always used clear definitions from the very beginning (Euclid).
[QUOTE=”DMartin, post: 5216520, member: 569662″]By the way, I did say, but probably should have emphasised more, that what applies to one pair of numbers doesn’t necessarily apply to the other. There are things that happen at or near 0 that don’t happen elsewhere. But this is surely relevant, nonetheless! Thanks for the discussion.[/QUOTE]
What are your definitions of ##0.9999…## and ##0.00…001##?
[QUOTE=”nuuskur, post: 5216492, member: 519618″]All countible sets are of equal cardinality. In other words, the infinite sequence we are talking about, its elements can be written in a sequence one after another. For the same reason there are exactly as many natural numbers as there are whole numbers. The different infinities you are talking about are countible and continuum. We have written the sequence of 9s out, this means they are countible. A set of continuum cardinality cannot be expressed as such: e.g you cannot write out every single real number between 0 and 1 in a sequence. Therefore there does not exist this 0.99….9, the sequence of 9s proceeds to countible infinity.[/QUOTE]
This is where I disagree because firstly, order is important: You have the fourth digit and the sixth digit and you know which precedes the other. So Cantor’s theory of infinity does not exactly apply since it doesn’t take order into account. What does apply is Cantors theory of ordinal numbers. And if you take that theory, then there are different types of countability. So it is not impossible to have sequences which are indexed over “more” than just the natural numbers, you can index them over any set of ordinals. This also implies you can index your sequences over a set of ordinal with continuum cardinality. Of course, this is not what we actually do in the real numbers, there the order type of the decimal representations is ##omega##, and nothing more.
But interestingly enough, there are other possible definitions of decimal representations. A very neat one is in the hyperreal number system. There you can define the following two decimal representations:
[tex]b_0.b_1b_2b_3b_4… = sum_{nin mathbb{N}} frac{b_n}{10^n}~text{and}~b_0,b_1b_2b_3b_4… = sum_{nin {}^*mathbb{N}} frac{b_n}{10^n}[/tex]
The first one is the classical one, but with summation interpreted over the hyperreal numbers. The second one sums over all the hypernatural numbers (which is a summation of continuum cardinality!). Curiously, it turns out that here ##1neq 0.99999…##, but ##1=0,999999…##.
But again, this is in the hyperreal number system. The usual way of intepreting ##0.9999…## is in the real number system, in which case ##1=0.9999…##. This is why I ask for definitions, since if you disagree that ##1=0.9999…##, you must be using different definitions than what is usual.
[QUOTE=”DMartin, post: 5216470, member: 569662″]
Incidentally, there have been distinctions drawn between different infinities, and it has turned out that they can be compared, and one infinity can turn out to be ‘larger’ than another. This might intuitively seem impossible, but ways to compare them were found. There is some loose similarity between that and what I’ve done.[/QUOTE]
All countible sets are of equal cardinality. In other words, the infinite sequence we are talking about, its elements can be written in a sequence one after another. For the same reason there are exactly as many natural numbers as there are whole numbers. The different infinities you are talking about are countible and continuum. We have written the sequence of 9s out, this means they are countible. A set of continuum cardinality cannot be expressed as such: e.g you cannot write out every single real number between 0 and 1 in a sequence. Therefore there does not exist this 0.99….9, the sequence of 9s proceeds to countible infinity.
We have already defined what ##0.99…## is. Micromass has proved the problem in his article without doubt.
[QUOTE=”DMartin, post: 5216470, member: 569662″]I never said it was a proof, surely even you noticed that. I said it was relevant, and that the context can make a difference. The point that the context makes a difference is borne out by micromass above saying about the context of surreal numbers:
“Whether 1=0.9999… in the surreals depends highly on the definitions for 0.9999… . Some definitions make it equal, others don’t.”
So it’s relevant to point out a sequence relating to the second pair of numbers I mentioned, in which there is a distinction between the two of them, because one infinite sequence lands somewhere different from the other.
Incidentally, there have been distinctions drawn between different infinities, and it has turned out that they can be compared, and one infinity can turn out to be ‘larger’ than another. This might intuitively seem impossible, but ways to compare them were found. There is some loose similarity between that and what I’ve done.[/QUOTE]
OK, so in order to proceed, you’re going to have to tell us rigorously what you mean with ##0.9999…## and ##0.0000….001##.
[QUOTE=”Mark44, post: 5216280, member: 147785″]
The subtraction gives 0.000…, which is 0, for short.[/QUOTE]
This is the most convincing piece of evidence, for we assume the sequence of zeros is countible (and it is), therefore there can never be that ..01 at the “end” for there is no end.
[QUOTE=”DMartin, post: 5216187, member: 569662″]Well, there’s a symmetry to be pointed out.
First, you have 0.9999….. and 1, and these two numbers look the same, or almost the same. People discuss whether they’re the same, and whether there’s a proof that they are.
[/quote]They [I]look[/I] different to me, which happens to be irrelevant. However, I recognize that they are different representations of exactly the same number. “Whether there’s a proof that they are [the same]” — this implies to me that you have some doubt that this has been proven (it has – see the link in the very first post of this thread). Have you read the rest of micromass’s article in that link?
[QUOTE=DMartin]
Then, if you subtract 0.9999….. from 1, you find another pair of numbers – that is, the result of the subtraction, and zero. This pair of numbers can be called 1 – 0.9999…. , and 0.
[/quote]The subtraction gives 0.000…, which is 0, for short.
[QUOTE=DMartin]
And there’s a symmetry between these two pairs of numbers. Each pair may well be in the same relationship, whatever that is. So anything shedding a bit of light on either pair might be relevant. And showing the second pair to be different from each other is relevant.[/QUOTE]
No. Again, I’ll point you to the rigorous proof that 1 and .999… are one and the same number. You seem to be stubbornly holding on to your mistaken beliefs, despite being shown that your arguments aren’t valid, and despite having been shown proofs that there is no difference between 1 and .999…
This is basically “Proof by confusion”
[QUOTE=”DMartin, post: 5216191, member: 569662″]That’s what one would expect it to be. I’ve shown that the second pair isn’t just 0 and 0.[/QUOTE]
That’s the problem. No you have not.
[QUOTE=”DMartin, post: 5216187, member: 569662″]And there’s a symmetry between these two pairs of numbers.[/QUOTE]
Yes. 1 is related to 1 in the same way that 0 is related to 0.
[QUOTE=”DMartin, post: 5216120, member: 569662″]Yes I knew it was 1 radian.
“This is by definition, not up for debate”. This implies that all our definitions are, by definition, correct.
[/quote]Definitions are automatically assumed to be correct. If you don’t agree with a definition, that’s the end of the discussion.
[QUOTE=DMartin]
What you learned in primary school may or may not be true. But I have said that the context makes a difference sometimes.
But the question of comparing a whole number, or a non-negative integer, and another nearby number that approaches it with an infinite series, is not always a clear cut question.
[/quote]A given number doesn’t “approach” anything — it is static, fixed. However, a [I]sequence[/I] of numbers can have a limit. Pertinent to this thread, the sequence {.9, .99, .999, …} does have a limit, and the limit is 1, which can also be written as 0.999… There is no difference between 1 and 0.999… other than how they are written. Both representations are at exactly the same place on the number line.
[QUOTE=DMartin] And what I’ve set out has bearing on it, because I’ve show the two looking different, and looking existent.
[/quote]With all due respect, you have shown no such thing. It has been repeated stated here that what you are calling a number (0.000…1) is in fact meaningless, since you can’t say at what position the 1 digit is.
[QUOTE=DMartin]
Don’t forget that we learn as we go, the mathematical structures we have are not just a given eternal structure, they were put together bit by bit by finding things, and what we have is, as always, incomplete.[/QUOTE]
DMartin, seriously, I learned about 0,999…=0,(9) in primary school, just get your facts straight please. You clearly do not know what rational numbers are.
Can we agree that by 0.99…. we mean [itex]9sumlimits_{j=1}^infty frac{1}{10^j}[/itex]? A series that converges (which is evident) has a sum equal to that of [itex]limlimits_{ntoinfty}9sumlimits_{j=1}^n frac{1}{10^j}[/itex]. Consequently it happens to be 1. (This is by definition, not up for debate)
Assume [itex]0.99… neq 1[/itex], then there exists [itex]alphainmathbb{R}[/itex] such that [itex](*)alpha + 9sumlimits_{j=1}^infty frac{1}{10^j} = 1 (*) [/itex]
If ##alpha leq 0## then ##*## does not hold (do not challenge the axiom of real numbers which states all real numbers are comparable), so we can, I hope, agree that ##alpha > 0##.
We have that ##9sumlimits_{j=1}^infty frac{1}{10^j} = 1 – alpha## which is ##0,9 + 0,09 + 0,009 +… = 1 – alpha ##.
Therefore ##alpha = 1 – (0,9 + 0,09 + 0,009 + …)## Addition is associative in ##mathbb{R}## and so we can group the summands(is that the word?) as we see fit.
##alpha = (1 – 0,9) – (0,09 + 0,009 + 0,0009 +…)## (Addition is also commutative so it doesn’t matter in which order we add them together).
##alpha = (0,1 – 0,09) – (0,009 + 0,0009 + …) = (0,01) – (0,009 + 0,0009 + …)##
Proceed similarly however many times you want.
##alpha = (0,0…01) – (0,00…09 + 0,000…09+…)##
As the sequence of zeros is countible we have nothing but
##alpha = (0,000….) – (0,0000… + 0,00000 +…)##
To assume the sequence ends (the ..001 everybody wants to get at), means a countible set is of finite cardinality. ##q.e.a## we get that ##alpha = 0## and ##0,99… = 1##.
[QUOTE=”DMartin, post: 5215910, member: 569662″]Sorry, our posts crossed. Yes, I know you think that whether or not a number exists is related to whether or not one can specify the positions of the digits.
[/quote]Numbers exist whether or not we can identify each digit in the decimal representation. However, in this thread we are exhibiting a number such that all of its digits are known. The one you showed, 0.000…001, is really meaningless, since you can’t say where the 1 digit is.
[QUOTE=DMartin]But the mathematical structure I’ve shown above hints at a relevant structure that is uncovered, and exists in some way other than just conforming to a made up set of rules.[/QUOTE]What structure? What “made up” rules are you referring to?
[QUOTE=”DMartin, post: 5215902, member: 569662″]Putting in 0.0001, I get x/(sin x) = 57.295779513111409697664737311509
try putting in 0.0000001
then 0.00000000000000001.
the result will approach 180/[pi], and to me this shows something that is discovered, rather than invented, and has bearing on the similar questions we’ve been looking at.[/QUOTE]
No it does not. [URL]http://www.wolframalpha.com/input/?i=%280.0001%29%2F%28sin0.0001%29[/URL]
You’ve somehow converted one x into “degrees” and kept another in radians.
[QUOTE=”DMartin, post: 5215902, member: 569662″]Putting in 0.0001, I get x/(sin x) = 57.295779513111409697664737311509
try putting in 0.0000001
then 0.00000000000000001.
the result will approach 180/[pi], and to me this shows something that is discovered, rather than invented, and has bearing on the similar questions we’ve been looking at.[/QUOTE]
This is unremarkable, and has no bearing whatsoever on the topic of this thread, that 0.999… = 1.
The expression ##frac x {sin(x)}## is undefined at x = 0, but has a limit of 1 as x approaches 0 (using real numbers, AKA radians, rather than degrees).
IOW, ##lim_{x to 0} frac x {sin(x)} = 1##.
Treated as a real number (radian), 1 radian = 180/π degrees or about 57.29577951 degrees.
[B]Edit[/B]: I should mention that mathematicians rarely work with degrees, and almost invariably work with radians, a sort of “unit-less” unit, being the ratio of two lengths.
[QUOTE=”DMartin, post: 5215883, member: 569662″]” 0.999999… exists because every digit in the decimal representation can be specified. If you ask, “what digit is in the 12th place?” Answer: 9. If you ask, “what digit is in the 59th place?” Answer: 9. If you ask, “what digit is in the 623rd place?” Answer: 9. No matter what specific digit you ask about, the answer is always “9”. ”
That’s true, and you believe that it’s significant.[/QUOTE]
It IS significant and has nothing to do with belief.
You quoted part of what I wrote, above, but made no comment about the rest of what I wrote, which was this:
[QUOTE=”Mark44, post: 5215870, member: 147785″]On the other hand, if I ask you what’s the position of the 1 digit in 0.00000….0001, you can’t possibly answer. For that reason, your number doesn’t exist.[/QUOTE]Do you have anything to say about this part?
[QUOTE=”HomogenousCow, post: 5215897, member: 435628″]What are you talking bout?
Let’s say we choose x to be 0.0001, x/(sinx) is approximately 1(~1.0000001).
This is nowhere near 180/pi.[/QUOTE]
To be fair, DMartin is apparently assuming degrees. The rest of the argument goes nowhere, of course.
[QUOTE=”DMartin, post: 5215880, member: 569662″]See posts above. There’s a series of increasingly small angles near zero degrees that approach a number at infinity, but that number is above zero. It’s clear that this number gives x/(sin x) = 180/π, because the values approach that number. Whether or not any other values give 180/π makes no difference, but it’s interesting to hear it.[/QUOTE]
What are you talking bout?
Let’s say we choose x to be 0.0001, x/(sinx) is approximately 1(~1.0000001).
This is nowhere near 180/pi.
[QUOTE=”DMartin, post: 5215860, member: 569662″]Well, it’s a matter of taste to some extent. You say you can prove that within a particular artificial system, a number 0.99999999…. exists, but 1 minus that number, or 0.00000….0001 doesn’t exist. And yet I’ve shown that the second number can be expressed as an angle between zero and 90 degrees.
This has bearing on the question of whether mathematics is invented or discovered. Are we inventing the rules, or discovering them. When I look at the points above about the digit 1 in my number, I think there’s a bit too much inventing going on.[/QUOTE]
[QUOTE=”HomogenousCow, post: 5215865, member: 435628″]There’s no philosophical issue here, you’re simply doing bad maths and making incoherent arguments.[/QUOTE]I agree completely.
0.999999… exists because every digit in the decimal representation can be specified. If you ask, “what digit is in the 12th place?” Answer: 9. If you ask, “what digit is in the 59th place?” Answer: 9. If you ask, “what digit is in the 623rd place?” Answer: 9. No matter what [U]specific[/U] digit you ask about, the answer is always “9”.
On the other hand, if I ask you what’s the position of the 1 digit in 0.00000….0001, you can’t possibly answer. For that reason, your number doesn’t exist.
[QUOTE=”DMartin, post: 5215793, member: 569662″]
If you’re not keen on how my number is expressed, perhaps you’d prefer it if I said:
an angle θ such that θ/(sin θ) = exactly 180/π. I can prove that this angle is not zero, because θ = 0 gives a different result.[/QUOTE]
How is this relevant to the discussion that 0.999… = 1?
There are an infinite number of solutions to the equation ##frac{x}{sin(x)} = 180/pi##, one of which is approximately 3.087675. It is obvious by looking at a graph that x = 0 is nowhere close to being a solution.
[QUOTE=”DMartin, post: 5215860, member: 569662″]Well, it’s a matter of taste to some extent. You say you can prove that within a particular artificial system, a number 0.99999999…. exists, but 1 minus that number, or 0.00000….0001 doesn’t exist. And yet I’ve shown that the second number can be expressed as an angle between zero and 90 degrees.
This has bearing on the question of whether mathematics is invented or discovered. Are we inventing the rules, or discovering them. When I look at the points above about the digit 1 in my number, I think there’s a bit too much inventing going on.[/QUOTE]
There’s no philosophical issue here, you’re simply doing bad maths and making incoherent arguments.
[QUOTE=”DMartin, post: 5215793, member: 569662″]If you say that one number exists and another doesn’t, you need to say what you mean by exists. If you mean exists within mathematics, Gödel showed that mathematics isn’t necessarily a self-consistent system, so existing within mathematics isn’t necessarily a meaningful concept.
If you’re not keen on how my number is expressed, perhaps you’d prefer it if I said:
an angle θ such that θ/(sin θ) = exactly 180/π. I can prove that this angle is not zero, because θ = 0 gives a different result.[/QUOTE]
This thread is about mathematical existence. I have no idea what other kind of existence you would want to talk about since numbers are inherently a mathematical concept.
[QUOTE=”DMartin, post: 5215793, member: 569662″]If you say that one number exists and another doesn’t, you need to say what you mean by exists. If you mean exists within mathematics, Gödel showed that mathematics isn’t necessarily a self-consistent system, so existing within mathematics isn’t necessarily a meaningful concept.
If you’re not keen on how my number is expressed, perhaps you’d prefer it if I said:
an angle θ such that θ/(sin θ) = exactly 180/π. I can prove that this angle is not zero, because θ = 0 gives a different result.[/QUOTE]
????
[QUOTE=”William White, post: 5215736, member: 562357″]
is not the simplest ‘proof’
1 ÷ 3 * 3 = 0.9999[/QUOTE]
No.
##0.9999 = frac{9999}{10000} < 1##Most of us have been careful to distinguish between such fractions as 0.99 and 0.99[B][COLOR=rgb(255, 77, 77)]...[/COLOR][/B], which some people have written as 0.9(9). Both notations indicate that the 9 digits continue indefinitely, following the same pattern.What you wrote, 0.99, is the same as 0.990000...
[QUOTE=”William White, post: 5215774, member: 562357″]they are mutually contradictory statments.
[/QUOTE]
They’re not.
[QUOTE]
If the line is continuous there is a number greater than 1 that is smaller than every other number greater than 1.
[/QUOTE]
That is false. The existence of a number greater than 1 smaller than any other number greater than 1 would imply the discontinuity of the number line, since then what is the average of 1 and that number greater than 1?
[QUOTE=”micromass, post: 5215767, member: 205308″]The number line is continuous and there is no smallest number greater than a number.[/QUOTE]
they are mutually contradictory statments.
You have a number.
Its exactly 1.
If the line is continuous there is a number greater than 1 that is smaller than every other number greater than 1.
If this is not the case, then all adjacent numbers are equal, so the line is not continuous.
[QUOTE=”William White, post: 5215754, member: 562357″]
If a smallest number greater than a number; or the largest number smaller than a number cannot exist; the number line is discontinuous.[/QUOTE]
The number line is continuous and there is no smallest number greater than a number.
[QUOTE=”micromass, post: 5215739, member: 205308″]There is no such number.
It’s a well-posed math problem with a good solution.[/QUOTE]
you have just contradicted yourself
if 0.999 is ‘no such number’ then the question goes away (so it Is NOT a well posed problem) and the answer is trivial.
If a smallest number greater than a number; or the largest number smaller than a number cannot exist; the number line is discontinuous.
[QUOTE=”William White, post: 5215736, member: 562357″]either .99…. is the largest possible number that is smaller than 1 (which makes them different)
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There is no such number.
[QUOTE]
surely a semantics rather than maths problem,
[/QUOTE]
It’s a well-posed math problem with a good solution.
either .99…. is the largest possible number that is smaller than 1 (which makes them different)
or
there is no number between 0.99…. and 1 (which makes them the same)
both are logically correct, so surely a semantics rather than maths problem, with zero real world difference (its an angels on a head of a pin debate)
is not the simplest ‘proof’
1 ÷ 3 * 3 = 0.9999
[QUOTE=”DMartin, post: 5215616, member: 569662″]I can show that whatever the meaning of the number 0.0000000 => 00001, with an infinite number of zeros, it is different from 0.
[/quote]It has already been pointed out by micromass, but it bears repeating. [B]You CANNOT define a number whose decimal representation contains an infinite number of zero digits, [U]followed[/U] [U]by[/U] a nonzero digit[/B]. In the number 0.999… of this thread every digit to the right of the decimal point has a defined position, which is not the case for the number you show.
[QUOTE=DMartin]This means that other similar numbers are probably the same, though it does depend on the context. The method involves showing that the two numbers 0.0000000 => 00001 and 0 give completely different output numbers when put into an equation.[/QUOTE]
[QUOTE=”DMartin, post: 5215685, member: 569662″]
About the position of the 1 in my number, out of all those 9s of yours, there must be one that corresponds to my 1.
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There isn’t. I’m afraid you don’t understand how infinity and positional notation works. Every decimal in my number is preceded by a finite number of decimals only. Your 1 is preceded by an infinite number of decimals and is therefore not allowed.
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So whatever problems I have with my number (and God knows it’s hard to keep them all in line), you must have the same problems with yours.[/QUOTE]
I don’t, and I can prove I don’t.
[QUOTE=”DMartin, post: 5215648, member: 569662″]So you have a rule that ‘each numerical digit must have it’s concrete position’. I suppose you know the positions of all the 9s in 0.99999….. then. But even if you argued that their positions are more concrete than the 1 in the number I used, it’s not clear where the rule came from.
[/QUOTE]
It is very clear since you can prove it.
[QUOTE]
what I’ve shown is a series that converges on, or approaches, a number at infinity, and the point is, whatever the existence status of that number at infinity, it isn’t zero. And surely whatever its existence status, it’s similar to the existence status of the numbers you’re talking about.[/QUOTE]
You can prove the existence of such numbers rigorously and formally. You cannot prove the existence of your number.
[QUOTE=”DMartin, post: 5215626, member: 569662″]Well, that may be so, but this thread is a discussion on the basis that such numbers are worth talking about, so we’re assuming they have some meaning before we start. If you say ‘there is no such number’, then presumably you think this whole thread is pointless.[/QUOTE]
Since I created the thread, I’d say that the thread is about the [B]real[/B] numbers ##1## and ##0.9999…##. And not about numbers which you have not rigorously defined and which certainly are not real.
[QUOTE=”DMartin, post: 5215626, member: 569662″]then presumably you think this whole thread is pointless.[/QUOTE]
No, because this thread is about rational numbers, that surely exist. What you wrote does not exist. Read this: [URL]https://en.wikipedia.org/wiki/Repeating_decimal[/URL]
If you have infinite number of 0s then you can not put 1 after that, bocause there is no place for it. That’s why what your wrote is wrong, it’s called positional notation, and each numerical digit must have it’s concrete position. In your example 1 does not have any position, since there are infinite number of zeros.
[QUOTE=”DMartin, post: 5215616, member: 569662″]I can show that whatever the meaning of the number 0.0000000 => 00001, with an infinite number of zeros,
[/QUOTE]
There is no such number. So the rest of your post is irrelevant.
[QUOTE=”nuuskur, post: 5215553, member: 519618″]It should be unanimous that 1 – 0.9(9) = 0.0(0) = 0. Assume the opposite, then 0.0(0) != 0, contradiction.[/QUOTE]
Unfortunately, that is not a proof. That is notational sleight of hand. If you were to carry out the subtraction in longhand, there would be a “borrow” propagated infinitely to the right. The notation has no way to record the existence of this unsatisfied borrow, so it is lost. A student would be correct to worry about whether that loss is meaningful or ignorable. One would have to dig deeper into the meaning of the notation and the properties of the real numbers to demonstrate that the result of the calculation is indeed justifiable.