Are you saying 0.999… is not a real number?
I have a problem with:
1 - infinitesimal = 0.(9)
But infinitesimal = 0 , therefore 1 = 0.(9)
because
1 - 0 = 1
Also,
if 1 - 0.9(n-times) = 1/10ⁿ
and n must equal ∞ to get 0.9(∞ times) , giving
1 - 0.9(∞ times) = 1/10^∞
If 1/10^∞ is not a number, yet it is the result of 1 - 0.9(∞ times), then either 1 or 0.9(∞ times) or both are also not numbers.
You’re using notation that isn’t correct. “0.9(n-times)” just alone makes no sense without specifying what n is an element of. If n a real number, then it can’t be infinity.
You keep using 0.999… as though it’s a real number, please state so … that seems an issue that we’re not clear on.
Either you switch to base 2 in which case it’s 0.111… or you don’t bother using the “…” notation. “…” is really just a sloppy notational short hand used for illustration, that requires that the writer and the reader agree have an understanding as to what it means to continue the series. When you wrote, 1/2 + 1/4 + 1/8 + …, you assumed that I would realize that the next two terms were + 1/16 + 1/32, but I guess theoretically they could be anything. Similarly when I write a decimal expansion ending in …, I expect it to indicate a limit of a series of the form Sum(a_n/10ⁿ) where a_n are integers between 0 and 9 inclusive. As far as what the a_n’s are that is left up to context. For 0.999… it is hopefully clear that all the a_n’s are 9’s. For pi=3.14159… it is assumed that the a_n’s follow the remaining decimal approximations to pi. With 0.875… I have no idea what this represents because there is no context as to what the remaining digits are, but if I wrote Sqrt(0.766)=0.875… Then it becomes clear what the digits should represent, and also gives me the added useful information that is between 0.875 and 0.876.
Numbers are distinct mathematical objects, while representations are tools mathematicians use to communicate the ideas of these objects.
For example 2, 1+1, 4/2, 1.999… are representations of the same concept.
Similarly “Hawaii”, “The last state to join the US”, and “The state where Obama was born” are all different representations of the same thing.
1.999… and “The birthplace of Obama” share the property that they are somewhat ambiguous. In the first case because I could possibly be talking about 1.99919991999…, while in the second case I could be possibly talking about Phil Obama the plumber, but with context and an understanding between the writer and the reader communication can still take place.
No. To be clearer about the problem. You are using the … notation in two different ways.
Conventionally:
0.875… = 0.875875875875875875875…
Conventionally 0.abcde… = 0.abcdeabcdeabcde… or 0.(abcde)
and 0.aaaaa… = a10[sup]-1[/sup] + a10[sup]-2[/sup] + a10[sup]-3[/sup] + a10[sup]-4[/sup] + a10[sup]-5[/sup] + a10[sup]-6[/sup] + …
As I have said more than a few times above. It would be a very good idea to restrict your arguments to the value of this series, and avoid ambiguous and misleading traverses into the decimal representation of this.
As has been noted above. The idea of a “decimal number” is not precise, and is not used in mathematics. It might be used in arithmetic, but that is simply as a way of denoting the base you are using. I could easily be using a different base. Two and 16 come to mind as common alternative bases.
Arguments about representation really need to be able to move from one base to another without any change to the results. Again, as noted above. These are representations - representations of of series. It would be an odd world if you “proved” that a number in one base was not equal to the number represented in a different base.
To be clear 0.999…[sub]10[/sub] = 0.1111…[sub]2[/sub] = 0.2222…[sub]3[/sub]
These are very good questions and highlight the problem clearly.
First you write a correct statement
1-0.999…= infinitesimal
but then you just set the infinitesimal equal to 0 and arrive at 1=0.999…
Consider this: don’t set the infinitesimal equal to 0, let’s say that it is equal to x instead. Now we get
infinitesimal=x, therefore 1 ≈ 0.999…
so that 1 - x = 0.999…
You write that “because 1 - 0 = 1”, therefore infinitesimal x=0
Consider this instead: because 1 - x = 0.999…, therefore infinitesimal x ≠ 0
What I am trying to say is: just because 1 - 0 = 1, you cannot conclude that therefore 1=0.999…
Think about this -1 + x = 0, the only answer is x=1, therefore x=0.999…is not
an answer. Also, you see, because 1 - 0 = 1 ,therefore 1≠0.999…
The only way that 1=0.999…is true is if you set the infinitesimal x equal to
zero. But as I have shown many times, setting x=0 is not justified because
there is no proof that 1=0.999…If we just assume that 1=0.999…it
is not a proof. On the other hand, there are many proofs which show that
1≠0.999…therefore it is justified to say that the infinitesimal x is not equal to 0.
Lets use the same infinitesimal x again, but don’t use ∞, in its place lets use Z.
Now we get,
instead of
1 - 0.9(∞ times) = 1/10^∞
we get
1 - 0.999…= 1/10^Z
1 - 0.999…= x
x = 1/10^Z
The question is :What is Z?
I have written that Z is the largest number satisfying the equation Z + 1 = ∞
Also, Z is not equal to ∞. We access infinity by adding 1 to Z but it is invalid
to write that Z = ∞
schooner26 please address 7777777 concerns.
You realize you’re risking total protonic reversal, right?
Please address my comments above … infinitesimal is not a real number therefore this is not the correct statement
1 - 0.999… must equal a real number by axiom.
1/3 = 0.333…
schooner26, is that a true or false statement? Using standard real numbers, can one third be represented in base 10 notation as a decimal point followed by an infinite number of 3’s?
Then we’re done. An indistinguishable amount that isn’t in the set of real numbers? Congrats, you’ve proven that they are, in fact, equal.
Also, pro tip: if you accept that lim(0.99…) = 1, then we’re in agreement, because 0.999… is a shorthand for the limit of that sum! That’s simply how the term is defined.
Most of them based on silly things like …999 (which is not a defined real number, and which you have failed to define). None of which have made any sense.
We’re not ignoring them. We’re pointing out clear and obvious problems, and you aren’t responding.
Because there is absolutely nothing within the rules of the real numbers that prevents a number from having more than one decimal representation.
Wait, weren’t you the one accusing us of assuming our conclusions?
Hey, you said something correct! …999 = -1 is false. In fact, …999 is, until you define it, meaningless! It’s not a meaningful way of writing a real number. It’s not a particularly sensible way of writing the sum sum(x=0, x->inf)[9*10^x] (which, by the way, is divergent, unbounded and therefore not equal to any real number; this is trivially proven via direct comparison with basically any divergent row you could choose). If you’re talking about a Borel Sum, then tell us - because in that case, the notation is flawed, and you cannot simply attach .9999… to the end of it.
Um… 1/3=0.33… It’s really not that complex.
The only part about any of this that’s hard for us to understand are the parts that you all but intentionally make obtuse - things like “…999999 = -1”. The list of people who have any idea what you’re talking about or what …999 even means, let alone how you reached that conclusion, is very short. Because, you know, you never take the time to explain it!
So what does this even mean?
I come up with x = n + ( 1 - n ) for any real number n.
Something quite sensible (though there’s a typo where “80” should read “8”). chingon is pointing out that there’s a well-studied interpretation of summation of infinite series according as to which 9 + 90 + 900 + 9000 + … = -1 [and thus, were we to want to interpret infinite leftward decimal notation via summation of infinite series in the straightforward way, this would give us an interpretation of such notation according as to which …999 does indeed equal -1, rather than being infinite or meaningless].
There are actually many such interpretations of infinite series summation of note. chingon points in particular to the notion of “generalized Borel summation”. We can delve into the specifics of this if you want.
Actually, the reason that the series corresponding to …999 in base 10 is generalized Borel-summable turns out to be because it is “absolutely summable” (i.e., summable to a finite value in the most familiar, conservative sense of infinite series summation) for sufficiently small bases (9 + 9b + 9b^2 + 9b^3 + … coming out to 9/(1 - b) for sufficiently small b), in such a way as admits a suitably well-behaved canonical extension even to larger bases (less fancily, we get to keep the formula 9/(1 - b); plug in b = 10 and see what you get).
The original linked proof that …999 = -1 is actually a valid proof (with perhaps some slight elaboration) on this interpretation.
Whoops; Chingon should be capitalized.
Indistinguishable has given a better description of summation of divergent series than I could give. A starting point for learing more is here (continued).
Admittedly, this is a broadening of the definition of “summation” but the definition of summation had to be broadened to allow it to be applied to convergent series too just like we had to broaden the definitions of addition and multiplication to allow x + 1 = 0, x + x = 1, x * x = 2, x* x = -1.
The reason people believe that 1 != 0.999… is because they are so in love with their prejudices that they have to dismiss anything that contradicts them. When arguing with them, it would be wise not to emulate that.
Well I disagree because limit 1.999… is the same ‘concept’ as 2, and not 1.999…
and
Σ 18/10ⁿ is a completely different concept since it involves infinity, of which we can voluntarily apply a limit, or any other mathematical operation, such as multiply it by a constant, but in no way is the original object equal to the one post-limit. Again, see my cake analogy. The ingredients aren’t automatically cake. We voluntarily mix them and heat them up to get cake.
I already addressed this issue [ HERE] on page 37
What, in your mind, is the difference between “limit 1.999…” and “1.999…”?
And you have proven we cannot discuss anything further if you arrive at that grand conclusion, “that because we cannot assign a number to the result, they MUST be equal”. Not sure what axiom this is you are using.
But you are using 0.999… as a number itself, when you mean the limit of it.
Several of my arguments have not been addressed either, and I am on vacation and don’t sit on my laptop all day.
Correct. You assume that 0.999… = limit 0.999… and that a / (1 - r) when “series shifting” was assumed to a legal move when it is not, etc.
Not when you are incorrect, since it is easy to make mistakes. [ LINK]
But that’s the misunderstanding. Every time you see someone here write x.999…, replace it mentally with lim(x.999…) and you’re done. Because that’s what we’re doing! We’re using 0.999… as a decimal representation of the limit of the sum of [0.9, 0.09, 0.009, …]. Which you seem to agree is equal 1. Or am I mistaken?
Well, yes, if there is no number between two numbers, then those two numbers are equal. If there is a number between them, there is an infinite number of numbers between them. It should not be hard to name or at least define at least one such number. Right? As much as 7777777 prattles on about infinitesimals, the fact is that they’re not real numbers - they’re not in the set, and within the concept of the set, they have no room to exist.
Err… No. I define 0.99… as lim(x->inf)[sum(y=1, y -> x)[9*10^-y]]. It’s not that I’m assuming them as equal, I’m defining the former term as the latter term. What meaning would it have otherwise? That’s simply what 0.99… means. It’s a crude decimal representation of the limit of an infinite sum and in any colloquial equation, that is how it is used.
Also, you’re not 7777777…