Correct me if I’m wrong, but what you’ve done here is defined a new operation. The plus sign should have a ticks around it, “+”, as to distinguish it from addition. Thus:
10 / 3 = 3 “+” 1 or 18 / 7 = 2 “+” 4 or 1 / 3 = 0 “+” 1 or
It’s like you’re saying that under minimal definitions and axioms, 0.999… = 1, but if we can make new definitions and new axioms, it’s not hard to make 0.999… ≠ 1. Alternately you can explicitly state the decimal representation of 1/3, which you didn’t do in post #1813.
No, I’m pretty sure schooner26 means the ordinary 10/3 = 3 + 1/3, or 18/7 = 2 + 4/7, or 1/3 = 0 + 1/3, or 1/3 + 1/3 + 1/3 = (0 + 1/3) + (0 + 1/3) + (0 + 1/3). After all, that’s the sort of thing they write in the rest of the post you’ve responded to.
0.999… is not the limit of anything. How can infinite 9’s be the limit ? Limits are used to remove the infinity concept. The infinity concept is still present in 0.999…
0.999… is not the number though. If you mean limit 0.999… = 1, then we have 1 = 1.
May I asked where did you learn that you *assign *limits their value ?? Limits output values after careful analysis. You don’t get to assign them something, and then after analyzing, which outputs something different, conclude they must be equal !!
There is none. Just like there is no exact decimal representation of PI. The […] is a lame attempt to do so.
1/3 is 1/3 and it is a downfall of the base-10 decimal system which prevents a decimal representation.
1/3 = 0.3333⅓ = 0.3333333333⅓
But that ⅓ always exists in an exact representation. If you leave it out, then you have an approximation. See my post on page 37 for what 1/3 equals, and what 0.333… is.
If that is your meaning and interpretation, then under that understanding 0.999… = 1, where […] means “the limit”, but there are many PhD’s who refer to […] as “and so on” or “endless”.
Where “endless” is meant, then 90% of the remainder gets endlessly small but never exactly 0 and endlessly far from doing so.
If you want to post a concept or something, please do so. I can’t read your mind about what you want me to address.
That’s the definition anyone actually doing math uses for 0.999…, that’s how. And limits are used to handle infinities, not to make the concept go away.
You may consider it a lame attempt, but anyone doing actual math considers it a perfectly useful notation.
What you have is not a mathematical argument based on commonly accepted standard notation, it’s your own set of odd misconceptions concocted to argue against your esthetic disagreement with commonly accepted standards.
0.333… meansthe limit of the sum of 310[sup]-i[/sup] as i goes from 1 and approaches infinity*, and that limit is equal to 1/3. Thus 0.333… = 1/3 , whether you like it or not.
You can make up your own notations if you want, but you wont find many rational people willing to play with you.
:rolleyes: oh my. I think we both know exactly what was meant.
“handle” “address” “deal with”
But infinity is usually (always) never present after being addressed and a finite number is used: L , if it exists.
I think one of the issues is that a lot of you can’t imagine infinity, ie, “endless”. You voluntarily invoke the limit, unnecessarily.
Σ (n=1 to ∞) 9/10ⁿ = 0.999…
and
limit Σ (n=1 to ∞) 9/10ⁿ = limit (a - arⁿ) / (1 - r) = a / (1 - r) = (9/10) / (1 - 1/10) = 1
In what I quoted above, are you talking about the expression “0.999…” or about the number represented by the expression “0.999…”? Because that, I think, is the source of the confusion. The number represented by this expression can indeed be (and is) the limit of something.
Well, 0.999… ain’t a number either. Never seen a number with those dots. :rolleyes: Look, the way 0.999… is used in mathematics is the limit of the infinite row I described earlier. That’s simply what it means. That is, as far as I am aware, the only sensible meaning you could apply to a decimal representation made up of an infinite number of nines to the right of the decimal point. No other usage is common in mathematics for this.
Where did I assign the limit its value? 0.9999999… is the decimal representation of the limit. That the value of that limit is 1 is something which can be proven in various ways; I was under the impression we were in agreement on that.
Ah … there’s the problem … you do not accept “0.333…” as a symbol for a value that cannot be written out completely. Then you do not accept the symbol “0.999…” as representing 1. Fair enough.
Why not? Substitute your definition for 1/3 earlier into your use of 1/3 in your last equation.
You assert that 1/3 = 0.333333[sup]1[/sup]/[sub]3[/sub]
So:
1/3 = Σ(n=1 to N) 3/10ⁿ + 1/ (3 x 10^N) … (your equation)
1/3 = ∑(n=1 to N) 3/10[sup]n[/sup] + 1/3 * 10[sup]-N[/sup]
In which case
1/3 = = ∑(n=1 to N) 3/10[sup]n[/sup] + ∑(n=N+1 to M) 3/10[sup]n[/sup] + 1/3 * 10[sup]-M[/sup]
for positive integers N and M
The point being that for any N you want to define, I can suggest an M for which the answer is arbitrarily closer to zero. You can never define a finite N that I can’t get closer to zero for. This is where you get infinitesimals. So we end up saying that there is an infinitesimal difference. But, here is the thing. There are no infinitesimals that do not equal zero in the reals. No matter what you do to define your infinitesimal I can make it smaller. I can do this so that you can never define any number between your infinitesimal and zero. There is never a gap that you can define that I can’t make smaller. Thus there are no numbers in the gap. If there are no numbers, in the gap, the gap has size zero, and your infinitesimal is equal to zero.
There is a common problem that all the critics of 0.999… ≠ 1 seem to share, although it manifests itself in different guises. In the end the problem is always a rejection or an inability to grasp the definition if infinity. schooner26 seems to have taken it back a bit further and manifested in a disagreement with the idea that any infinite series can converge. 7777777 has the more common issue of trying to make some form of infinity finite in some construction.
Almost all arguments seem to involve, at some point, the idea that we can’t ever “arrive”, or you can “never” reach the needed result. Again, imposing some notion of finite time on infinity. Programs executing, traversing a number line, and so on. All eventually being expressed as a finite “process”. Which of course is where Zeno’s fallacies come into play. The problem being that there is an underpinning notion that mathematics takes place in some physical universe, and takes time to happen. So despite one being able to clearly walk from A to B in finite time, Zeno generates a fallacy in which the mathematical representation of an expression of an infinite series seems to take “forever” - a fallacy trivially resolved by pointing out that the time taken for each step is not constant, but also shrinks with the distance it must cover, and thus velocity is constant, and non zero. Mathematical representations take zero time. Perhaps there is a relationship between people not being able to resolve Zeno’s paradox and those unable to understand finite sums of infinite series.
No doubt. Infinity is not a trivial concept. The notion most people carry around in their head is not adequate for mathematics. Cantor spent most of his life wrestling with it, and came up with some astounding results in order to cope with it.
Simple and yet mind bending at the same time. For N in the above series, when we run it to infinity, we are using Cantor’s Infinity: the cardinality of the set of Integers.
How many Integers are there? That is Infinity.
How many Rational numbers are there? That is Infinity as well.
There are the same number of rational numbers between 0 and 1 as there are integers, and yet this means that there are the same number of rational number between any pair of integers, yet there are an infinitude of such intervals. Yet the definition remains consistent and useful. If you want to understand how different Infinity is, and why so much of the preceding thread contains arguments that miss use and misunderstand infinity, you need to get past this first.
No, you obviously don’t. You have a problem with infinities, yet seem to accept limits going to infinity because they make the infinity “go away”. Real mathematicians accept infinities and use limits to discover things related to them.
Again, this is your misunderstood interpretation of the notation. The “…” notation implies the limit. I’ve seen the misunderstanding before, but usually accompanied by an allergy to limits involving infinities. Are you sure you wouldn’t be more comfortable arguing Limit 0.333… ≈ 1/3, or Limit 0.333… = 1/3 + ε ?