Is 1 = 0.999... ? Really?

obsrvr524 · February 9, 2021, 5:46pm

I don’t think that is true.

It is meaningful to say that 1/3=0.333…

But that does not mean that 0.333… is truly an exact match to 1/3.

What 0.333… represents exactly is a number that is within an infinitesimal of being 1/3 - and that is a useful meaning.

phyllo · February 9, 2021, 6:22pm

But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.

obsrvr524 · February 9, 2021, 7:49pm

That is not true.

Magnus_Anderson · February 9, 2021, 8:27pm

No matter how many 3s are implied by the ellipsis, the expression is never equal to (\frac{1}{3}). In other words, even with a (3) located at the position with an index of (-infA) (or any other negative infinity), the number would still be less than (\frac{1}{3}).

The larger the number of (3)s, the closer the number is to (\frac{1}{3}). But it can never be exactly (\frac{1}{3}).

phyllo · February 9, 2021, 8:42pm

That is basically contradicted by all the mathematical results shown in this thread.

Sure, you can say that but at the same time you are saying that the sum of a converging geometric series is wrong. (For example)

Which result do you prefer to retain?

I prefer not to throw away the work on geometric series.

obsrvr524 · February 9, 2021, 8:56pm

Do you think that you can Resolution Debate that proposal?

Magnus_Anderson · February 10, 2021, 1:26am

No problem. But let’s not start a new debate until we review and summarize our first debate. Also, I’d want to make certain things clear in advance. (And I won’t be available until the weekend or so.)

But I’m surprised you disagree. What’s your position on the issue?

Magnus_Anderson · February 10, 2021, 2:06am

I think you’re presenting a false dichotomy. You either accept that (0.33\dot3 < \frac{1}{3}) and throw away the work on geometric series or you reject that (0.33\dot3 < \frac{1}{3}) and keep the work on geometric series. In reality, you can accept that the two numbers aren’t equal AND keep the existing work by making an adjustment to it.

What they have shown is that the limit of (0.3 + 0.03 + 0.003 + \cdots) is (\frac{1}{3}). I don’t dispute that. But then they erroneously conclude that (0.33\dot3 = \frac{1}{3}). It’s like someone correctly showing that the ceiling of (1 + 5 \div 10) is (2) but then erroneously concluding that (1 + 5 \div 10 = 1).

wtf1 · February 10, 2021, 2:08am

But that is the DEFINITION of the sum of the infinite series.

It’s not an erroneous equating. It’s a definition.

Magnus_Anderson · February 10, 2021, 2:19am

What’s the point of repeating this point? I’ve already responded to it. Why not address my response instead?

I am fully aware that’s how they define it. The thing is that that definition conflicts with existing definitions.

If we define symbols “2” and “+” the way we normally define them, then the meaning of “2 + 2” can be deduced to be the same as the meaning of “4”. You are thus not free to declare that “2 + 2” means “10” and not “4”.

“0.3 + 0.03 + 0.003 + …” has its own meaning and someone (whoever that is) declaring that it means something else is introducing a contradiction in their system of thought.

That expression DOES NOT stand for a limit.

(In the same exact way that “1 + 2 / 4” does not stand for the ceiling of “1 + 2 / 4”.)

wtf1 · February 10, 2021, 3:31am

It does stand for a limit, that’s the ONLY thing it can stand for.

Once you agree that the limit of .3, .33, .333, … is 1/3, we’re done.

I truly don’t understand your point. Addition is a binary operation. The only way to define addition of infinitely many summands is to define it as the limit of the sequence of partial sums. Having done that, we’re done. You agree to all this … then you say no. This I truly don’t get.

obsrvr524 · February 10, 2021, 4:16am

Wikipedia:

Limit
In mathematics, a limit is the value that a function (or sequence) “approaches” as the input (or index) “approaches” some value.[1] Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit of a function is usually written as
\({lim_{(x\to c)} f(x)=L}\)

obsrvr524 · February 10, 2021, 4:20am

Who said I disagree?

So who would you want to debate?
I can moderate.

wtf1 · February 10, 2021, 4:28am

Look up the definition of the sum of an infinite series. It’s defined as the limit of the sequence of partial sums.

Secondly, you posted a handwavy Wiki definition of limit, not the technical definition. What is the point of flaunting your lack of mathematical understanding? If you don’t know what a limit is, copypasting Wiki won’t help.

obsrvr524 · February 10, 2021, 4:58am

A series sum is a function as long as there are variables for the limits (which there are).

Wikipedia:

Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) “approaches” as the input (or index) “approaches” some value.

Series (mathematics)
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is,
\(\sum_{i = 1}^{∞} a_i = \lim_{(n → ∞)} \sum_{i = 1}^{n} a_i \)
When this limit exists, one says that the series is convergent or summable, or that the sequence ({ (a_{1},a_{2},a_{3},\ldots )}) is summable. In this case, the limit is called the sum of the series

They say that the limit is “called” the sum. That admits that it is actually a limit and that it is merely “called the sum”.

So they provide a definition for “limit”
They describe the infinite series of partial sums as “approaching” the limit.
And they admit that limit is merely “called” “the sum” for convergent series.

Add to that -

obsrvr524:

The only definition I could find for the ellipsis “…” is in Wikipedia -

Wikipedia:

The latter formula means the sum of all natural numbers from 1 to 100. However, it is not a formally defined mathematical symbol. Repeated summations or products may similarly be denoted using capital sigma and capital pi notation, respectively:

I have never seen “…” being used to mean anything other than “continue indexed throughout the natural numbers”. So without a formal definition in our debate I would have stated -
Ellipsis - “…” = “repeated throughout the natural numbers with index starting at either the first digit to the right of the decimal (when a decimal is to its left) or repeated throughout the natural numbers with index starting at the first left hand digit indicated at the left of the symbol.”

and I think we have every definition issue raised throughout this entire thread.
And it seems that James has agreed with Wikipedia’s definitions throughout.

If you don’t like Wikipedia, show us a better source.
Or better yet - go talk to them and correct them.

Certainly_real · February 10, 2021, 7:41am

Then your ‘=’ must be taken to mean near to but not an exact match to, or, nearly equally to but not truly equal to.

I’m looking at this from a non-mathematical point of view. I don’t think you can have multiple infinitesimals. To me, infinitesimal is that which separates all possible measures and things from each other. To me, only one existing thing can take the measure of infinitesimal and infinite. I call It Existence/God.

Magnus_Anderson · February 10, 2021, 1:04pm

Yes, addition is a binary operation. It is a function that takes two values and outputs a single value. We know that. Do you have anything else to say? Because you are repeating yourself.

You seem to think that an expression such as “2 + 2 + 2” represents an operation of addition that takes three values and outputs a single value. If that’s what the expression stands for then it contradicts the definition of the word “addition”. As you say, addition takes two inputs; but this expression, you also claim, represents a function that takes three. How can it then be addition? It cannot be – unless you change the definition of the word “addition” that is. Fortunately for us, that’s not what the expression stands for. It stands for a chained series of additions where the result of the first addition (which is “2 + 2”) is taken as the input of the second addition. This means that you DO NOT have to redefine the concept of addition. You DO NOT have to extend it so that it can take three (rather than merely two) values as its input.

wtf1 · February 10, 2021, 10:11pm

Exactly what I explained to you in detail a couple of posts back.

Yes you STILL have no definition of an infinite sum, since you can only use induction to reduce a finite sum to a succession of binary operations.

Well folks it’s been fun, I’m done here. All the best.

Magnus_Anderson · February 11, 2021, 1:21am

I can debate anyone: Certainly Real, phyllo, wtf, Ecmandu, etc. But we’ll have to find a set of rules that both me and the other participant are willing to abide by. The job of a moderator (you or anyone else) would be merely to check that the rules are respected and to intercept when violation is suspected.

obsrvr524 · February 11, 2021, 1:36am

No one can agree to rules until they hear them. Why don’t you propose some on the General Resolution Debate thread. We can talk about it.