Is 1 = 0.999... ? Really?

What makes you think that I cannot use categories of numbers?

What makes you think that (b) in (a^b) cannot be a category of numbers?

You think that exponentiation is limited to specific numbers? (Sort of like how wtf thinks it is limited to real numbers?)

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

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.”

From there, most of our argumentation would never have occurred - although perhaps you want to debate a “better” definition - but that is a different debate.

Where .333… is meaningful, it is hypothetically possible to divide the meter into 3 equal parts. But the equality of those parts must be expressed by saying that each part is .333…m long. .333…m long is either meaningful (therefore it has an exact meaning, which means it can serve as an exact answer), or it is not meaningful (which means we cannot divide the meter into three meaningfully equal parts). It would be like saying:

A: I’ve just divided x into 3 equal parts.
B: How big is each part
A: Unknown in terms of exact size
B: How do you meaningfully know they are exactly equal?

or

A: I’ve just divided x into 3 equal parts.
B: How big is each part
A: It is exactly 1/3 of the whole of x
B: How big is 1/3 of the whole of x
A: I can’t give an exact answer to this
B: Does an exact answer actually exist but you just don’t know it, or is it the case that no exact answer exists?
A: No exact answer exists
B: You are saying 1/3 of x is meaningful, yet you are also saying that there is no objective truth regarding what 1/3 of x actually amounts to in terms of size/measure. How can one of those three pieces of x not have a measurable value? If it has a size, it has a measurable value.

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.

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

That is not true.

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}).

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.

Do you think that you can Resolution Debate that proposal?

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?

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).

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

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

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”.)

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.

Who said I disagree?

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

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.

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

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 -

  • 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. :smiley:

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.

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.