Is 1 = 0.999... ? Really?

I rejected a definition that mathematicians developed because they didn’t know how to properly deduce the meaning of certain category of expressions from existing definitions.

An analogy would be a man who due to his inability to calculate the result of “2 + 2” says “Fuck it, let’s just define it to be 10!”

When I say that an infinite product is this or that, I am not merely saying “This is a personal definition of mine”. I am actually saying “This is what the term means according to the standard mathematical definitions”.

If you think you are going by “standard definitions” then we disagree on that too.

That could only apply to convergent partials.
The limit of divergent partial products would remain undefined.

I agree with you. But I want to be charitable and allow extended reals so that we can call the limit of 1, 10, 100, 1000 … (\infty). But @Magnus will not even agree to that. So I’m at a loss.

Great. Then tell me what (10^\infty) means “according to the standard mathematical definitions,” and please make sure to reference those definitions. Because frankly I’m highly familiar with “standard mathematical definitions,” and you have not presented one yet.

Yes, but the question is can a circle be divided into 3 wholly equal parts? I know I can divide a circle into 3 non-equal parts. I am not certain if I can divide it into three wholly equal parts. I think it’s doable, and given pi, this is supposed to be doable. As far as I can see, if you accept pi, then consistency would have it that you accept 1/3 to = 0.333…

I don’t see how we can say that pi only contains a finite amount of information. I understand we can represent this infinite amount of information in a finite manner. But this is not the same as pi itself containing a finite amount of information. If pi consists of an infinity of digits, then that is an infinite amount of information because each number is a bit of information.

Where we can divide a circle into 3 wholly equal parts using pi, then pi is infinite. 1/3 is a finite part of something. Pi is more like a tool to do something with than a finite part of something.

I meant that it says that it consists of an infinity of digits.

See my above point on the representation of pi versus the value of pi.

I mean no offence by the following, just constructive feedback: I think you are too focused on representation and not focused enough on semantics.

I agree that you can get as close as you want. We were never in disagreement on this. And that was my point to you when you first said 1/3 = 0.333…

If you remember, I said that 1/3 cannot equal .333… because an infinity of 3s are impossible and an infinity of 3s are needed to fulfil the semantic of 1/3 as opposed to just get close to it. But then you made that point about circles and I reconsidered my position.

To emphasise, you say you do not need to talk about infinitely many 3s, but then semantically speaking you have not fulfilled the semantic of 1/3 have you? When the 3s are finite, you have only fulfilled close to 1/3. The only way you can say 1/3 = .333… is if you are saying that an infinity of 3s follow.

Certainly - 120° each slice = exactly 1/3 of a circle.

Still - you are confusing the digits involved with the value they represent. A very small number might be represented by an infinity of digits - that does NOT make the number infinite. And it does not stop anyone from dividing something into 3 equal parts.

The decimal system of listing numbers has issues - problems. Some values (such as 1/3) cannot be entirely and precisely represented with decimal numbers. The issue is only in the language being used (base_10 digits) - not with the real values.

Pi is a value that cannot be exactly represented by decimal digits ever. The value of Pi doesn’t care whether you try to represent it in decimal form. The value of it doesn’t change to match your estimations. The value is a little over 3.14 and always will be even if you list an infinity more digits in the effort to be 100% exact. Pi will always just be a little over 3.14.

Similar with 1/3. It doesn’t matter how many digits your want to throw together to try to represent it. It will always remain merely 1/3 or a little over 0.33. Add a billion more 3’s at the end and you do not change what 1/3 is at all. You only change the accuracy of your string of decimals.

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

The language keeps saying that 1/3=0.333… and 1=0.999… through series convergence and the many other techniques that have been demonstrated in this thread.

The results are completely consistent.

Take the semantic of ‘21’ and the semantic of ‘divide’ and the semantic of ‘3’ and the semantic of ‘equals’. What does 21 divided by 3 equal? The answer is not 21/3 because that is the question that is being asked. It is not the answer. The answer is the semantic of 7. 7 is the answer in its actual purest form.

Now, what is 1 divide by 3 equal to? The answer is either absurd or .333… (where … stands for an infinity of 3s)

You cannot say the decimal system of listing numbers has issues. It is the semantic of ‘one’ and the semantic of ‘three’ and the semantic of ‘divided by’ such that when you ask what is one divide by 3? the answer is either you cannot divide 1 by 3 or it is zero point followed by an infinity of 3s.

Yes but semantics dictate that either 1/3 is absurd, or 1/3 is 0.333…
It is semantically inconsistent to say that 1 divide by 3 is 0.333… (follow by a vary large but finite number of 3s).

These forms are interchangeable because they are equivalent. They refer to the same value. 21/3=7=7/1=700/100=49/7

I see them as having the same identical answer. The semantic of 7. I don’t see them as being semantically identical. As in 21/3 is not semantically identical to 7/1. Their answers are semantically identical. This does not make them semantically identical.

That’s part of the point and usefulness. You can move from one form to another. Gaining understanding and insight in the process.

Like

the cat = le chat

Let’s you move from the society of English speaking people to the society of French speaking people. And back again.

If you focus purely on the differences in the words, then you limit yourself.

Certainly real,

It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as “semantics”, “infinite”, “…”, and “infinity”.

I don’t have any doubt that this whole thread is nothing but arguments over words and symbols.

Would you agree that if the string of 3s in 0.333… actually got to infinity then 0.333… = 1/3 ?

I already covered this. There is no time or process involved.

There is no “got to infinity” or “getting to infinity”. It’s at infinity. It’s always equal.

That is all I meant. So if 0.333… represents the 3s being at infinity then you agree that 0.333… = 1/3.

Semantic = meaning
Infinite = that which has no beginning and no end
Infinity = that which no greater quantity can be conceived of

If I ask the question what is the answer to a third of 21 exactly, the answer I would get is 7. Not 21/3. Perhaps someone might say you will find the answer when you divide 70 by 10, but they are still saying you will find the answer after you do this division.

To convey what I’m trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that’s just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it’s .333…m or 33.333…cm

Are you asking the same question that I just answered?

I’ve already stated multiple times that 1/3=0.333… exactly.

But that doesn’t mean that you can’t divide the meter. It only means that you cannot say it using decimal digits.

Bend that meter into a circle an cut it at exactly 120 degrees and you have your 1/3 meter. Ask someone how long those portions are in centimeters and they cannot give you an exact number.