Is 1 = 0.999... ? Really?

Leibniz can still be felt everywhere today. In mathematics, in physics, in technology in general, in science in general, and not least in philosophy.

Actually, I didn’t want to address this rather general topic at all, but only address the topic of this thread (“Is 1 = 0.999… ?”) and just say that Leibniz, to solve a problem, used a trick, which was ingenious. This has had effects until today and will have effects also in the future, unless we will forget all science.

I was just pull’n your leg, mate.

But seriously do you agree that maths is actually just logic applied to quantities?

Mathematics is a subset of logic.

This is going to sound kinda silly…

Think about bijection with the convergence:

0.9 : 1

0.99 : 1

0.999 : 1

But for some reason, people use the concept of INFINITY to claim that 0.999… equals 1.

Actually, if you don’t take rounding into account (which is what this is), you literally have a number infinitely larger than the number 1.

Don’t let the decimal fool you.

1 is simple, finite.

0.999… can take the space of the whole universe!

So which one is bigger?

Maybe it’s like wave/particle duality.

Maybe not.

Just some thoughts.

That is what I meant.

So the issue here is about sticking to the logic when forming maths theorems.

And on a related note - do you understand the relation between Logic and God?

THAT is silly.

It is NOT “for some reason” but only for one specific reason why “people use the concept of INFINITY to claim that 0.999… equals 1”. And this one specific reason is a mathematical one (a mathematical-technical one), which is based on experience with the environment and one’s own thinking.

And certainly the whole thing has no resemblance to the wave/particle duality.

1 is bigger than 0.999… . But this equation can be used mathematically in order to solve a lot of problems, if not of most problems of all problems.

The number 0.999… can never reach 1 or will reach 1, when it is too late. If it could reach 1, then the equation would be redundant. 0.999… makes only sense, if it is different from 1. But mathematically, i.e. from the mathematic task and the result, it makes sense, because it solves a problem with the help of a trick. So one pretends that “1 = 0.999…” in order to solve something important - a problem. And it works. That’s the important thing about it all. In the non-mathematical realm, this equation is wrong. But that means it must be wrong in the mathematical realm as well. Right? That is absurd. Isn’t it? But if we think of the trick, which is mathematically allowed, if a problem can be solved with it, then at least it is sensible, sensibly correct. Right?

So again: It is useless to think about this equation or non-equation in the mathematical sense, because there is a trick which mathematics itself cannot solve, apparently not even logic, its superset. It can be solved only linguistically.

Do you mean “The Real God”?

Whichever.

I cannot agree with that. There is a logic.

I see the difference between Logic and God as the same as the difference between Truth and Reality - you can hide from one but never escape the other.

But eventually God will lead Logic (Reality will lead Truth) to find you. :smiley:

In this thread the idea of infinity allows people using wtf’s theory (as he said - not actually his) to hide the evidence (to allow committing the fraud). But “seek and ye shall find”.

For me, this means that actually nothing endures, because everything passes, cannot remain what it is.

I have noticed that you are impressed by James, but I do not know now whether this also applies to James’ concept of “The Real God”.

Read edit above. O:)

And really I think he should have said - “earnestly seek and ye shall find”. Disingenuously evade the evidence and you merely lose sight of the Truth - but still have to face Reality.

Nobody of us is saying there is no logic. There is logic in linguistics as well, although linguistics is more than logic. Logic is a subset of linguistics. (One can tell nonsense too).

We need to find the right wording. We have to get our wording right. The logical imperative (cf. Kant’s categorical imperative, which is meant rather ethically) could be: “Get your wording right!”

Yes, “seek” is the most right word in that wording.

Again read edit.

I know the logic (the wording). I am merely seeing if wtf (and Certainly real) is willing to be earnest and interested.

Yes, addition is commutative, so order doesn’t matter. (9 \times 10^{\infty - 1} + 9 \times 10^{\infty - 2} + 9 \times 10^{\infty - 3} + \dotso) is the same as (\dotso + 9 \times 10^{\infty - 3} + 9 \times 10^{\infty - 2} + 9 \times 10^{\infty - 1}). But that number is not the same as the one that you presented and the one that you presented is most definitely NOT (99\dot9). The most significant digit in a number is the leftmost digit. Conversely, the least significant digit is the rightmost digit. (999\dotso) has the most significant digit (the one it starts with) but it has no such thing as “the least significant digit”. Yet, your number does have such a digit. Your number is actually (\dotso999).

Like Certainly real, perhaps you confuse the representation with the reality - the value.

The values, even the one you denoted, are going to be the same regardless of how they are listed. I was objecting to trying to coherently justify (\infty-1). In maths that isn’t an identifiable number - so how is anyone to start from there?

The listed sum would begin -
(9*(\infty-1)) = ?
+
(9*(\infty-2)) = ?
+
(9*(\infty-3)) = ?
.
.
.

The number of digits in (999\dotso) is an infinite number. (That’s what “(\dotso)” indicates.)

A decimal number (d_1d_2d_3 \cdots d_n) were (n) represents both the index of the last digit as well as the number of digits is equal to the following number:

(d_1 \times 10^{n-1} + d_2 \times 10^{n-2} + d_3 \times 10^{n-3} + \cdots + d_n \times 10^{n-n})

Here’s an example:

(345 = 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0)

Thus, if we use (\infty) to represent any infinite number then:

(999\dotso = 9 \times 10^{\infty - 1} + 9 \times 10^{\infty - 2} + 9 \times 10^ {\infty - 3} + \cdots)

My point is that (999\dotso) is NOT the number you said it is. That’s regardless of what you think about (\infty - 1).

As for (\infty), in maths, it has the same meaning as the word “infinite”. Thus, if (999\dotso) has an infinite number of digits, which it does, then (999\dotso) is equal to (9 \times 10^{\infty - 1} + 9 \times 10^{\infty - 2} + 9 \times 10^ {\infty - 3} + \cdots).

How do you think mine was incorrect? I thought I stated to sum from (9*10^n) starting at n=0 to infinite. How is that wrong?

Yours was sum from (9*10^n) starting at n=infinite-1 to 0 (you really should have started with infinite, not infinite-1 but same issue).

I’m already on the defensive on this but will keep an open to try and understand your position. Immediate objection is: You cannot count to infinity. So how can you expand to the point of becoming infinite? I do not deny that x can expand or count endlessly. I deny that x can count to infinity or that x can expand to the point of becoming infinite (no matter how fast or long this process was happening). Do you see where I’m coming from?

Just as I think it semantically inconsistent to expand to the point of becoming an infinite, I think it semantically inconsistent to shrink to the point of becoming an infinitesimal. What separates these two infinitesimal items from each other such that no semantical inconsistencies occur?

I read the rest of your post and tried to understand, but could not.

I understand your defensiveness. That is where I would have started too.

But perhaps what you are missing is the issue of “counting”. We need not count the reality that an infinity of somethings entered an area for it to occur. We can perhaps note that there were only two and then note later that it had become infinite. The question would only be how we discovered the reality, not how we counted the items. And that is addressed in the following -

Let me give you one of James’ ontological examples -

In free space (free from mass particles) there is randomly directed and sizes of affectance waves - totally filling literally ALL space. That is what is between any and every identified items.

But in a small area of all of that there can be a limits number of “high peaks of affectance” - our “identified items” - 2 in this case - two high peaking waves - spikes.

But it is possible that an infinity of high peak spikes were heading toward that area before we noted those two. They literally surround the area in 3D.

As soon as those enter that area, our former 2 items (the spikes) become and infinity of items because they were already headed that way. All we had to do is wait and reexamine (I have no idea how we would physically accomplish that).

So as with the idea of a single point in space raising from a finite affectance value to infinite value being impossible (as in his picture) - a single counting cannot raise to infinity either. But it isn’t a single counting. We are not gating in the spikes one at a time. Literally an infinity of them enter together from an infinity of angles - a process of infinite counting occurring by reality itself - they just merge into the area (as shown in his picture).

Does that help?

Your number is actually a smaller one. It’s not so easy for me to explain why. But I’ll try.

Every whole decimal number can be represented in the following way:

(d_n \dotso d_3d_2d_1)

(d_x) represents a digit where (x) represents the index of that digit.

The above stands for this:

(d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^{n - 1})

The non-zero digits in YOUR number are digits whose index is greater than (0) but not greater than every integer.

The non-zero digits in (999\dotso) are digits whose index is less than (\infty) but greater than every integer.

Do we agree so far?

Let me know before I proceed.