Is 1 = 0.999... ? Really?

It is a common maths symbol for bijection.

Is there a bijection between them?

“countable” does not mean having to increase the cardinality else every set is countable.

James explained that the use of his “inf(Variable inserted)” is an arbitrary choice made during a calculation. The variable “A” was merely his most common example - “infA = 1+1+1+…”. Any variable can be chosen. InfB can be any other infinite set of interest. They are not fixed labels for predetermined sets.

I chose that infB, during this discussion, was representing the even naturals. I could have chosen infQ.

For sake of keeping it clear, I suggest not using the word “infinite” in this discussion and replace it with “endless”. And when discussing the “very end of the endless series or sum” perhaps call that “infinity”.

And as wtf explained, we normally do not refer to the endless length of decimals of a number but rather the value those decimals represent. The value of 0.999… is very nearly 1.0 - a very far distance from being infinity (almost the exact opposite). As I tried to explain earlier - each of those right-of decimal digits gets increasingly smaller and when added we have an infinite sum of infinitesimal amounts which finally sums to a finite value.

The idea that a finite in the physical universe cannot ever become infinite is not actually true. Take this case -

In a small area in space there exists only 2 infinitesimal items (whatever they might be or be called). If it is also true that speeding toward that area are an infinity (infA) of other similar items that just happen to reach that area at the same time - viola - suddenly those 2 items are joined by an infinity of others.

So we go from merely 2 to infA+2 in merely moments.

James used that little jewel to explain how it is that particles, having an extreme mass density ever get formed. He had a pictorial of such an event converging at a single point in space -

I think his MCR = Maximum Change Rate. He was explaining why waves of affectance (energy waves) delay to the point of becoming stable particles of mass. With an infinity of converging waves from literally all directions of even an infinitesimal average value, an infinite value must be summed at the convergent point. But because the single value cannot grow from finite to infinite (as you stated), a delay in wave propagation occurs that ends up growing and creating a traffic jam of affectance known as a “subatomic particle” - made sense to me.

But if we are just talking about the total count within an area, that number can easily grow from a finite number to infinity under the right circumstances (no value summation at a single point required).

Gottfried Wilhelm Leibniz invented and founded the differential and infinitesimal calculus (1665) seven years earlier than Newton (1672), which suggests that Newton stole from Leibniz. Leibniz invented a multiplying machine (1673), Newton the reflecting telescope (1669). Newton also founded the laws of gravitation (1666).

Leibniz’ causal principle and final principle, his theorem of the reason and his differential and infinitesimal calculus led in the application to physical processes to the interpretation of the laws of nature as extreme principles (differential, integral or variational principles), the binary number system with the digits 0 and 1 (dual system) developed by him to the computer technology which he already initiated with his constructed calculating machine. Even more: Leibniz had an effect in all fields of knowledge and on all fields of science.

In the public it is always said: Leibniz invented and founded the differential and infinitesimal calculus 7 years earlier than Newton. Newton would have done it independently of Leibniz, but this is not true, because Newton stole from Leibniz.

Invention/foundation of the differential and infinitesimal calculus: 1665 (Leibniz) and 1672 (Newton). So Leibniz was the inventor/founder.
Publication of the differential and infinitesimal calculus: 1684 (Leibniz) and 1687 (Newton). So Leibniz was also the first publisher.


Yeah yeah, mate. But what’s he done lately?
You’d think after 400 years – I mean really.


That’s right. But the “equation” works mathematically, so mathematically said it is an equation (without quotation marks); but the “equation” does not work philosophically,so philosophically said it is either no equation or an “equation” (with quotation marks). Thus the problem is a philosophical one but not a mathematical one. And that means that it is discussable as a philosophical but not as a mathematical phenomenon. It works mathematically. It works!

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”?


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)