[split] 0.999... equals 1

[Tiberius]

Didn't I just explain this? Infinite = infinite. It is the same concept. If something is endlessly long, then it will continue to increase in size, as you cannot increase length without increasing size (Which is L*W*H).

The fact that there are infinite .1s, .9s, or .14637281946827s doesn't change the fact that the value is endlessly getting longer (and as a result: bigger).

It's the same concept, but you can't just apply one instance of infinity to another. I used your own example to show that; a blue car is not a blue computer, although both are blue. An infinitely long number is not necessarily an infinitely large number, although both are in some way infinite.

I am equally as confused as Meatball with your assertion that size (did you mean volume?) = L*W*H, because not only is it completely irrelevant, it's not even true. It works for cuboids, and not much else.

The value isn't getting longer, the number representing the value is. The value is always constant, since we have not said it is changing (nor have we said the rate at which it is changing, ergo the rate must be taken as 0). You are thinking of the number as constantly adding 9s to the end, but this isn't the case...the 9s already exist in the number, since it is described by 0.9r. There are already an infinite amount of 9s in that number, and it isn't changing value.

Furthermore, if you do add values to a number repeatedly, you are decreasing the rate at which the value grows by a factor of 10 each time. If you tend to infinity, the rate tends to 0, and hence there is a limit to the value. This is why we invented infinite series sums!

An infinite number is still a number. oo + 1 is an expression like any other... but it is unsolvable because of the nature of infinity. If you thin about it, A + 1 is just as impossible to solve. However, by giving value to A... we can solve it. Infinity cannot be given a simple value though... because it is endlessly changing value. For example, take oo + 1 - oo = 1, because we cancelled out that which previously could not be solved. The number of oo + 1 exists... but it cannot be calculated.

An infinite number ∞ is defined such that there can be no number greater than it. ∞ + 1 is a number greater than it, since adding 1 to any number increases its value by 1. So either ∞ is not the largest number, or ∞ + 1 is an invalid concept.

Since the largest number is defined as ∞, the first cannot be true. Therefore ∞ + 1 is an invalid number.

Secondly, lets quickly look at the ramifications of ∞ + 1 if we allowed it to exist, because it reveals itself as meaningless. If you can add 1 to ∞, you could add 2 as well. If you could add 2, you could add 3...etc. If you could add any number, you could also add ∞, and get ∞ + ∞. By the same reasoning you used for your assertion of ∞ + 1, we could say you could also have ∞ + ∞ + 1. Then we could say we could have ∞ + ∞ + ∞, and so on and so on.

So the ∞ symbol loses all meaning we gave to it.

Again, I repeat myself when I say that the value ∞ isn't changing. It's just endless.

How do you mean?

I mean learn the difference between values and the numbers that are used to represent them.

They are not equivalent. I think we are discussing this?

You asked, I provided the equivalent numbers. You have failed repeatedly to show any proof that the numbers are not equivalent. All your proofs so far have relied on an almost complete inability to do simple mathematics, and a rejection of standard concepts and methods in order to continue to support your assertion.

2) By definition, an infinitely large number cannot have a number of a value greater than it. your misconception. If something is endlessly getting bigger... then it is endlessly big. .9 < .99 < .999 = infinite increase in size. It is endlessly growing bigger, and it is big without limit... therefore infinitely big. It is endlessly big, because it forever grows in size. That it never reaches an end means it is infinite.

Consider it also this way: When one says 'Infinity'... they usually are referring to a number of endless size. Endless ≠ greatest. That's how I can argue this.

Again, it gets bigger by a smaller amount each time. Tend to infinity and the rate of change tends to 0. It has a limit. This limit is 1 (in this case), and the sum of an infinite series can be calculated on the series of 0.9 + 0.09 + 0.009... as I showed a couple of pages back. The sum of the infinite series of 0.9r is 1.

You contradict yourself yet again when you say "It is endlessly growing bigger, and it is big without limit... therefore infinitely big" and then follow it up with "Endless ≠ greatest.". Your whole argument revolving around why infinitely long numbers are infinitely large is that they are endless, and so they are the largest numbers.

There is no difference, once again. You can put it on a color, on how rocky something is, on a lot of things... and the concept of infinity does not change.

So a blue car is the same as a blue computer?

The blueness of the computer is equal to the blueness of the book. The infiniteness of a length is equal to the infiniteness of a large number. Infinitely long denotes infinitely large, because size is Length*Width*Height, and something infinitely long Length^*Width*Height makes the size infinite.

Yes, the 'infiniteness' of length is the same 'infiniteness' of largeness. However this doesn't mean the two are equal things. That would be like saying a blue computer is the same as a blue car just because they are both blue.

Again, why you bring volume into this is beyond me. Let's try this again.

Every number on the number line is infinitely long (since all numbers can be represented as ...00000002.50000000...). What is not true is that an infinitely long number is infinitely large, since all of them are infinitely long, and we can have many instances of a < b < c.

To give some examples.

1r is an infinitely long number (infinite amount of 1s)
2r is an infinitely long number, yet is larger than 1r, and so 1r cannot be infinitely large.

What we can say is that the number 9r is equal to ∞, since it is infinitely long and there are no numbers greater than it. However 9r is the only number the fulfills this definition.

00000.250000 is the same number as 0.25, because nothing changes when you add nothing... you are adding no description other than there isn't further description. You could call the number .25 'infinitely finite' if you don't want to just simplify it to being 'finite'. The number is the same number, and because of that: its value has not changed.

Edit reason: Trying out responding in a different way, needed to fix a number of things.

ARGH!!! We aren't adding zeros!!! They are already there! The reason ...000000.2500000... is the same value as 0.25 is because we ignore all the 0s on the sides. Just because we ignore them doesn't mean they aren't there. They need to be there in order for us to multiply by 10, because otherwise we are moving the decimal point into nothing.

As I showed in my example: .9 < .99 < .999 < .9999, it is endlessly getting bigger in smaller increments

Please do the sum of the infinite series for 0.9r. I am begging you. Find the formula, do the calculation.