[split] 0.999... equals 1

[Tiberius]

Wake up Saerules. He said the value isn't infinitely long, but the number is. If you don't know the difference between the two, you haven't been following the topic at all.

0.9r is a different numerical representation of the value 1.

Now technically speaking, all numbers are infinitely long. The number line is infinite, and the value 2.5 is actually:

...0000000000000002.5000000000000...

There are an endless number of 0s either side of the 2.5. Multiplying by 10 moves the decimal place to the right, dividing by 10 moves the decimal place to the left. Every number is infinitely long (in both directions) on the number line. The value isn't however.

[Violet]

Again, we aren't talking about infinite values, we are talking about infinite lengths. The fact that there are an infinite amount of 9s on the end of the number does not mean the number is changing value. The fact that it is infinitely long is a constant.

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).

Indeed, and yet you cannot have the infinity + 1th number of the sequence 0.9r, since it doesn't exist. The number 0.9r is the "last" number in that sequence (for sake of argument, since there isn't technically an end to the sequence, since 0.9r is infinitely long).

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.

Now you are talking about values again. Stop it.

How do you mean?

0.9r is represented as a 1 in the decimal system.
0.5 is represented as 0.49r in the decimal system.
Any other examples you'd like?

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

Please explain to me how you can even argue this!

1) 0.9r is an infinitely long number. yes.
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.
3) You argue that any infinitely long number is infinitely large. It is.
4) Therefore by 3 and 2, 0.9r is infinitely large, and cannot possibly have a number larger than it. See bold after 2
5) 2 is a larger number than 0.9r. It is
6) Therefore either 0.9r isn't an infinitely long number (which it is by definition), or your assertion that all infinitely long numbers are infinitely large is wrong. See above.
7) Since 0.9r is infinitely long by definition, your assertion is wrong. See above

We aren't placing it on the value, we are placing it on the length. The difference is so astronomically important I can't believe you still don't get this simple fact.

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.

Yes, but each way you use it means something different. A blue book isn't equal to an blue computer, neither is an infinitely long number equal to an infinitely large number just by definition. The only case this applies to is 9r which is both infinitely long and also infinitely large.
never end.

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.

---

Wake up Saerules. He said the value isn't infinitely long, but the number is. If you don't know the difference between the two, you haven't been following the topic at all.A number IS a value by the definition of number

0.9r is a different numerical representation of the value 1. Not well enough justified for me to accept yet... although I have no doubt that the mathematicians are probably right

Now technically speaking, all numbers are infinitely long. The number line is infinite, and the value 2.5 is actually:

...0000000000000002.5000000000000... Zero is the value of nothingness. No value changes by adding zeros.

There are an endless number of 0s either side of the 2.5. Multiplying by 10 moves the decimal place to the right, dividing by 10 moves the decimal place to the left. Every number is infinitely long (in both directions) on the number line. The value isn't however.

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.

[fr0d0]

9r is infinitely big. 0.9r infinitely grows towards the value '1'. 0.9r can't be infinitely big.

[Violet]

Endlessly big ≠ endlessly getting bigger? They would have to be equal, because otherwise there would be no such thing as Endlessly big

[fr0d0]

How can it endlessly get bigger if there's a bigger number? (0.9r < 1.1) There is a limit to it's 'bigness' so it can't be endlessly big.

[Meatball]

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).

Firstly, since when did this thread have anything to do with volume? The formula for volume of a rectangular prism is entirely irrelevant and the fact that you think otherwise shows you have a very poor grasp of math.

You were discussing limits earlier in this thread, so you should know that while adding values after the decimal DOES increase the magnitude of a number, it does not do so endlessly. It always approaches a limit as you add more.

This is embarrassing.

[Violet]

How can it endlessly get bigger if there's a bigger number? (0.9r < 1.1) There is a limit to it's 'bigness' so it can't be endlessly big.

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

Firstly, since when did this thread have anything to do with volume? The formula for volume of a rectangular prism is entirely irrelevant and the fact that you think otherwise shows you have a very poor grasp of math.

You were discussing limits earlier in this thread, so you should know that while adding values after the decimal DOES increase the magnitude of a number, it does not do so endlessly. It always approaches a limit as you add more.

This is embarrassing.

I was representing the concept of size with that formula... you could do the same with the volume of a sphere, cylinder, pyramid, whatever If you increase a number by any of our spacial three dimensions... then you have increased a numbers' overall size.

As far as I am aware... this is very much based on limits. When I say endlessly getting bigger, but doing so in smaller increments: the infinite limit is exactly what I am referring to I'm just rewording it (Much like many of you have reworded the same 'proofs' for the last 20 some pages?). As it approaches the limit, it doesn't stop moving forward... but if one was to graph it: the line would curve further and further away from the limit... as it continues to get endlessly closer. Simply: the more you add, the bigger the number gets. If you add endlessly, then the number will be endlessly big

I don't deny that it's embarrassing. It's like I'm the only person in class who raises my hand to ask questions because I do not understand what the teacher is saying. You guys all understand perfectly well... but I'm not convinced. I believe that you guys are right in .9^ = 1... but I just don't see why. I'm taking that equality on faith, and providing counterpoints to see why you guys believe it is an equality.

[fr0d0]

How can it endlessly get bigger if there's a bigger number? (0.9r < 1.1) There is a limit to it's 'bigness' so it can't be endlessly big.

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

That didn't address the question. You didn't show why getting endlessly bigger is the same as endlessly big.

[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.

[fr0d0]

I get it... it's an infinite subject

That's a mindblowing concept... it's a fixed value yet never ending.