[split] 0.999... equals 1

[Tiberius]

That is quite a nice proof

[Violet]

I saw the thread metaphorically explode... and was actually a little intimidated by it growing page after page after yet another page I'm back though . After reading a number of repeated statements (and a couple new ones), it seems the general consensus can be summed up by Evie's statement here:

So there is an endless end of 9s on the end of 0.9, and then you are suggesting there be a gap?? There isn't a gap, there logically can't be - it is endless it goes on forever, you can't put anything on the end of it to make it =1...so it is 1.

My question: where's the room for it to endlessly go on... if there isn't a space in which to do it? There's your gap.

It's endlessness... makes the number nonfinite (aka infinite). The state of something being finite, is that it has limits... By the very definition of .9^... it can never be finite, because it is limitlessly expanding. This means that that the gap between it and one is limitlessly getting smaller... if the number were to ever finally converge: then it wouldn't be infinite. It would be 1, which is finite. But it never converges, because it is forever increasing in size... but every increase is smaller and smaller.

I'll get onto the other things in good time... because I remain unconvinced after reading many of these pages. Anyway, what are your thoughts to the bold?

[Tiberius]

It's not limitlessly expanding; it's not expanding at all. It is a constant number, just infinitely long. 0.9r is a fixed value, it doesn't change.

Your question is meaningless, we are not talking about limits in reality (as in, the physical world), we are talking about infinity as a mathematical concept. Technically speaking, there is no space for any number, since numbers don't exist except in the mind. The concept of infinity is an endless supply of something, be it size or length.

There isn't an end for it at all, so you can't put a number on the "end" of it. Your misconception is that infinitely long numbers aren't finite values, when they are. 0.9r is infinitely long, but it has a finite value, since 2 is a larger value (and by definition an infinite value is the largest value). Your second misconception is that there is only one way of representing a value in the decimal system, when there are many ways of doing so.

[Violet]

If the runner never stops running... then he will always be running more than before.

How can you have something without limit... and not be expanding? You say it is endless... and yet you also think it is constant? That which does not end, does not stop going, isn't that the idea of infinite...? That it doesn't ever end?

.9 is less than .99, continuing to expand this brings you bigger and bigger numbers... while their increase is getting smaller and smaller. .9^ is a continual state of 'almost there's... in that it cannot converge with one lest it stop expanding.

You are adding a ridiculous definition to infinite. Infinity is the concept of never-ending. In example: an infinite supply of food... is not necessarily an ever-increasing mountain of food... it could easily be a single banana that continuously replenishes itself.

That banana could easily be smaller than another banana... but that banana will last you much longer than the other banana.

Another example of infinity: is my runner above. He is infinitely running (see, running without end)... and therefore he is always running more. Now say that he is running towards a gate... and he is always getting closer to that gate... but he never reaches it. Not in all of eternity and beyond does the runner reach the gate. Are you saying that the runner did not run infinitely? How can you suggest that he isn't always running more? He is running an infinitely long distance... and he is always running more. He cannot stop.

Neither can 0.9^. This is because to be infinitely long: is to be infinitely increasing in length.


If it is a misconception... then represent .9^ a different way in the decimal system. In fact, do something easier: represent .5 in a different way in the decimal system.

[theblindferrengi]

Now you are contending the value 0.9^ is infinitely large, and therefore not a constant.

It is a value, and on the number line, its distance from one is less than the planck length, therefore it is one.

Your infinite runner is analogous to a number that is infinitely large, and therefore does not apply.

[Violet]

An infinitely long number would be infinitely large. This is because it is continuously getting bigger. I am contending that it is not a constant, because infinites cannot have constant value.

Infinity is not a value... it is a concept. And it has the effect of making what it is applied to endless. When you place endlessness on a value: you make the value endless.

It applies perfectly to my point. Infinity is a concept... infinitely long, infinitely large, infinitely grave, infinitely dead, infinitely advanced... all of those are valid uses of the concept. It is an adjective... it modifies the meaning of that which it is defining.

In example, blue rock, blue sky, blue human, blue book, blue computer. The word blue modifies rock, sky, human, book, and computer. Infinity is no different. It is a concept, and not a number.


Practically, it is one. In the realm of mathematics however: it forever doesn't end. This is because numbers never end.

[theblindferrengi]

THIS IS NOT INFINITY, ITS BLOODY 1

The value .9^ is not infinitely long, to express it in standard form is however. It is a repeating number, and therefore not an irrational number. It is not a variable either, so it is not a function. It does not get continuously larger, not even by your logic either, as by your logic it just gets closer and closer to one.

And how does the concept of infinity have anything to do with adjectives?

[Tiberius]

If the runner never stops running... then he will always be running more than before.

How can you have something without limit... and not be expanding? You say it is endless... and yet you also think it is constant? That which does not end, does not stop going, isn't that the idea of infinite...? That it doesn't ever end?

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.

.9 is less than .99, continuing to expand this brings you bigger and bigger numbers... while their increase is getting smaller and smaller. .9^ is a continual state of 'almost there's... in that it cannot converge with one lest it stop expanding.

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).

You are adding a ridiculous definition to infinite. Infinity is the concept of never-ending. In example: an infinite supply of food... is not necessarily an ever-increasing mountain of food... it could easily be a single banana that continuously replenishes itself.

Now you are talking about values again. Stop it.

If it is a misconception... then represent .9^ a different way in the decimal system. In fact, do something easier: represent .5 in a different way in the decimal system.

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?

An infinitely long number would be infinitely large. This is because it is continuously getting bigger. I am contending that it is not a constant, because infinites cannot have constant value.

Please explain to me how you can even argue this!

1) 0.9r is an infinitely long number.
2) By definition, an infinitely large number cannot have a number of a value greater than it.
3) You argue that any infinitely long number is infinitely large.
4) Therefore by 3 and 2, 0.9r is infinitely large, and cannot possibly have a number larger than it.
5) 2 is a larger number than 0.9r.
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.
7) Since 0.9r is infinitely long by definition, your assertion is wrong.

Q.E.D

Infinity is not a value... it is a concept. And it has the effect of making what it is applied to endless. When you place endlessness on a value: you make the value endless.

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.

It applies perfectly to my point. Infinity is a concept... infinitely long, infinitely large, infinitely grave, infinitely dead, infinitely advanced... all of those are valid uses of the concept. It is an adjective... it modifies the meaning of that which it is defining.

In example, blue rock, blue sky, blue human, blue book, blue computer. The word blue modifies rock, sky, human, book, and computer. Infinity is no different. It is a concept, and not a number.

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.

[Violet]

The value .9^ is not infinitely long, to express it in standard form is however. It is a repeating number, and therefore not an irrational number. It is not a variable either, so it is not a function. It does not get continuously larger, not even by your logic either, as by your logic it just gets closer and closer to one.

Oh, it isn't infinitely long? Then it isn't endless (synonym), and doesn't equal 1.

And how does the concept of infinity have anything to do with adjectives?

Considering that it is one... a great deal. Infinity is the quality of being infinite. A quality is an adjective (A distinctive attribute). Being blue is a quality... just as being infinite is a quality. Infinity is often used to imagine a number endlessly huge.

It is also an adverb when used to describe another quality, such as infinitely small. Either way, it is just a descriptive word meant to convey something's limitlessness (endlessness).

[fr0d0]

Coo.. 9... *wonders in amazement*