If 0.9r and 1 are not equal, there must be another number between them.

What is the average of 0.9r and 1?

Also, from Cambridge: http://www.dpmms.cam.ac.uk/~wtg10/decimals.html

Well, that would be (1+0.9r)/2.

Irrelevant. This is personal blog, and person might have less idea about math, then I - about chinese pottery from period of Ming dynasty.




Read carefully. It says, that "A great many question or reject the equality, at least initially." There is a good reason for that, ultimately.
Why? Because, infinitesmall value does NOT equals zero. If it would, it would ruin big portion of mathematical analysis. Therefore, this claim is made simply to ease the understanding of "infinite approximation" concept. It`s much easier to say "equals", then explain, that it "Is closer, then any other number, but still does not equals.".




Both irrelevant. This is what scientists call "anecdotal evidence". Sorry, I don`t buy it just because "Doctor Tim" says it. He might be plumber, who haven`t even finished elementary school just as easily.



Now, this is more like it.

Let`s take 0.9(9) as an infinite series.

A1=0.9, An=An-1/10

Now, obviously, series CONVERGE to 1. It means, that they are forever approaching it, but do not become it.

1/3*3 = 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
therefore 0.9999999999999....999 is 1.

Exactly. Let's play with variables. X=1, Y=0. Let's use White instead of X, and Black instead of Y. If White=1, and Black=0, then 0.9r does not equal white because it has an infinitesimal amount of black.

[Edit - than != then]

0.3(3) converges to 1/3 just as 0.9(9) converges to 1.

Neither is actually equal.

If you look at the 9s as a finite series, then yes, it converges to 1.
But, uhh, it's not.

Ex-cuse me? So infinite series do not converge?

Ok, I know the proper explanations and that 0.999... = 1 but it is confusing when you try to actually do some maths with it.

For example:

X = 1 - 0.9999...

Y = 2 - 2*0.999...

Y > X. (or even we could say that Y=2X).

Um... And? This is an explanation, why 0.9(9) != 1

If 0.9(9) = 1, then X = Y

Screw it, let's just say that 0.9r and 1 are equivalent but not equal.

They can be used interchangeably without any loss of accuracy but the values of each are different. That's basically what the big Cambridge brains seem to be saying.

Alice Shade, your use of the first comma in your last post was definitely wrong.

Alien, yes. Wrong, no.

And no, again. Sorry for ruining peacemaking here, but irrational value can not be equated to the natural number. Approximated, yes. Converging, yes. But not interchangeable.

Those are different things. Perhaps, better would be to say, that for laymen, subtle difference between two is irrelevant, and in common-day calculations, 0.9(9) can be rounded to 1 without any noticeable loss.

Can I agree with everything you said in that post but change "equivalent" to "approximate"?

No, because no matter how many .999999999999999999999 (infinity) it will never actually become the number 1. However, it will obviously be so close that no human mind could fathom the difference, thus making this poll ridiculous.

NO NO NO NO NO NO NO NO NO NO.

GOD.

No.

Because I still know that 0.9... equals 1.

First let's look at infinite series. If a series converges to 1, it means that should we reach the infinite(th?) entry in the series the value will be exactly 1.

0.9... is not saying "here is an infinite series," it's saying "here is the result of the infiniteth value in the series". 0.9... is not a process it's the result and it equals 1.

Secondly, let's not get smart and go on about other numerical systems. In the standard set of real numbers, there are no nonzero infinitesimal values that are members of set. This means that your precious- infinitesimal-difference-is-still-a-difference argument is using a difference of 0. Now, if the difference between 0.9... and 1 is 0, wouldn't that make them equal?

There are so many different proofs that 0.9... equals 1, just pick one that you like the best.

I voted no because .999 repeating doesn't fit into memory and therefore I couldn't program my computer to prove it.

Also, becuase I know this looks stupid now.

I know .9~ = 1. I'm disagreeing with all the people that can't grasp why.

Oh, really?

I`ll have to note, that any series converging to certain value become infinitely close to it, but NOT it. This is definition of converging series.

Therefore, we can get as close to 1, as we went, but there will STILL be another number between any we can concieve, and 1.


P.S. YES. 0.9(9) ~ 1. But not precisely 1.

Yes.

My maths teacher convinced me with an explanation of the equation.

I'm still waiting for someone to accurately tell me what:

1 - 0.999... = ?

(Preferably from the ones who are trying to argue mathematical proofs, aka ones who say no.)

1 - 0.9(9) = 0.0(0)1

Wasn`t that obvious? ^_^

Sorry. Silly question = silly answer.

Okay, 1 - 0.9r = 0.0r1

Now imagine an infinite train track. This train track goes on forever. There is no end to the train track. However, there is a train station at the end of the train track. Is this possible?

That 1 at the end of 0.0r1 is that train station. It's not there. It can't be there. Thus, 1 - 0.9r = 0.0r

As for those of you who say 1/3 ~ .3r, do this for me. Remember how to do long division? Divide, by hand, 1 by 3. Here, I'll start it off. Now continue it for yourself. When do you think you will stop writing 3s? Never, right? So 1/3 = 0.3r, there is no approximation.

Exactly.

Silly question? Then why did you provide an answer that doesn't work? :p

Prove to me that you can tack a number onto the end of an infinite number of zeros after the decimal. :p

Of course it`s possible.

There is infinite number of carriages in the train, yes, but it still starts at the station`s semaphor, and ends at the station`s gate.

That ending 1 is the one sticking out of gate. Add as much of carriages, as you wish - your train is still confined to station by initial axiom.


Look, if you want practical examples, let`s look from other end.

Let us suppose, that we have some number, which has infinite amount of nines (0.9999999999999....), and equals 1. So, what happens, if I add another 9 at the end of it? It becomes bigger then 1, or what?

You won't ever reach that "station's gate". Ever. Ever. That's what infinity does.

As soon as you add an endpoint, it no longer has an infinite amount of 9s! You can't add a 9 on the end because you don't have an end because the 9's repeat infinitely.

You can either say that there aren't an infinite number of 9's, or you can say you can't add a 9 on the end, but you can't say both.

Also, where did this train carriage come from? That was not part of my analagy. Think only of the length of the train track. As soon as you add that station, it is by definition not infinite.

But I can always say, that no matter how much of infinity of nines you put, there`s always room for one more.

Therefore, it FOREVER gets closer, and closer, and closer - and CONVERGES to 1. Getting forever closer, but never crossing over. Because if it would then it would be more then 1.

And Monty? If what you said was true, then 0.9(9) would be equaling infinity, not 1. It`s not how many cars we can string, it`s how many small cars we can chop up out of finite value.

Wow, I just cannot believe the blatant ignorance here. Read the damn articles I posted until you understand, especially the wikipedia ones.

You do realize that there is no mathematical difference between something converging to 1 and equaling 1? When a series converges to a number the series equals that number.

Can I ask what's the smallest real number that doesn't equal zero? Perhaps this will better my understanding.

Show me one instance where someone said that 0.999 repeating > 1

Cause I sure haven't seen it. We are arguing if it EQUALS 1.

And of course you can say "no matter how much of infinity of nines you put, there's always room for more". Which is exactly why you can't state 0.0...1. The 1 might as well not exist because no matter how you look at it, you will never reach that one. EVER.

I still don't understand how you're arguing this. What you keep stating is that infinity terminates, which is totally, completely, mathematically wrong.

Ok, by points.

1) Converging is NOT equaling. If converging was equaling, then people would just say equaling, and not use converging as term at all.

2) That`s exactly my point. It can`t be more then 1, therefore, it`s forever less, then 1, approximating with each number, but never becoming. It can`t be equal, because there is NO definition of how big is infinity. So, there is always "bigger" infinity between ours, and 1.
Let me illustrate it like this.

Let me take 0.9(9), and 0.99(99). 0.99 > 0.9 - right? So, if we take infinite cases, 0.99(99) > 0.9(9) by induction. So, if 0.9(9)=1, then 0.99(99) > 1.

1/infinity

aka

1/0

aka

undefined

So, what will we get, if we`ll do 1-(1/inf) ?

P.S. Flawed. inf=1/0

0.99(99) = 0.9(9), so failure?

How so?

0.99 > 0.9
0.9999 > 0.99
0.999999 > 0.999

Continue, or it`s enough to see the pattern?

If there`s infinity of nines in 0.9(9), then only half of infinity of 99 in 0.99(99), for them to be equal. Ready to define half of infinity?

Absolutely. infinity / 2 = infinity. That is a mathematical fact. If you don't believe me, which I'm sure you won't ( :downs: ) look it up in a calculus text or ask a real professor.