Here Free', I'll help them out a bit:

http://www.google.com/search?hl=en&lr=&client=firefox-a&q=0.999+%3D+1&btnG=Search

What...? I thought high school geeks would be right and university professors and professional mathematicians would be wrong.

.9999 (repeats forever) * 10 is not 9.9999

That's the fundamental flaw with the equation. You can't multiply by a number that never ends.

The "proof" is cool, but a shoe is a shoe.

I give up.

Then, I never tried too hard either.

False proofs prove it correct. HOORAY

Do you have some sort of random fact that spews forth false information from bodily orifices? Follow the links posted.

HELLO?! Is there a brain in there?

Pi is a number that never ends. We multiply by it all the time. Ever heard of (pi)r^2? 1/3 is a number that never ends. We multiply by it all the time. You are really a piece of work. Talk about not being able to admit when you're wrong. Your prime example is here, folks!

LonelyDagger: Give it up.

Dude, .99999~ can't be equal to 1. Proof?

1 = 1
.9999~ = .99999~

1 - .999999~ = .111111~

Haha. Owned. I win. Take that professors and experts in the field!

Excuse me? Are you braindead? You realize that .111~ > 1/10, no?

I think what you're looking for is 1 - .9999~ = .00000~ (after an infinite number of zeros, add a one on the end. oh wait....)

What a coincidence: Just like last time, this topic boils down to understanding the concept of infinity.

One thing people may fail to realize: How often do we deal with infinite quantities in real life?

Only in math in itself can 0.999.... = 1. To say that in the real world, you'd have to be dealing with something infinite, right?

Some of you guys seem to think numbers actually exist; probably because people tend to confuse numbers and quantities.

[/something useless]

How often do we deal with infinite quantities in real life? Never. The only thing that might be considered infinite is space. But to say that .999~ = 1, I don't understand why I'd have to be dealing with something infinite. They are equal quantities. 0.999~ = 1 = 1/1 = ((sin(x) - 1) / (sin(x) - (sin^2(x) + cos^2(x))) = (1.000~ with a 1 tacked on the end). There is no significant difference between a number and a quantity for the given problem. A number is a quantity, but a quantity isn't necessarily a number. But these are numbers..

You can't put a 1 on the end of the infinite digits. There is no end.

Exactly, that's yet another method we can use to prove that 0.999~ = 1. Because 1 - 0.999~, if you think about it, is 0.000~ with a 1 as the last digit.

But you forget that math is just a constuct, created by your so-called professors. Today I tear down that wall of tyranny with a proof of my own.


Note to self: include emoticons at the end of my joke. Apparently the reference to Emon's post wasn't enough.

Although the subtraction error was accidental. I suck at math.

You don't multiply pie out and write it in decimal form to get an exact answer though. You use pie as a constant and write 2(pi), if you write it out it's only an approximation.

.. see below

Yes. I don't know how to properly convey what I'm thinking, but I'm going to try.

We use infinite decimals to represent a fraction because they're much easier to use mathmatically. Many fractions, when converted to decimal, end up going on infinitely (like 1/3). Of course, the actual fractional quanitity does not go on infinitely. And that is what distinguishes quantities from numbers.

That's nothing even close to what I'm trying to say, but hopefully someone understands what I'm trying to say. :-\

BLEH.

Wouldn't it be more accurate to say that 0.99999... approaches one? Kinda like in graphing when a line continually approaches zero, but will never touch it.

how can you say that? "it never reaches(???) infinite"

there are no special dimensions or time continuoums in math (if there are, than show me). numbers ARE numbers, there is no "reaching" it, So 0.9¯ Is 0.999999999999999999999...(infinite nines). there ARE infinity nines. Like 1.0¯ with infinity zeros IS 1. I hope we don't need to write any proofs that 1.0¯ = 1.

also what who that mentioned: quantities != math numbers, said

.
edit: hit the reply with quote button instead of edit

Yes.

It's not a line, though. If you made a graph by continually adding nines after the decimal, it would approach .9̅ (and 1 as well, since they are the same number)

*cough*analogy*cough*

Last I checked, our number system was linear.

Your analogy doesn't make sense because a single number is not a line.

If you were to attempt to write .9̅ on paper in its entirety (without using anything to denote a repeating 9), the number on the paper would approach 1. By using something to denote a repeating decimal (such as the overline or ellipsis) the number written is at that limit.

If you attempt to write .3̅ in its entirety, the number on the paper would approach .3̅ and 1/3, but using the overline, .3̅ is equal to 1/3 - it doesn't approach it.

The approaching thing is how infinite decimals are referenced, like on a line. That was my point.

What do you mean by "referenced"?

Wow, interesting. I never looked at something that simple but contrary.

*applauds*

Can we all agree that 1/3 = 0.3~ and that 2/3 = 0.6~? If so, can we all agree that 0.3~ + 0.6~ = 0.9~? If so, this thread should have ended days ago.

1/3 = 0.3~
2/3 = 0.6~

0.3~ + 0.6~ = 0.9~
1/3 + 2/3 = 0.9~
1/3 + 2/3 = 1
0.9~ = 1

The whole point of an asymptote is that should you ever reach infinity the result will be whatever the asymptote tends towards. 0.9~ means to take the result from your graph where you've reached infinity.

Obviously on a finite graph it won't ever reach 1, but we're not talking about finite, we're talking about infinity.

You could quite easily just argue that 0 is an infinitely small difference between 2 numbers.

Damn you people. Why are you arguing this still? What you're saying is no better than saying the sky is red at noon on earth. LEARN. Don't say things that are false. No, it is not more accurate to say that 0.999~ approaches one. 0.999~ is one. Don't argue with me. You're wrong. It's a fact. There is no interpretation here. It's as much a fact that 0.9~ = 1 as it is that 1+1 = 2. If you think otherwise, you can think that, but you're wrong.

.9999999999 does not equal 1!!!!

I don't see how hard it is to understand when two numbers are right before you'r eyes. No matter how many nines, it comes close but IS NOT 1. Forget the equations and just look at the numbers, they are not the same!

(Correct me if I'm wrong :D)

There are plenty of posts that correct you...read them.

Yeah, okay... That's like saying that 1/3 != 0.333~ because they don't look the same. Though that's a bad analogy because if you don't think 0.9~ = 1, then you don't think 1/3 = 0.3~ either.

What is your point? You said that we can't multiply a number by 0.999~. Pi is irrational. 0.999~ is rational. If you can't multiply a number by a rational number, then how in the hell can you multiply a number by an irrational number?

(which in reality you can, of course)

Aside from that, what the hell is your point, really? Okay.. you're saying that if you write it out, then it's only an approximation. Umm.. well.. duh. Because it's impopssible to write out Pi. Computers that crank out a billion digits of Pi per second will never be able to accomplish that task. The question was can you multiply Pi by stuff, and the answer is of course! Whether or not you're multiplying by the perfect value of Pi with a symbol or an apporximation is irrelevant. The fact is, you were wrong. WRONG. WRONG!

We went through this a week ago.

0.99999... is the sum of the infinite series where the first term is 9/10 and the ratio is 1/10.
an = a1 * (r ^ n)
an = 0.9 * (.1 ^ n)

The sum of an infinite series is S = a / (1-r).
S = 0.9 / (1 - 0.1) = 1

If we want to prove this sum of an infinite series, there are a few ways. We could integrate our series function and use limits to find what it approaches as n approaches infinity. I can prove this summation formula.

Just know that n here has no relevance to our other function, it's just used to define this function.
S = a + a(r^1) + a(r^2) ... a(r^n)
Let's also write down a series rS where each term is multiplied by r. This will yield a formula which gives us the sum multiplied by r.
rS = ra + a(r^2) + a(r^3) ... a(r^n+1)

Subtracting S from rS:
rS - S = a - r^(n+1)
This function is equivalent to our sum times (r-1). Why? Remember that rS is r times our sum. If we call our sum x, rx - x = (r-1)x.

So if we want our sum:
S = (a - r^(n+1)) / (r-1)

However, as n approaches infinity, r^(n+1) will approach zero (because we're holding r to be less than 1, multiplying a number between 0 and 1 will make it smaller), so that term essentially cancels out.

S = a / (r-1)

There you have it. Not only have I proven that 0.9999... = 1 a few times, I've proven the sum formula used for it. If you don't believe this, at this point, I'm going to murder you.

This explaination proved it to me.

You have an ego problem. And your an *******, but mainly the former.

I agree, I may appear to have an ego problem, but it's a side-effect of the desire for truth. That side of me only comes to light when there's some misconception that needs to be cleared up.

And thus follows that:

.... =