It's all about the symbols and their properties and understanding them in their singular format.

For example: 9 and 0 are synonymous in all forms of numerical operations, they are mirrors and also happen to exist on each side of the number line. Where 0+5 = 5. 9+5 = 14, and 1+4 = 5. It's called digital root addition. 9*4=36, 3+6=9. 0*4=0. They mirror each other and theirselves. The number 10 is merely an aggregation of the symbols 1 and 0. Now, in the human mind and it is observable as well, we can count "10" things, we just don't have a mathematical system that represents this aspect in 100% representative clarity. We do however have our imagination, that which is suffice enough to allow for a base 10.

The conundrum is not if .9999=1, because it obviously doesn't, and yet 1 can never be proven. The mystery is where is the proof of 1? So in a philosophical aspect since 1, or a closed and whole system can never be proven this results in its definitive properties being equivelent to .99999 or any other repeating decimal as an infinite can never be poven, or shall we say ever proven. Whereas ever proven and never proven are now synonymous. It would take infinity to ever prove a proof of ever, and a proof of ever attempting to be ever proved can never be proved in totality, only through infinity, and proof to us, so far, is a finite concept. So we arrive at a contradiction of proof, and proof being only a matter of faith regardless of how discerning the evidence may ever appear, and that my friends is 1 and infinity in a nutshell and an infinite nutshell.

The elaboration could go on but deep down I feel as many won't comprehend this lecture as it is, simple as it may seem to me.

Einstein said of mathetmatics: As far as they are certain they do not apply to reality, and as far as they apply to reality they are not certain. I think that sums this up.

On your 4th line down:

Where you go from

10x-x=9x to
9x=9

What you are doing is going from
10x-x=9x
9.999...-0.999...=9x

Now this would be all nice, except, that you have to remember that the quantity 9.999... was actually defined as .999...*10, so it does appear to repeat an infinite amount of times, but in reality it only repeats infinity-1 times. A very abstract concept, yes, but when you are working with infinities, this logic seems to be valid.

When you subtract the .999... quantity from the 9.999... quantity AS YOU DEFINED ABOVE (This part is important), the correct answer would be
9.999...-0.999... = 1.000...001.
At least I think that is correct. Since you defined the 9.999... quantity as 10*.999..., it therefore stands that there must be one less number to the right of the decimal than there is in the .999... quantity.

I think it all comes down to the this:

.999... * 10 = 9.999...990
instead of
.999... * 10 = 9.999...

This mistake was made in the second line when you did the multiplication.

This is why, as far as I can tell, Mathematicians prefer to keep numbers as fractions rather than write them out as decimals. You would NEVER see a mathematician write
2^.5 out as 1.4142. They leave it as 2^.5. Likewise, it stands to reason that they do the same thing for repeating rationals- they keep them as a fraction.

Hope that helps you- I know it gave my brain a twist for a little while.

[edit on 14-1-2008 by erkokite]

ok, infinity is not a number, it is a term, for something that goes on forever and never stops, so you cannot subtract or times or do any mathematics with infinity except if you are only labelling something as infinite.

you cannot say that 0.9r = 1-0.0r1, if you say 0.0r1 then you are saying that you lied previously that the 0 was reccuring, if it ends in a 1 do not lie plz as it is obviously not 0, unless you are saying 1 = 0

we are not approximating this reccuring number, we are saying that the number does go on for ever, not approximate.

if you end up with 1 = 0.9r, you cannot say that this equals is a lie, and that it is approximate, the fact it is an equals sign sorts that out.

i hope i have cleared at least some stuff up, and personally, to save confusion, we should change to a purely fraction based number system of soem sort.

I don't know if anyone tried this yet (haven't read all the way through,) but you can solve this using basic calculus.

9/10=0.9
9/100=0.09
9/1000=0.009

This is called a geometric sequence, where each consequtive number is multiplied by the same thing, called the common ratio (r) - in this case 1/10. Now, if we infinitely add all the numbers in this particular sequence:

9/10 + 9/100 + 9/1000...=0.9 repeating, correct?

The way to find the sum of a geometric sum is like so:

n
∑ a(r)^k
k=m

In k=m, m is our lower limit, the first integer we plug in for k in the equation on the right. n is the upper limit, where we stop plugging in numbers. a is our first number in the sequence, in this case 9/10, r
is the common ratio, in this case 1/10. The sigma is just a symbol meaning sum.

Now for our particular problem:

∞
∑ 9/10 x (1/10)^k
k=0

k=0 means that the first number we use is 0, and the infinity means that we keep plugging integers in for infinity - the next one would be 1, then 2, etc. If we do it out we get:

x=0 9/10 x (1/(10^0))= 9/10 x (1/1) = 9/10 =0.9
x=1 9/10 x (1/(10^1))= 9/10 x (1/10) = 9/100 =0.09
x=2 9/10 x (1/(10^2))= 9/10 x (1/100) = 9/1000 =0.009

Look familiar? If we continue this process infinitely, adding all the terms, we would get 0.999..., this we have already established. Now, there has been a formula around for centuries to show what the infinite sum would be for a geometric sequence whose common ratio was less than 1, which ours is. It is as follows:

_a_
1 - r

With me still? All right, last part. Now a is the first number in the sequence, ours is 9/10, and r is the common ratio, ours being 1/10. Here we go lads:

__(9/10)__
1 - (1/10)

1 minus 1/10 is 9/10, and:

(9/10) A number divided by itself most undoubtedly equals 1.
(9/10)

And there we have proof that 0.999... equals 1. Of course, one could always argue that a mathematical equation cannot encompass the concept of reality. However, I, and the spirits of all established mathematicians of the past, present, and future, would have laugh at you anyway.

But hey, we could be wrong.

...he...hehe...hehehe...

[edit on 1-4-2008 by 00Einstein]

This is an easy one to understand, you must not think complex or you will get lost. The simple answer is the correct one.

.9 can NEVER = 1

Everything on the RIGHT SIDE OF THE DECIMAL, is only a PART.

A fraction, a part, a piece, a section of.

.9 is 9/10ths of 1

The more numbers you add after the decimal, makes it SMALLER. What people don't understand is you can ALWAYS DIVIDE.

No matter how small something is, you can always divide it into smaller. (hence infinite)

Case Closed

[edit on 2-4-2008 by ALLis0NE]

maybe this was said. i'm lazy so i don't feel like reading every page...
_
.9 != 1

it approaches 1 and becomes infinitely close to it but never technically becomes 1 unless you're rounding.

Originally posted by an0maly33
maybe this was said. i'm lazy so i don't feel like reading every page...
_
.9 != 1

it approaches 1 and becomes infinitely close to it but never technically becomes 1 unless you're rounding.

Or this....

.9 < 1


The way I think of it. Is that a decimal point means something was chopped into pieces.

For example, I have 1 pizza. If I cut the pizza into 10 slices I have "1.0" pizzas. If I eat 1 slice of that pizza, I now have .9 of a pizza left over.

Now lets cut the pizza into a hundred slices, then I would have "1.00" pizza. If I ate 1 slize of that pizza, I would have .99 slices left.

tens, hundreds, thousands, millions, billions


The reason .9999 is infinite is because you are working with a really really really really small piece of pizza.



[edit on 2-4-2008 by ALLis0NE]

Originally posted by an0maly33
maybe this was said. i'm lazy so i don't feel like reading every page...
_
.9 != 1

it approaches 1 and becomes infinitely close to it but never technically becomes 1 unless you're rounding.

No, .9... = 1

If it's not equal to 1, can you tell me what number is between .9... and 1? Hint, there isn't a number between .9... and 1 because the two are equal.

it may be so infinitely close to 1 that it may be safely considered 1, but like i said, unless you're rounding it never becomes 1.

this is high school math. look up limits, graphs, calculus. you'll see that there are equations that when graphed will produce a curve that approaches but never becomes some number. that's what we're talking about here - 1 is just a specified limit for this discussion. the closest you can get to it is .999 repetend. it's not 1, it's inifitely close to 1.

you could also say that repeating zeros with a one at the end is not equal to zero. it may be infinitely small, but it's something. if you claimed it was equal to zero then it would be nothing, which is false.

[edit]

after reading that wikipedia link i see where you're coming from, but i'm part of that group mentioned that sticks to this:

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals.

i'll just agree to disagree on this one. =)

[edit on 2-4-2008 by an0maly33]

I know it can seem to be a hard idea to wrap one's head around.

There have been numerous proofs posted that show that .9... = 1. Here';s the one that got me to understand.

1/3 = .3...
2/3 = .6...

1/3 + 2/3 = 3/3 = .9... = 1

Originally posted by nataylor
I know it can seem to be a hard idea to wrap one's head around.

There have been numerous proofs posted that show that .9... = 1. Here';s the one that got me to understand.

1/3 = .3...
2/3 = .6...

1/3 + 2/3 = 3/3 = .9... = 1

You are WRONG. 3/3 = 1 not .9

geezzzz don't just forget what whole numbers are people..

Originally posted by nataylor
No, .9... = 1

If it's not equal to 1, can you tell me what number is between .9... and 1? Hint, there isn't a number between .9... and 1 because the two are equal.

Yes, the number between .9... and 1 = .9...9

You just add another 9 at the end.

case closed. wiki is wrong.



[edit on 2-4-2008 by ALLis0NE]

Originally posted by ALLis0NE

You are WRONG. 3/3 = 1 not .9

You misread what I wrote. I didn't say 3/3 = .9. I said 3/3 = .9.... The "..." is important.

Originally posted by nataylor
No, there is no such number as .9...9. The ... means the nines go one forever, so there's no room to slip in another 9 at the end. It's endless.

What do you mean there is no room? This is where your understanding of the universe ends, and mine begins.

YOU CAN ALWAYS DIVIDE, NO MATTER HOW SMALL.

I already know that your ... stands for infinity.

You asked, what number is between .9... and 1. The correct answer is technicaly STILL ".9..." but with an extra 9 at the end. I'm sorry you can't grasp that concept.

Let me explain further....


1.0 = 1 whole sliced into 10 pieces.

1.00 = 1 whole sliced into 100 pieces.

1.000 = 1 whole sliced into 1000 pieces.

1.000... = 1 whole sliced into infinite pieces.

---------------------


0.9 = 1 whole MINUS 1/10th of a slice.

0.99 = 1 whole MINUS 1/100th of a slice.

0.999 = 1 whole MINUS 1/1000th of a slice.

0.999... = 1 whole MINUS 1 infinitly small slice.

No matter what.. if there is a 0 before the decimal, IT CAN NOT EQUAL 1 WHOLE.

Damn no wonder we still use oil for fuel.

[edit on 2-4-2008 by ALLis0NE]

Originally posted by nataylor
You misread what I wrote. I didn't say 3/3 = .9. I said 3/3 = .9.... The "..." is important.

No I didn't misread it.

You said 3/3 = .9................................................................

YOU ARE STILL WRONG.

3/3 = 1 DUH!!


Please tell me, what do I have to subtract from 1 to get 0.9...?

Keyword SUBTRACT.



[edit on 2-4-2008 by ALLis0NE]

Originally posted by ALLis0NE
Please tell me, what do I have to subtract from 1 to get 0.9...?

Actually, that question proves my point. What you subtract from 1 to get .9... is 0. Because .9... and 1 are equal.