They are equal. src @ http://mathforum.org/dr.math/faq/faq.0.9999.html "...The first thing to realize about the system of notation that we use (decimal notation) is that things like the number 357.9 really mean "3*100 + 5*10 + 7*1 + 9/10". So whenever you write a number in decimal notation and it has more than one digit, you're really implying a sum. So in modern mathematics, the string of symbols 0.9999... = 1 is understood to mean "the infinite sum 9/10 + 9/100 + 9/1000 + ...". This in turn is shorthand for "the limit of the sequence of numbers..." Read the whole page for more proof.

All numbers are infinite. For example, the number "1" is infinite. Why? Because technically, "1" is

"...000000000001.00000000000..." We just simplify it to "1." "0" is still a number, and it exists before, and after, all integer numbers. This number is a "normal" "1". It is off by "0."

"...00000000000.9999999999..." is not a "normal" "1". It is not a "1" at all. In fact, it's a "false" "1." Why? Because unlike the "normal" "1," it is not off by "0," (or "...000000000.00000000...") but is off by "...0000000000.000000000...1"

Notice how there is no "1" at the end of the infinite number of "0s," before the decimanl point, found in the number "0," which happens to be the amount the "normal" "1" is off by.

There is a "1" at the end of the infinite number of "0s," before the decimal point, found in the number "...000000000.000000001...," which happens to be the amount the "false" "1" is off by.

The number "1" has value, unlike the number "0." If the number "1" exists in the number "...00000000.0000001...," which the amount the "false" "1" (remember, which is "...000000.99999...") if off by, which it does, than the "false" "1" truly is "false," since a "normal" "1" cannot have any other number, besides the number "0," before and after it.

Just look at them:

...000000.99999...

1

Does this look the same to you? Does it?

0

0

That looks the same.

1

1

That looks the same.

How about this?

2

2

That looks the same.

How about this.

...00000000003.0000000000...

3

They may not look the same, but they are the same. Why? Because the only number in front of the 3, and behind it, is the number 0, which has no value.

This?

...00000000004.0000000000...
---------------------------------------
...00000000001.0000000000...

(That means 4 over 1)

4

Also doesn't look the same, but it is. Why? Again, eliminate the number 0, and divide 4 by 1. You get 4.

Let's go back to the first question:

...000000.99999...

1

Does this look the same? Does the number 9 exist anywhere in the number 1? Last time I checked, there is only an infinite amount of 0s before, and after the 1.

How the fuck do you get the number 9 anywhere in the number 1? I mean, the number 1 is the smallest number next to 0, and 9 is the largest number, before previous numbers are combined to form larger numbers (such as 10, which is only a 1, attached to a 0).

So how do you get the largest number in existence, in a 1? The answer is, you don't! Why, because ...000000.99999... is not a 1, since it is off by ...0000000.0000001... I don't care how many 0s are placed before that 1. At least the 1 exists.

I mean, take a look at the number 5, which can also be displayed as ...0000005.000000... Is there a 1 anywhere in that number? At the end of the infinite amount of 0s, is there a 1 anywhere? Nope. At the end of the infinite amount of 0s is... Yes... A ZERO! At the end of ...000000.000000...1, the number that ...000000.999999... is off by to equal 1, is a 1. Sure there is an infinite amount of 0s before that 1, but the 1 still exists, even though it is at the end.

Are these numbers the same?

...000000000.00000000...
...000000000.00000000...1

What happens when you eliminate all of the 0s?

1

That's what happens. Let's see that again.

...000000000.00000000...
...000000000.00000000...1

Minus the 0s.

1

Oh, look at that. I thought the 1 didn't exist. How come it exists when you remove the 0s?

Let's bring back the infinite amount of nothing, and the 1, lost, in the infinite amount of nothing:

...000000000.00000000...
...000000000.00000000...1

The top number is the amount the normal 1 is off by before becoming a normal 1, and the bottom number is the amount the false 1 is off by before becoming a normal 1. You saw in the previous demonstration that after you eliminate all of the 0s, a number of actual value still existed. This number is the only number found in the infinite amount of 0s, in the number that is the difference between the false 1, and normal 1.

Oh, and it gets worse from here. The only reason why people think that the number 1 does not exist is because there is an infinite amount of 0s. If that were true, would this mean? Yes it would!

...0000000.000000...3857349573489573948573948 also doesn't exist.

If the infinite amount of 0s is the problem, then would this mean?

0349580958340958...0000000.000000.... doesn't exist? This is a true, infinite number. Does this number not exist?

If all this is true, that would mean that 1 could be off by ...0000.0000...934857398, and still be a number 1.

Sorry, this is complete bull shit. 1 = 1. ...0000.9999... does not equal 1.

Done!

So basically, the theory is that any number, behind the decimal point, behind an infinite amount of 0s, does not exist. If this was true, then these numbers would be the same:

4.000...903457349857430598734

4

Last time I checked, there is an an infinite amount of 0s before the decimal point of the 4, and not this load of crap, "903457349857430598734."

Remember, every single integer has an infinite amount of 0s in front of the number, and behind the decimal place of the number, and no, there is no load of crap like this, "903457349857430598734" after the infinite amount of numbers.

If this theory was true, the next time you see a math question such as:

2+2=?

If all numbers after an infinite amount of 0s don't exist, you might as well write this:

...00000000004.000...903457349857430598734

instead of 4.

Or if you really want to mess things up:

...0000(...000.000...)00(...000.000...)00004.000...903457349857430598734(...00043.34054... divided by 1)343
-----------------------------------------------------------------------------------------------------------------------------------------------
...0000(...000.000...)0000001.0000000(...000.000...)000...4839(...00045.4556... times 45) 3485748458674707

to the power of ...0001.000...48957

times ...0001.000...454454545

instead of 4.

The conclusion: Absolute bull shit. 0.999... does not equal 1.

Hey everyone, I have ...0000000000999.00000000000...4594754898354353474 posts. Isn't that wonderful?

There is no other avatar best suited to match my posts of extreme logic rather than Eggman himself!

 Of course the problem with that notation is that the "903457349857430598734" part comes after the "...", which would mean it would follow an infinite number of zero's. But something can't follow an infinite amount of zero's, because it is inifite, it doesn't end, there's nowhere for the "903457349857430598734" to begin.

Yes, but to simply put it: Those numbers still exist!

By the way andamanen, you have the pride of knowing that my 1000th post, was made to your response. That's right! One-thousand posts!

Or more like...

...000...1000.000...094674867686786736345734560745 posts!

I'm coming for you, Happy Squirrel, and when I do, you wont be so happy anymore! In fact, you will have ...000.000...57930475307 happiness! Ha, ha, ha! The Eggman is so mean!

@ a.l.e.x59:

not saying that it's not educational to think of theories... but you could have spent all the time writing it up to just LEARN about it.

take it from someone who knows--you'll be greatly rewarded! 

i think you misunderstand, 0.999... IS a rational number.

a.l.e.x. -- Your incredibly drawn out theory is flawed because the instant you put anything after the ... like 0.99....1, you specifically reduced the INFINITE number of nines to A VERY LARGE number of nines. As soon as a series of decimal places becomes non-infinite, then the rules change. Kytiara - You talk as if limits are not an equality. They are, though. You can read it as "function equals something as x approaches infinity." What that means is that, when x reaches infinity, the function ACTUALLY EQUALS the "something." Not that it has gotten infinitely close. This is the concept behind a lot of calculus. For example, you can take the integral (area under a curve) for a curve that never reaches the x axis (like 1/x).