0.9999999...=1.0

It's a fact. Deal with it, bitches.

False, 0.9999.... is a limit, though for all intents and purposes it can be substituted with 1, it is not the same thing. There is an iota of difference. Example, an object can accelerate infinitely towards the speed of light but never reaches it.

>>2
Fail.
Not going to say why, just letting you know that you fail.

>>2
The Calculus taught me that as long as the function converges, the limit is a value that is the same thing as other values.  Hence, since 0.999... converges, it is equal to 1, which is the limit of its convergence.

A line has no area, yet is exists on a plane.  I see the two situations being fairly similar.  There is zero difference between a plane, and a plane with a line subtracted.  Hence, there is zero difference between 1, and 1 with an infinitesimal subracted.

>>2
Divide 1 by 3. You should get .333333... Multiply that by 3. You should get .999999... Under all circumstances, and I mean ALL circumstances, .9999... is 1.0. After all, zero in itself is the reverse of infinity. Where as infinity never ends, so does zero at being infintisimaly small. And for the record, it is possible to reach the speed of light. It is just not within human grasp, and very well may never be.

>>2
0.999... is a limit in the sense that EVERY real number is a limit (of a cauchy sequence of rational numbers), and 0.999... = 1.000...

>>5
1.) It requires an infinite amount of energy for a particle with mass to accelerate to the speed of light.
2.) The universe contains a finite amount of energy.

∴ It is impossible for any particle with mass to move at the speed of light.

>>6
0.999... isn't a number, 8 isn't a number, IV isn't a number, none of these things are numbers.  I think people are arguing over what symbols mean, not numbers.  Numbers are concepts unto themselves.  Obviously the symbols 0.999... and 1 aren't the same thing, because christ, look at them.  Not so obvious is what is meant by the symbol 0.999..., series or limit.  While it is not intuitive, the symbology of decimal fraction digits has come to be intertwined with the definition of real numbers as limits, and so on that basis we say that 0.999... and 1 are symbols for the same number.

>>8
It's not what the numbers look like that's the issue. It's what the numbers represent.

>>7
In the creation of the universe, matter had in fact traveled faster that the speed of light. The matter had reached it's destination before it's own image. Interestingly enough, it's because of this that the universe was created in 3 min rather than near instantaneously. Time distortion caused weird side effects.

>>9
Yes, but people see what they see, and if they aren't operating under the same idea, or under the formal real number definition, they aren't going to agree because what the symbols look like leads to their idea of what is meant.

If they aren't operating under the formal definitions and axioms, they should make up new terminology for their attempts to construct a new system and not confuse everybody by using the same words to mean different things.

When something is understood in general to mean one thing, the onus is on the person using the word to use it correctly, not the person reading it to cover every single possible thing that it could mean if the user was an idiot.

>>12
Hardly anyone learns or gives a shit about formal definitions and axioms, hence the disconnect.  When all is said, 0.999... is a shit way to represent the unitary quantity, as far as most people are concerned.  Face it, the formal definition of real numbers is not even close to common knowledge.

>>4


That's just not true.

There's a distinct topological difference between the x-y plane and the x-y plane with the line x=0 removed.

One's connected and the other isn't.

All of you, you fail. Asking if 0.999... equals 1 is pointless. There is no answer. Accept it, bitches. It is like asking "why does a spatial point have zero dimensions?"

It's like the square root of a million, we'll never know.

>>16
Wha?  The sqrt of 10⁶ is 999.999..., isn't it?

>>17 [i]did you mean the sqrt of 1⁶?

1/3 = 0.33333~
2/3 = 0.66666~

1/3 + 2/3 = 3/3 or 1

0.33333~ + 0.66666~ = 0.99999~

Ergo: 0.99999~ = 1

Q.E.D.

>>17


Still not replying to >>14 ?

>>20
Niggaz on my dick.

This has something to do with the geometric sum to infinity.

0.9999999... = 0.9 + 0.09 + 0.009 + ...

r(ratio of geometric progression) = 0.09/0.9 = 0.1
and a(the first term)

Sum of a geometric series approaching infinity is given as S=a/(1-r) if r<1, and S=a/(r-1) if r>1.

Therefore, given that r is less than 1...

S= 0.9/(1-0.1)
 = 1

Convergence of numbers is when an total infinite number of sums is a finite number. 0.9 + 0.09 + 0.009 + ... will be 1 as the number of the sum approaches infinity.

Grah. Correction.

*This has something to do with the geometric sum of a series approaching infinity as well.

*Convergence of numbers is when the total of an infinite number of sums is a finite number. Well, at least that is how I think of it.

>>22
sums of infinite series are based on the limit of the general formula for sums of series.

0.99999........ = ~1
0.99999........ != 1
GTFO.

>>24 (Am I doing this quoting thing right? I'm new.)

I guess it is healthy to state that as well.

Just showing another way of looking at it through something with the same basic idea. Limits.

>>25
brainwashed

>>27
dropped out

>>28
wets bed

First point of contention: 0.99999... is approximately equal to one.

Draw a number line and place exactly where 0.99999... is. If its approximately equal to 1, how close would you place it to the number 1? One may think that the number moves closer and closer to 1 but never reaches there; however this is not the case. Since 0.99999... is a fixed number and not a sequence/series, it will not move on the number line, which brings me to my next point.

Second point of contention: 0.99999... as a geometric sequence/series.

You're not adding (0.9)/10^n where n is 1,2,3... to this number an infinite number of times. That is nonsensical. The number 0.99999... is a concrete and fixed quantity, not a sequence. Since the number extends infinitely, it is EQUAL to its limit, not "approaching" it. The limit in this case happens to be 1.

[0.9999999.........] = 0
[1] = 1
GTFO now.

Check out

http://en.wikipedia.org/wiki/P-adic_number#Introduction

you damned dirty apes

>>30
Wha?

9/10 + 9/100 + 9/1000 + ... = 0.999...

Tell me again how that's not a SERIES.  It converges, and the limit is ONE.

>>33

Wha?

tell me how

>>4 " There is zero difference between a plane, and a plane with a line subtracted."

connectedness lol fail

>>33
The left hand side is a series. The right hand side is a number. Or textual representations of a series and a number, or whatever.

>>35
ah so, that is the question then, what does 0.999... mean to the reader?  (We already know what it means to the fairly advanced student of mathematics, so everybody shut the fuck up for a second.)  The reader's meaning will shape the reader's perception.  A series or a number?

I think there will be other perceptions on how to view the subject, one way or another.

Looking at it as a form of a series is one of those views. It is a form of series, anyways. ._.

>>35

Both are series of numbers. Their ratio of the the two consecutive terms works the same.

>>30

Huh, whut? On number two. What is with the adding 0.9/10^n where n is a positive integer approaching infinity thingy? Is this what you meant?(but if you want it to work in there, the value of n must be a zero to represent the first term and going towards a positive integer when going up to infinity): 0.999... = 0.9+0.09+0.009+...+0.9/10^n, but the last part that you wanted to add is really not necessary. Anybody with a mathematical inclination can see that it's a fixed series/sequence of numbers without the extra appendage.

0.99999... is not a solid and concrete quantity, in my opinion. Can you write the number down in its entirety? I certainly can't. It's a geometric sum too. A sum that works its way up geometrically in sequence where the number of terms is approaching infinity. 0.999... = 0.9+0.09+0.009+...+0.9/10^n(Really, I don't think it's necessary).

I don't know how to modify post, if there is a function like that in here anyways... so, yes. When the number of the geometric terms(or however you may call it) reaches infinity(which is the limit, as you said), it'll be 1 when summed together(a finite number), not approach 1. My apologies, I was too lazy to clarify myself, and I suck at explaining what I mean. I will get my terminology wrong many times.

>>37
If 0.9999... is a series, then so is e, and π, and 1 and every damn number in existence. Just because two things have the same value doesn't mean they can't be differentiated and classified based on their composition. All series aren't fourier series simply because they can be rewritten to be one.
There are no 'consecutive terms' in 0.9999...

These "proofs" are all flawed.

1/3 is not the same as 0.333...

0.333... only tends to a limit of 1/3 but will never reach it.

So 0.999... != 1

0.999... + 0....1 = 1

If 0.999... = 1 , then

 1 + 0....1 = 1
     0....1 = 0 , which is false

Proof by contradiction, 0.999... does not equal 1

Now where can I pick up my Fields Medal?