Poll: Does 0.999.. equal 1 ?

[Kingsman]

You're asuming the function f(x)=x (because that's basically what we're talking about here, just a line of real numbers) is discontinuous?

The function was never stated to us, so I was ASSUMING that it was discontinuous. If the function's really nothing more than f(x) = x , well, that changes things. I didn't read this thread all the way through because I found that doing so on the Escapist is a good way to get ulcers, but if he mentioned that the function is f(x) = x, then I retract my original statement.

IF the function is discontinuous, however, then there is potentially a very, very important distinction between x being less than 1 and x being 1, which anyone who went through a Calculus class on limits can understand.

 

[Zukhramm]

You're asuming the function f(x)=x (because that's basically what we're talking about here, just a line of real numbers) is discontinuous?

The function was never stated to us, so I was ASSUMING that it was discontinuous. If the function's really nothing more than f(x) = x , well, that changes things. I didn't read this thread all the way through because I found that doing so on the Escapist is a good way to get ulcers, but if he mentioned that the function is f(x) = x, then I retract my original statement.

IF the function is discontinuous, however, then there is potentially a very, very important distinction between x being less than 1 and x being 1, which anyone who went through a Calculus class on limits can understand.

There was never any specific function mentioned, but if 0.999... != 1 the line of real numbers must be discontinuous since the limit and the value in the same point would be different.

 

[mps4li3n]

x=.9999...
10x=9.9999...
10x-x=9.9999...-.9999...
9x=9
x=1
.9999...=x=1
.9999...=1

The flaw with this is that you have one less than infinity 9s after the decimal in 9.999 so it would not be 9 but 8.999...1

OT they are not equal. The reason they always appear to be is that the difference is so negligible that it can be ignored. You could say .999... ≈ 1 but that is because it is approximation. Also 1/3 does not equal .333... it is just a common approximation like pi ≈ 3.14

Protip: Infinity minus 1 is still infinity, Subtraction does not make it a finite number

Hey, i wanted to say that...

Infinity minus anything but Infinity is always Infinity... also, plus and x and / ....

Just don't divide by Zero... that's how black holes get made.

 

[Winthrop]

By definition, it does (0.9... anyway, not 0.9999). There cannot exist proofs both for and against the same thing, so the proofs at that link are necessarily false; sadly I don't have time to pick over them in detail.

Your proof is much better than most of the other proofs. However when working with an infinite set one more 9 will always be appended to the end of the equation. So the number between it and 1 would continually be itself. If that doesn't make sense I am sorry I am less confident in this then the counterexamples of the other proof. I agree that proofs can not exist for both sides, however I have found flaws in the proofs saying they are the same and have yet to find them in those supporting it. I merely supplied those because you implied I had no proofs to back my claims. My statement that by definition they are not equal is actually what is up for debate here so I apologize for using that as defense.

 

[Winthrop]

I agree the proofs which work with statements like 1/3=0.(3) do not actually give new insights, but only show that the statement 1/3=0.(3) is equivalent to 0.(9)=1. However, I'd like to know what proof(s) you (or the mathematicians (what are their credentials by the way?) you know) can give that "1/3 is not .(3)".

My brother is a math PHD and his friends have degrees in math(not sure what level sorry). They are the ones I was referring to. I do not have access to any of them atm so I can not provide you with there proofs however some of them are similar to the ones on this page http://en.wikipedia.org/wiki/User:ConMan/Proof_that_0.999..._does_not_equal_1#Proving_that_1_does_not_equal_.9.._using_the_definition_of_number_sets:

 

[cahtush]

i meant 0.333 X 3, tnx for telling me.

 

[Winthrop]

Implying that the set of decimals will end 1 digit before the other set is also implying that either one will end at some point. Because they are infinite they will never end.

Neither has to end for them to be different sizes. There is an infinite amount of whole numbers and an infinite amount of integers, that said there are more integers than whole numbers. The same can be applied here.

 

[jakefongloo]

x=.9999...
10x=9.9999...
10x-x=9.9999...-.9999...
9x=9
x=1
.9999...=x=1
.9999...=1

Nice

 

[SomethingUnrelated]

Yes, it does, and there are a variety of ways to prove it (though I wont list them, the community here already has you covered from that aspect).

 

[mew4ever23]

Only if you round. Even though the gap between 0.99 repeating and 1 is so incredibly small that is usually matters very, very little, it's still not 1.

 

[Extragorey]

Sorry to end this discussion so abruptly, but 0.9 recurring equals 1. It's got many mathematical proofs.

 

[mjc0961]

No it doesn't. The gap is so infinitesimal that it hardly counts, but 0.9999 recurring does not equal one.

This is the correct answer. All other answers are wrong. Basic mathematics right here. You can round it up to 1 and not really screw up whatever it is you are doing, but it is still not actually 1.

Also, I feel this is relevant:

 

[Shirokurou]

If you're theoretical and abstractly talking... then of course not, why have different definitions of the same thing. 0.999 is not 1, just as 0.999999 is not 0.999998.
Also equals can be reversed, does 1 equal 0,999?

But if it's something domestic, like measuring your wang, then why not?

In simple math, anything over 0,50 is 1, and under 0,49 is 0.

 

[karplas]

I agree the proofs which work with statements like 1/3=0.(3) do not actually give new insights, but only show that the statement 1/3=0.(3) is equivalent to 0.(9)=1. However, I'd like to know what proof(s) you (or the mathematicians (what are their credentials by the way?) you know) can give that "1/3 is not .(3)".

My brother is a math PHD and his friends have degrees in math(not sure what level sorry). They are the ones I was referring to. I do not have access to any of them atm so I can not provide you with there proofs however some of them are similar to the ones on this page http://en.wikipedia.org/wiki/User:ConMan/Proof_that_0.999..._does_not_equal_1#Proving_that_1_does_not_equal_.9.._using_the_definition_of_number_sets:

I haven't read the page you linked completely (yet?), but from what I've seen so far the proofs of 0.(9) =/= 1 have their flaws pointed out already. Could you please specify a proof which as of yet seems completely sound?

 

[Dags90]

If you're theoretical and abstractly talking... then of course not, why have different definitions of the same thing. 0.999 is not 1, just as 0.999999 is not 0.999998.
Also equals can be reversed, does 1 equal 0,999?

If they aren't the same number then there should be a number between them, right? What number is .99.. < X < 1?

 

[Darth Crater]

By definition, it does (0.9... anyway, not 0.9999). There cannot exist proofs both for and against the same thing, so the proofs at that link are necessarily false; sadly I don't have time to pick over them in detail.

Your proof is much better than most of the other proofs. However when working with an infinite set one more 9 will always be appended to the end of the equation. So the number between it and 1 would continually be itself. If that doesn't make sense I am sorry I am less confident in this then the counterexamples of the other proof. I agree that proofs can not exist for both sides, however I have found flaws in the proofs saying they are the same and have yet to find them in those supporting it. I merely supplied those because you implied I had no proofs to back my claims. My statement that by definition they are not equal is actually what is up for debate here so I apologize for using that as defense.

I, in turn, apologize for my lazy dismissal of said proofs (while true, my statement made no further arguments to support my case). I'm not sure what you mean by the number between being "itself". There is never "another 9" added; it is, from the beginning, an infinite series. The number is not changed at any point after this is established.

In any case, the number 0.9... (hereafter called x) cannot be "between" itself and any other number. No matter the value of x, (x = x) is true, so (x > x) and (x < x) cannot be true. You assert that x < x < 1, which cannot be the case.

 

[Darth Crater]

Sigh, double post.

 

[Skratt]

The gap is infinitely small, but there is a gap.

Plus anyone who says they are the same is clearly a pretentious retard desperately trying to look clever to cover the fact that he has below average IQ and has a small penis.

So yeah >.>

Find the flaw in this logic

1/9 = 0.111111111111111...
0.111111111111111... * 9 = 0.99999999999999999...
Therefore 9/9 = 1 and 0.9999999999999...

hmm, I'll have to think about that.

Got it.

1/9 = 0.111111111....
0.111111111.... * 9 = 0.999999999999....
Therefore 0.999999999... = 0.99999999999

By way of:

1/9 * 9 = 0.11111111111... * 9

If we know that 1/9 already equals 0.1111111...
Then 1/9 * 9 = 0.999999999999...
So, 0.9999999... does not equal 1.



It's just like the x = 0.99999..., x is already defined as equal to 0.99999.....
So if 10x = 9.99999..., x still equals 0.999999...
So (10x - x) does not equal (9.9999... - 0.99999) it equals (9.9999... - x)
Therefore X does NOT equal 1, as by solving for X on only ONE side of the equation, you incorrectly CHANGE the value of x.

0.99999... does not, and never will equal 1.

Unless you round.

 

[mps4li3n]

If you're theoretical and abstractly talking... then of course not, why have different definitions of the same thing. 0.999 is not 1, just as 0.999999 is not 0.999998.
Also equals can be reversed, does 1 equal 0,999?

If you're talking theoretically, if they aren't the same number then there should be a number between them, right? What number is .99.. < X < 1?

Over here we'd spell it 0.(0)1 , except that it's not a number that can exist because you can put a 1 at the end of infinity because infinity has no end.

But if you really want to be a purist in math when doing 1+0.(9) you use 1.(9) as the result... i'm pretty sure most math teachers would at least look funny if you used 1+0.(9)=2, the more anal ones would even dock some points for it (hint: you use &#8771; instead).

 

[Shirokurou]

If you're theoretical and abstractly talking... then of course not, why have different definitions of the same thing. 0.999 is not 1, just as 0.999999 is not 0.999998.
Also equals can be reversed, does 1 equal 0,999?

If they aren't the same number then there should be a number between them, right? What number is .99.. < X < 1?

Theoretically there isn't such a number or else 0.99 would be called 0.98 and that mysterious number would in fact be 0.99
The logic and theory is that 0,1,2,3,4,5,6,7,8,9 are the only single-digit numbers with none in between them. That's the whole point to their order

Or in the problem of ".99.. < X < 1?" where X exists, we can assume.
.991 < .992 < 1