Poll: Does 0.999.. equal 1 ?

[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

 

[Zukhramm]

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

No. How did you come to that conclusion?

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

You're making a logic jump here that is incorrect.

The algebra may be correct, but the logic is flawed and the value of 9/9 is 1, not 0.99999...

If the logic is flawed but the algebra is correct that must mean that algebra itself is flawed. In that case, why has no one noticed until now?

 

[Darth Crater]

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).

Then they are the same number, by Dags90's logic, yes?

 

[the.chad]

the big issue here is our modern counting system uses a base of 10

In a discrete mathematics class i had at uni, my lecturer told me the Babylonians used a base of 60 instead of 10. so with their counting system, they could get more whole numbers from their fractions.

1/5 = 12
1/4 = 15
1/3 = 20
1/2 = 30

Possibly the most interesting thing I learned during my short time at uni =p

 

[Shirokurou]

OK, I'm a lawyer, so I can't say for math majors, but here goes...

Philosophy says... 99,99% is still not 100%. Because 100% 0r 1 is classified by it's completeness, so any 0.999...9 is still incomplete for it lack something.

Law says... 0,99 can be considered 1, unless the missing 0,01 alter the attributes of 1. Also see it your contract allows for minor inconsistencies. Otherwise sue the shipper.

Logic says... Both are used to denote different amounts, so they are not one and the same. Just as "a" does not equal "b" just because it is close to "b"

 

[Winthrop]

10x-x=3
9x=3
x=1/3
0.3...=1/3

By your logic, 0.3 recurring does not equal one third.

It does not it is just an approximation from what I learned in high school (ironically our teacher used sort of an inverse of your proof that .99... was = to 1.

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?

Try the definition of a number set one. Its not perfect but its pretty good. Also the last one is decent.

snip

You are completely right on second thought that does not make any sense. I have no counter example to the density proof but I know very little about it. I do not understand fully how it is possible since every number could in theory be set equal with it right? Ugh damn now I am confused. I suppose I always viewed the whole thing as a postulate rather than a theorem. I could not find good proofs that stated they were equal so I assumed they were not. You do have a pretty decent proof though.

 

[Skratt]

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

No. How did you come to that conclusion?

1/9 as a fraction = 0.1111111...
so that is the same as saying 1 divide by 9 = 0.11111...
So 1/9 * 9 is 1 divide by 9, multiply by 9 = 0.99999999


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

You're making a logic jump here that is incorrect.

Explained it poorly, my bad.


When I said that "The algebra may be correct, but the logic is flawed and the value of 9/9 is 1, not 0.99999..." I again, explained it poorly. The value of 1/9 is being changed halfway through the equation, and now that I look at it again, I am not entirely sure why.

 

[maninahat]

Yes, but that is where it deviates from real life.

Actually that is real life. If you're measuring distance up to an infinite point, you can't reach it. That's the point of infinite.

The best way of knowing whether a number can be rational is whether it can be converted to a simple fraction.

Remind me, what is 0.(9) as a fraction? Because 1/3*3 is 1.

Exactly. To convert a recurring decimal to a fraction, do the following:

x = 0.999...
10x = 9.999... (multiply both sides by 10)
9x = 9 (subtract line 1 from line 2, (i.e. 9.999...-0.999...))
x = 9/9 = 1/1 = 1 (reduce to lowest terms)

Mind boggling, isn't it? You said yourself that 1/3*3 = 1, even though you know that 1/3 is 3.333..., and therefore 3*3.333... should be 9.999...

...repeated posts on a subject that has an official answer and a differing scientific answer are always good for post boosting. See the Triple Point of Water, How Many Moons does Earth have, Is Pluto a Planet and others.

The official answer is the scientific answer in all those cases: The Earth technically only has one moon (the other, recently discovered bodies do not qualify as moons), the triple boiling point of water is 0.01 degrees Celcius at 6.1173 millibars, and Pluto is officially a "dwarf planet", not a "planet". I think by "official" answer, you mean "commonly accepted", which isn't necessarily correct at all. That was the point that QI tries to make.

 

[PxDn Ninja]

As a software engineer I can tell you this,

0.999... != 1.0;

However, depending on the system you are using, it's possible that indeed the two will be equal. It also depends on the current state of the system. I have seen our code say it was true on one run, and false on the other.

As for mathematics in general, no they are not equal.

As a computer scientist, I can tell you that what you're seeing is a floating point rounding error. In general mathematics, they are equal, but (being infinitely long) the series is impossible to represent in digital form.

I have my degree in CS as well, so very much aware of what the cause of that is in the computer environment. I was just saying that is the only way I can see the two being equal: when the floating point gets rounded off once you get to the end of your precision.

I don't see how in standard mathematics they would be considered equal. In proper math, everything is as is, and as such 0.999... != 1.0.

 

[Zukhramm]

1/9 as a fraction = 0.1111111...
so that is the same as saying 1 divide by 9 = 0.11111...
So 1/9 * 9 is 1 divide by 9, multiply by 9 = 0.99999999

But you're dropping the "...", how does the infinite number of decimals suddenly go finite?


How about this, try the summation of the sequence 9/(10^n) from 1 to infinity. If it converges to 1 then 0.999... = 1.

 

[The_root_of_all_evil]

10x = 9.999... (multiply both sides by 10)

You can't use an operand on a recurring decimal without accepting equivalency, because to do so would be to accept that the last figure of the recurring digit would become 0. Not a valid proof.

I think by "official" answer, you mean "commonly accepted", which isn't necessarily correct at all. That was the point that QI tries to make.

Commonly accepted 0.(9)=1 : Correct answer 0.(9)&#8801;1

 

[mip0]

Those who voted no simply didn't know the answer to this. Makes me feel like a whole bunch of those who voted yes didn't know it either :|
Can't blame 'em for voting though, I mean.. it IS a poll

0.9999...=1

You say you've had many in-depth conversations on this. Why didn't just ask a teacher? There's gotta be a high school somewhere not to far away, right? You should ask one, even if you've already gotten the answer from many of us. They might be able to explain it to you better (more in-depth) than just proving it mathematically.
glhf

P.S. Another way of writing "0.9999..." (1, derp) is to draw a straight line over the nines (and remove the dots). It means they're repeated infinitely. It's really an American way of writing it and you can find it on the TI calculators, my teacher didn't really approve of it and so I have no idea of how to make that symbol on a qwerty.
|| ___ ___
0.999=0.333*3=1/3*3=3/3=1

 

[maninahat]

10x = 9.999... (multiply both sides by 10)

You can't use an operand on a recurring decimal without accepting equivalency, because to do so would be to accept that the last figure of the recurring digit would become 0. Not a valid proof.

I'm not sure what you are saying as I am not familiar with most of these terms. Could you please explain in layman's terms? From what little I have gleaned from reading around the issue, I don't see why a recurring decimal would have to terminate at a 0 (i.e become finite) for it to be multiplied, and I haven't seen that criticism directed at the proof before. Could you provide an explanation, or direct me to an article that explains your argument?

I think by "official" answer, you mean "commonly accepted", which isn't necessarily correct at all. That was the point that QI tries to make.

Commonly accepted 0.(9)=1 : Correct answer 0.(9)&#8801;1

I'm not familiar with the use of a triple bar. What does it mean?

 

[Kinguendo]

Yes, some people have trouble understanding this so I explain it like this...

0.999(r) has no knowable end so think of it as taking a piece of pie away each time and every 9 represents another piece taken, eventually you will have eaten every single piece of that pie no matter how small thus you have eaten 1 whole pie.

when will eventually come ??? do you have a reoccurring pie?

It will come when all of the pie is gone... and then you get 1 whole. And no you dont have infinite pie or you would never get 1! T_T

Its just how you explain the concept of 0.999(r) being equal to 1.

 

[Skratt]

1/9 as a fraction = 0.1111111...
so that is the same as saying 1 divide by 9 = 0.11111...
So 1/9 * 9 is 1 divide by 9, multiply by 9 = 0.99999999

But you're dropping the "...", how does the infinite number of decimals suddenly go finite?


How about this, try the summation of the sequence 9/(10^n) from 1 to infinity. If it converges to 1 then 0.999... = 1.

It is quite likely that I don't fully understand an infinte number. They way I understand it, if:

3x = 2x +1
solve for x

Then 3x-2x = 2x-2x+1
1x=1 or x=1

So,

x = 0.999...
10x = 9.999...
10x-x = (9.999...)-x
9x = (9.999...)-x

If you convert x to the value originally defined, then 10(0.999...)=(9.999...)
So, 10(0.999...)-(0.999...)=(9.999...)-(0.999...)
Thus 9(0.999...)=9
Divide both sides by 9 and you get 0.999... = 1

However, isn't the equation on the right side being reduced incorrectly?

x = 0.999...
10x = 10(0.999...)
10x-x = 10(0.999...) - (0.999...)
9x = 9(0.999...)

It changes the value of x when you reduce the equation on the right. Are you supposed to do that? I know I have forgotten a lot of math, and I won't say I am corret anymore, but, where is the flaw in my example? In my example x = 0.999~ OR 0.999~ = 0.999~.

When you say it converges to one, meaning it's so darn close, it's practically 1, thus 0.999~ = 1?

Just trying to understand. Thanks for not flaming.