Poll: 0.999... = 1

[crudus]

Actually, it didn't :-/ That proof works.

It does. Between step 5 and step 6 it divides both sides by (a-b). a=b therefore (a-b)=0.

[There is an even simpler proof.

1/3 = 0.333...

1/3 + 1/3 + 1/3 = 3/3 = 1

But 0.333... + 0.333... + 0.333 = 0.999...

Hence 0.999 = 1

If I remember right you are almost begging the question there. You are using the concept to prove itself. Again, don't remember the exact reasoning behind it. So far the most satisfying reason I have seen was the infinite series proof.

 

[Snarky Username]

There is a pretty standard proof for it. I can't remember what it was because I wasn't a math major in college and I had it explained to me once. I found Graham's number more mind blowing that .9999... being equal to one.

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

Every math major I have talked to and showed that to has described that as "shady".

Whenever someone puts that up to me, I point to this shirt:

a-b=0

Can't divide by 0.

OT: Your problem is with infinity. Let's try this with just .9.

x=.9
10x=9
10x-x=8.9
Which implies nothing.

Because anything to an infinite variable can not reasonably exist, and infinity is the basis for your proof, I say that your equation is not accurate.

 

[Rubashov]

Whenever someone puts that up to me, I point to this shirt:

Spoiler

It's funny because that shirt divided by 0.

Actually, it didn't :-/ That proof works.

The shirt divides both sides by (a - b). Because a = b, (a - b) = 0. Thus, the proof is invalid.

 

[Naheal]

You know, I'm not sure that makes true logical sense. The thing here is that you have 0.9999... and one, and math decides it need to jump through some hoops to say that they equal each other. Well, in the strict sense, it is CLOSE ENOUGH to consider it that, but every unyielding calculator in the world says you're wrong, because they don't think in terms of 'close enough'. They think in terms of 1 = 1. Inventing a whole set of equations to prove that curve that never really touches 1 IS actually 1 is essentially like continuing to count towards infinity in the hopes of actually getting there, which you can't.

The infinite loop of the x/y fraction is one that is just rounded off whenever it becomes no longer relevant. We do it all the time, actually. Even with "terminating" fractions.

 

[SturmDolch]

Actually, it didn't :-/ That proof works.

Third last step, at 2a(a-b) = a(a-b). a = b, so dividing by (a-b) is impossible. Yet they still do. So no.

The third last step is still valid, because a-b = 0, so both sides equal 0.

You could just as easily say

532531521215a(a-b) = a(a-b)
532531521215 = 1

which is equally as wrong.

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

You're doing it wrong, too. At step 4, 9x = 9, that is not true.

9x = 8.99999999



Edit: I'm not even a math major, but the amount of ignorant false intellectualism in this thread is about to make me cry.

 

[Naheal]

*cut*

*snip*

*slash*

Right. I always screw that shit up. Thus drunk deriving.

 

[SimuLord]

2003 called, it wants its math meme back.

 

[FalloutJack]

You know, I'm not sure that makes true logical sense. The thing here is that you have 0.9999... and one, and math decides it need to jump through some hoops to say that they equal each other. Well, in the strict sense, it is CLOSE ENOUGH to consider it that, but every unyielding calculator in the world says you're wrong, because they don't think in terms of 'close enough'. They think in terms of 1 = 1. Inventing a whole set of equations to prove that curve that never really touches 1 IS actually 1 is essentially like continuing to count towards infinity in the hopes of actually getting there, which you can't.

The infinite loop of the x/y fraction is one that is just rounded off whenever it becomes no longer relevant. We do it all the time, actually. Even with "terminating" fractions.

Yes, I know we do that, because we as humans CAN. But if we program the rules of math into a machine - which we have - then the machine will literally produce a number that is less than one FOREVER. The fact is that math is yet to be a perfect animal. It can only carve out so much of the universe into something understandable. But because we, as humans, made it...then we believe it logical to take the shortcuts which actually bend its rules a smidge. That's basically how it really is.

 

[Feste the Jester]

1/3 = .3333...

therefore 3/3 = .9999... = 1

 

[quhouv vhqwiufv qwiu]

2003 called, it wants its math meme back.

This has nothing to do with any meme. I'm not some 4chan idiot. I want to see how many people reject the concept.

 

[crudus]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

You're doing it wrong, too. At step 4, 9x = 9, that is not true.

9x = 8.99999999

I think you are going to have to explain that one.

 

[Naheal]

You know, I'm not sure that makes true logical sense. The thing here is that you have 0.9999... and one, and math decides it need to jump through some hoops to say that they equal each other. Well, in the strict sense, it is CLOSE ENOUGH to consider it that, but every unyielding calculator in the world says you're wrong, because they don't think in terms of 'close enough'. They think in terms of 1 = 1. Inventing a whole set of equations to prove that curve that never really touches 1 IS actually 1 is essentially like continuing to count towards infinity in the hopes of actually getting there, which you can't.

The infinite loop of the x/y fraction is one that is just rounded off whenever it becomes no longer relevant. We do it all the time, actually. Even with "terminating" fractions.

Yes, I know we do that, because we as humans CAN. But if we program the rules of math into a machine - which we have - then the machine will literally produce a number that is less than one FOREVER. The fact is that math is yet to be a perfect animal. It can only carve out so much of the universe into something understandable. But because we, as humans, made it...then we believe it logical to take the shortcuts which actually bend its rules a smidge. That's basically how it really is.

Most modern computers will do it automatically in the language, so no extra programming is necessary. If you're not specific with the machine, there's a chance that it won't even give you a decimal point anyway, so it isn't a big deal.

 

[Rubashov]

Actually, it didn't :-/ That proof works.

Third last step, at 2a(a-b) = a(a-b). a = b, so dividing by (a-b) is impossible. Yet they still do. So no.

The third last step is still valid, because a-b = 0, so both sides equal 0.

You could just as easily say

532531521215a(a-b) = a(a-b)
532531521215 = 1

which is equally as wrong.

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

You're doing it wrong, too. At step 4, 9x = 9, that is not true.

9x = 8.99999999



Edit: I'm not even a math major, but the amount of ignorant false intellectualism in this thread is about to make me cry.

It's true that 9x = 8.999..., but since 8.999... = 9, the proof remains correct, strictly speaking. However, one could say that the proof thus relies on the very concept it aims to prove, thus committing a logical fallacy.

 

[quhouv vhqwiufv qwiu]

To quote a math major who is a friend of mine:

I'm letting a_n = .99...9, with n 9s. So a_1 = .9, a_2=.99, etc.

Consider the closed interval [a_n, 1]. I want to show that .999...=1, and so I'm going to show that as n goes to infinity, there is exactly one number in this set. Since 1 is obviously in it, and .999...= a_n as n goes to infinity, this will imply the equality.

Note that for n < m, a_n < a_m. So, a_m is in [a_n, 1]. It's clear then that [a_n, 1] as n goes to infinity is the same thing as the intersection over all n of these closed intervals. That is, I'm going to only look at numbers that are in every such closed interval.

Set A to be equal to the set of all numbers in every such interval. Since 1 is in every interval by definition, A is not empty. If A has more than one point, then the diameter of A is larger than 0. But a_n --> 1 as n--> infinity, and diameter of A is smaller than the diameter of [a_n, 1].

So, we end up with the inequality 0 < diam(A) < diam( [a_n,1] ). The left and right sides go to 0, implying that diam(A) does as well. This contradicts the assumption that two points are in A.

But note that for any n, .999... > a_n, and so .999... must be in A as well (since it's in every interval). Thus, .999...=1.

 

[zfactor]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

Uh, wait, the second part will have one less 9 after the decimal point than the first part.

x = 0.9999999999
10x = 9.999999999

So the 10x - x would actually equal 8.9999999991, not 9.

Back on topic, .999999999999999999999999999999999999999999999999999999999999999999999999999(and so on) does not equal 1 because 1 is 1.0000000000000000 not (above .999...). They are two different numbers which can be rounded to each other, but are not equal to each other.

 

[SturmDolch]

I think you are going to have to explain that one.

I'm not going to try to prove something I have no clue about, as way too many people have tried before me.

But it probably has to do with subtracting infinites, which isn't possible with rational numbers.

 

[enriel]

Double Post, sorry

 

[FalloutJack]

You know, I'm not sure that makes true logical sense. The thing here is that you have 0.9999... and one, and math decides it need to jump through some hoops to say that they equal each other. Well, in the strict sense, it is CLOSE ENOUGH to consider it that, but every unyielding calculator in the world says you're wrong, because they don't think in terms of 'close enough'. They think in terms of 1 = 1. Inventing a whole set of equations to prove that curve that never really touches 1 IS actually 1 is essentially like continuing to count towards infinity in the hopes of actually getting there, which you can't.

The infinite loop of the x/y fraction is one that is just rounded off whenever it becomes no longer relevant. We do it all the time, actually. Even with "terminating" fractions.

Yes, I know we do that, because we as humans CAN. But if we program the rules of math into a machine - which we have - then the machine will literally produce a number that is less than one FOREVER. The fact is that math is yet to be a perfect animal. It can only carve out so much of the universe into something understandable. But because we, as humans, made it...then we believe it logical to take the shortcuts which actually bend its rules a smidge. That's basically how it really is.

Most modern computers will do it automatically in the language, so no extra programming is necessary. If you're not specific with the machine, there's a chance that it won't even give you a decimal point anyway, so it isn't a big deal.

Well, I didn't mean it like with the programming of the shortcut itself. I meant a machine programmed to pursue the number to its straight logical conclusion. Still, this was never the point. I adopted this line of thinking because the OP wanted to know what I think. And what I think is that the rules in math are largely accepted, but I'm not quite sure if they'rte intrinsically right. That's all.

 

[Singularly Datarific]

In this case, you have an infinite number of decimal places, all filled with 9's.
This means, at each place, the possible difference is 10x less, so an infinite number of places will equate to an infinitely small difference, meaning zero. There is an infinitely small difference between 1 and .999..., so 1 = .999

 

[enriel]

.9(infinite) = 1 because my brain can't reasonably comprehend the difference between the two numbers.

This isn't a math puzzle as much as a logic puzzle. What's being invoked is a Zeno's Paradox. .9(infinite) gets as close as possible to 1 without ever actually being 1.

Therefore, technically, no, they are not the same number. However, for simplicity's sake they may as well be, since nobody can really understand an infinitesimally small difference.