48÷2(9+3) =

[rumborak]

The difference is that Pi is an irrational number which cannot be represented as a ratio of two whole numbers.  A repeating decimal is still a decimal.

My point is that neither 0.9999... nor 3.1415... describe through their notation the number correctly. Using a flawed notation to prove something leads to a weak argument. From Wikipedia:

William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.[2] Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".[3]

Some may argue that, but the proof is meant to show that they are in fact equal to each other.  

I am not disputing the truth of the statement. I'm saying the proofs need to be better than just the superficial proofs mentioned above.

rumborak

[Jamesman42]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

I agree that it is more of a definition, but these "proofs" are a good way to see why it's true.

Then what does 1/9 equal in decimal form?

0.111... but if that's the case you might as well cut out the first two steps in your proof. I dunno. I feel like it's cheating. I'm actually not an expert.

Eh, I just like to start with 1 = 1 in those proofs, it makes it more elegant to me. That's just me, though.


Edit: This thread has been burned. Also I put "nonrepeating" facepaaaaalm

[Implode]

And it's all my fault! 










()

[lonestar]

You guys have gone over my head so many times in this thread, I have no hair left up there.

[Jamesman42]

You guys have gone over my head so many times in this thread, I have no hair left up there.

One divided by zero equals infinity.

[Nic35]

How is it possible

[King Postwhore]

You guys have gone over my head so many times in this thread, I have no hair left up there.

Maybe if they explained in tbsp and tsp.............

[Implode]

 

[glaurung]

You guys have gone over my head so many times in this thread, I have no hair left up there.

"Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo." is a grammatically valid sentence.

[lonestar]

You guys have gone over my head so many times in this thread, I have no hair left up there.

One divided by zero equals infinity.

[rumborak]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

[lonestar]

[glaurung]

[lonestar]

Huh.  Didn't know we had that one.

[blackngold29]

I also posted this on facebook. Got two people that said 2, posted it step by step showing how to get 288. And then someone else answered 144.

[lonestar]

OK, last one of the night...

[Jamesman42]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

What does your calculator tell you then? I'm not getting your logic here.

[kári]

What they do in Germany a lot is to have the third digit be slightly different:

holy shit. and i thought gas was expensive here.

and in europe, they go by the liter right? which is less than a gallon.

Those prices are outdated.. Diesel is € 1,45 or so at the moment, and gasoline is about 1,65... That's $7,858 and $8,672 per gallon.

[jsem]

The reason gas is pricier in most european countries is because of taxes.

[kári]

Well of course... But that doesn't make it any better or does it?

Only DTF could argue four pages in six hours about a fucking math problem.

This thread reached 27 pages in 2 days on some other forum and was locked because there was too much flaming etc...

[kári]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

How about
x = 0.99...
10x = 9.99...
9x = 9.99... - x
9x = 9
x = 1

[XJDenton]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

Forgive me, but can't you arrive at 1/9 = 0.1... through simple long division?

[kári]

I think rumborak's point is that 0.11... isn't a correct notation for what you get when you divide 1 by 9. Or something like that.

[XJDenton]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

[RuRoRul]

Well this thread has moved on quite a bit from when I saw it yesterday...

All I would say is (and all this has probably already been said) this is just poor notation, it should be written using fractions rather than the ÷ sign as it leads to the ambiguity.

Multiplication and division have the same priority, even though the common acronym to remember the order might put them one in front of the other (BODMAS or PEMDAS). But this is just a way of remembering it, there is no rule saying a direct multiplication should go before division because they are essentially the same thing. Going left to right technically leads to 288 as it is 48 * 1/2 * (9+3).

However, intuitively I would be inclined to say that the 2(9+3) is supposed to be grouped together because of the way it is written. The 2(9+3) represents (9+3) with a common factor of 2 taken outside, which is 24. Then it becomes 48 * 1/24 which is 2. But I see in this thread people saying that the 2(9+3) notation isn't accepted as a form of grouping any more than 2 x (9+3), so using a left to right convention would lead to 288.

[Jamesman42]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

This is my belief as well. I understand that the proof may have some shakiness to it, but it still works to support the definition that 0.999... = 1.

[YtseBitsySpider]

B - brackets
E - exponents
D - division
M - multiplication
A - addition
A - subtraction


BEDMAS.
Learn it.

[Xanthul]

48/2(9+3) = 24 * 12 = 288
48/(2(9+3)) = 48 / 24 = 2

That's what I've been taught and what I've seen in any math book I've ever used/read.

[RuRoRul]

B - brackets
E - exponents
D - division
M - multiplication
A - addition
A - subtraction

BEDMAS.
Learn it.

Yeah, except the whole point of this is that division and multiplication aren't done in any certain order the same way addition and subtraction aren't...

Also, that's BEDMAA.

[rumborak]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

To elaborate on my point: In the end, 0.999... = 1 makes a statement about the infinitely small remainder between the two numbers, i.e. that the remainder *is* indeed zero and they are thus the same numbers.
Now, 1/9 = 0.1111.... makes exactly the same point! Just as in 0.9999... the argument goes there is always another "9" that makes it closer to 1, in 0.1111... there is always another 1 that makes it closer to 1/9.

So, my point is that you are starting out with the thing you are actually trying to prove, and then, duh, you magically prove it!

Again, I do not question the overall truth of the statement, I'm just commenting on that a lot of those "easy proofs" for this are inherently flawed.

rumborak

[kári]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

To elaborate on my point: In the end, 0.999... = 1 makes a statement about the infinitely small remainder between the two numbers, i.e. that the remainder *is* indeed zero and they are thus the same numbers.
Now, 1/9 = 0.1111.... makes exactly the same point! Just as in 0.9999... the argument goes there is always another "9" that makes it closer to 1, in 0.1111... there is always another 1 that makes it closer to 1/9.

So, my point is that you are starting out with the thing you are actually trying to prove, and then, duh, you magically prove it!

Again, I do not question the overall truth of the statement, I'm just commenting on that a lot of those "easy proofs" for this are inherently flawed.

rumborak

Then what is the "real proof"? I think we proved it last semester in one of my courses called "Proving and reasoning" using Dedekindsneden, whatever that means in English.. Dedekind cuts?

[blackngold29]

B - brackets
E - exponents
D - division
M - multiplication
A - addition
A - subtraction

BEDMAS.
Learn it.

Yeah, except the whole point of this is that division and multiplication aren't done in any certain order the same way addition and subtraction aren't...

Also, that's BEDMAA.

Correct. In school I was taught "Please Excuse My Dear Aunt Sally" which reverses division and multiplication.

[rumborak]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

To elaborate on my point: In the end, 0.999... = 1 makes a statement about the infinitely small remainder between the two numbers, i.e. that the remainder *is* indeed zero and they are thus the same numbers.
Now, 1/9 = 0.1111.... makes exactly the same point! Just as in 0.9999... the argument goes there is always another "9" that makes it closer to 1, in 0.1111... there is always another 1 that makes it closer to 1/9.

So, my point is that you are starting out with the thing you are actually trying to prove, and then, duh, you magically prove it!

Again, I do not question the overall truth of the statement, I'm just commenting on that a lot of those "easy proofs" for this are inherently flawed.

rumborak

Then what is the "real proof"? I think we proved it last semester in one of my courses called "Proving and reasoning" using Dedekindsneden, whatever that means in English.. Dedekind cuts?

There's many of them, https://en.wikipedia.org/wiki/0.999... has a lot of them, including Dedekind's cut.

rumborak

[7StringedBeast]

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

How about
x = 0.99...
10x = 9.99...
9x = 9.99... - x
9x = 9
x = 1

Correct me if I am wrong but, you can't just subtract x from one side of your equation without also subtracting it from the other.  you should end up with 8x=9  with x being = to 8/9  not 1.

[kári]

Well it definitely is the correct notation for that fraction. I think Rumborak's point was that the proof was relying on two unproved axioms, each a different wording of the same problem, whereas in the case of 1/9, the answer of 0.111... can be found using simple mathematics (long division). Admittedly its not the best proof in the world but its not a tautology as Rumborak suggested.

To elaborate on my point: In the end, 0.999... = 1 makes a statement about the infinitely small remainder between the two numbers, i.e. that the remainder *is* indeed zero and they are thus the same numbers.
Now, 1/9 = 0.1111.... makes exactly the same point! Just as in 0.9999... the argument goes there is always another "9" that makes it closer to 1, in 0.1111... there is always another 1 that makes it closer to 1/9.

So, my point is that you are starting out with the thing you are actually trying to prove, and then, duh, you magically prove it!

Again, I do not question the overall truth of the statement, I'm just commenting on that a lot of those "easy proofs" for this are inherently flawed.

rumborak

Then what is the "real proof"? I think we proved it last semester in one of my courses called "Proving and reasoning" using Dedekindsneden, whatever that means in English.. Dedekind cuts?

There's many of them, https://en.wikipedia.org/wiki/0.999... has a lot of them, including Dedekind's cut.

rumborak

I see, thanks. It also includes the proof I gave a few posts above... Does it also make use of the same tautology you are referring to?

Rumby, the thing is, we KNOW that 1/9 = .11111... or .1 with the bar over it. It is a repeating decimal. Then, since we know that, by multiplying .11111....by 9, we can logically conclude that all those 1's turn into 9's, because (1)(9) = 9.

Err, no, James, we DON'T know that 1/9 = .11111..., because it is just a rewording of what you are trying to prove!! (that 0.9999... = 1)
A tautology is not a proof.

rumborak

How about
x = 0.99...
10x = 9.99...
9x = 9.99... - x
9x = 9
x = 1

Correct me if I am wrong but, you can't just subtract x from one side of your equation without also subtracting it from the other.  you should end up with 8x=9  with x being = to 8/9  not 1.

You are correct, but I did subtract x.. 10x - x = 9x.