0.999...=1 and why people believe it is false.

[tintin113]

Haha this thread is so stupid. It should be 0.999...=1 and why people believe it is true. There are enough ignorants in the world today without things like this attempting to spread the cause.

[Zierro]

You know, aside from those two debates, I generally tend to agree with you. Before that, I used to think of you as someone who mirrored my ideas\beliefs fairly closely. Then again, if I was right, your stubborn and abrasive when arguing (just like me) so it makes sense that the tiniest difference would spark an explosive argument :lol:

 

 

 

Good point. It's just that the last time this thread was up I seemed to be the only one arguing that these "proofs" aren't legit and saying 0.999.... =/= 1 because it would be saying 1/infinity = 0. Everyone was calling me a troll for it and things got pretty ugly. Good to see people on the same debating level as me agree. :P

[Faux]

Jaziek, ignore all the [puncture]s nitpicking the fact that you don't have an ellipse after the nines.

 

 

 

Christ, I thought everyone here was supposed to be a nerd.

 

Are you being sarcastic?

 

 

 

[hide=]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

99.9999 / 10 = 9.99999 so this whole thing is wrong.

[/hide]

 

 

 

what? thats what I wrote.

 

 

 

and sworddude as well. I fail to see your problem. you're just repeating whats in my post.

 

 

 

9.9999 != 9.99999

[compfreak847]

 

There's a difference between backing up an entire argument on "collage professor > teenager" and not ending a sentence with a point. Especially when the former was wrong :thumbdown:

 

I'm afraid I missed where I claimed I was a collage professor, perhaps you could explain that to me? I'm a computer consultant with no particular background in math :roll:

 

 

 

 

Good point. It's just that the last time this thread was up I seemed to be the only one arguing that these "proofs" aren't legit and saying 0.999.... =/= 1 because it would be saying 1/infinity = 0. Everyone was calling me a troll for it and things got pretty ugly. Good to see people on the same debating level as me agree. :P

 

I don't venture into OT much, so I rarely read beyond the first page of a thread :)

[Jaziek]

Jaziek, ignore all the [puncture]s nitpicking the fact that you don't have an ellipse after the nines.

 

 

 

Christ, I thought everyone here was supposed to be a nerd.

 

Are you being sarcastic?

 

 

 

[hide=]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

99.9999 / 10 = 9.99999 so this whole thing is wrong.

[/hide]

 

 

 

what? thats what I wrote.

 

 

 

and sworddude as well. I fail to see your problem. you're just repeating whats in my post.

 

 

 

9.9999 != 9.99999

 

 

 

if all you can do is argue over how many decimal places I use to represent an endlessly recurring decimal (because its pretty difficult, and ultimately pointless, to use the actual symbol for recurring on this forum), then it is pretty clear you dont have much of a leg to stand on in regards to the actual debate going on.

 

 

 

My proof is right.

[Rebdragon]

The jaziek proof argument was all just a misunderstanding of whether or not the nines were infinite. Still, in the context of this thread you'd think it'd be obvious that he was using .999... .

[mmmcannibalism]

Im surprised so many people are arguing against this; most of you are making legitimate points but why are there nuts arguing the fact that .999...=.999999999999999999999999999999999999(9's forever)?

 

 

 

The very simple proof of this as shown in the first post involving no limits

 

 

 

3/3=1

 

 

 

3/3=3*(1/3)

 

 

 

1/3=.333...

 

 

 

3*(1/3)=1

 

 

 

3*.333...=1

 

 

 

if we distribute the 3

 

 

 

.999...=1

 

 

 

its true that for any measured amounts of nines there is a difference between the 9's and 1; however, when we say there are an infinite amount of 9's the two numbers are equal.

[Laura]

Im surprised so many people are arguing against this; most of you are making legitimate points but why are there nuts arguing the fact that .999...=.999999999999999999999999999999999999(9's forever)?

 

 

 

The very simple proof of this as shown in the first post involving no limits

 

 

 

3/3=1

 

 

 

3/3=3*(1/3)

 

 

 

1/3=.333...

 

 

 

3*(1/3)=1

 

 

 

3*.333...=1

 

 

 

if we distribute the 3

 

 

 

.999...=1

 

 

 

its true that for any measured amounts of nines there is a difference between the 9's and 1; however, when we say there are an infinite amount of 9's the two numbers are equal.

Didn't we state that .333... isn't 1/3 of 1? It would be 1/3 of .999...

 

 

 

There's a difference between backing up an entire argument on "collage professor > teenager" and not ending a sentence with a point. Especially when the former was wrong

A mistake is a mistake.

[Quoi_Tu]

Im surprised so many people are arguing against this; most of you are making legitimate points but why are there nuts arguing the fact that .999...=.999999999999999999999999999999999999(9's forever)?

 

 

 

The very simple proof of this as shown in the first post involving no limits

 

 

 

3/3=1

 

 

 

3/3=3*(1/3)

 

 

 

1/3=.333...

 

 

 

3*(1/3)=1

 

 

 

3*.333...=1

 

 

 

if we distribute the 3

 

 

 

.999...=1

 

 

 

its true that for any measured amounts of nines there is a difference between the 9's and 1; however, when we say there are an infinite amount of 9's the two numbers are equal.

 

You just showed "3*.333...=1" not ".999...=1". Connect the two or use tryto's first proof.

[mmmcannibalism]

[hide=]

Im surprised so many people are arguing against this; most of you are making legitimate points but why are there nuts arguing the fact that .999...=.999999999999999999999999999999999999(9's forever)?

 

 

 

The very simple proof of this as shown in the first post involving no limits

 

 

 

3/3=1

 

 

 

3/3=3*(1/3)

 

 

 

1/3=.333...

 

 

 

3*(1/3)=1

 

 

 

3*.333...=1

 

 

 

if we distribute the 3

 

 

 

.999...=1

 

 

 

its true that for any measured amounts of nines there is a difference between the 9's and 1; however, when we say there are an infinite amount of 9's the two numbers are equal.

Didn't we state that .333... isn't 1/3 of 1? It would be 1/3 of .999...

 

 

 

There's a difference between backing up an entire argument on "collage professor > teenager" and not ending a sentence with a point. Especially when the former was wrong

A mistake is a mistake.

[/hide]

 

 

 

.333...=1/3 of .999 and 1/3 of 1

 

 

 

dividing one by 3 manually results in an infinite amount of 3; therefore, .333 is 1/3 of 1

 

 

 

multiplying 3 by 3 yields 9 so 3*.333...=.999...

 

 

 

since .333... is 1/3 of 1 and .999... is 3 times .333...; .999... must equal one because

 

 

 

1/3*3=1

 

 

 

the threes cancel so 1=1 true statement.

 

 

 

edit--rocco thats incorrect

 

 

 

.333...(an infinite amount of 3's) is the result you get when dividing 1 by 3; therefore, 1/3=.333...

 

 

 

.999 is only less then 1 when using a finite number of 9's, the 1/3 proof shows that.

 

 

 

2nd edit--distributing the 3 through .333... yields .999...

[Zierro]

2. 0.999... is always 0.000...1 less than 1, and therefore not 1. Saying 0.999... is equal to 1 is like saying 9 is 10... except on a scale with many more numbers.

 

 

 

I see where you're coming from, but that's the wrong way of showing that number because it implies that the 0's aren't infinite - they stop before the 1.

 

 

 

0.<--0001 is the same concept but it is mathematically possible, because the 0's run into the decimal point instead of an actual number.

[Rebdragon]

Ha, laura, that proof's pretty funny. It's a legitamite proof, but it seems a little weird that part of the proof utilizes the fact that 1/3 = .333... when it's attempting to prove that 1 = .999... . It's almost like circular logic, except the maths right so it really isn't.

[mmmcannibalism]

2. 0.999... is always 0.000...1 less than 1, and therefore not 1. Saying 0.999... is equal to 1 is like saying 9 is 10... except on a scale with many more numbers.

 

 

 

I see where you're coming from, but that's the wrong way of showing that number because it implies that the 0's aren't infinite - they stop before the 1.

 

 

 

0.<--0001 is the same concept but it is mathematically possible, because the 0's run into the decimal point instead of an actual number.

 

But doesn't the phrase infinite indicate that it will never, ever, ever, ever.... ever, ever, stop? You would never get to the 1, and therefore be left with 0.999...

 

 

 

not to be a jerk, but your just not understanding infinity

 

 

 

.999... indicates that the one never occurs because the 0's from subtraction go on forever. The one would appear after the infinite 9 but since you cant go past infinity all you can read the number as is 0.0...

 

 

 

 

 

laura--how is .333... not a third of one? If I divide 1 by 3 manually I get .333...

[Mr_Adam]

I showed this to my math teacher, specifically the second point.

 

 

 

x = .999...

 

10x = 9.999...

 

10x - x = 9.999... - 999...

 

9x = 9

 

x = 1

 

 

 

He looked at it for a bit, and found an error in 9x = 9.

 

Since x (at least at that moment) is not equal to one, .999... x 9 isn't equal to 9.

 

 

 

So I guess there's something impossible that happens?

[mmmcannibalism]

2. 0.999... is always 0.000...1 less than 1, and therefore not 1. Saying 0.999... is equal to 1 is like saying 9 is 10... except on a scale with many more numbers.

 

 

 

I see where you're coming from, but that's the wrong way of showing that number because it implies that the 0's aren't infinite - they stop before the 1.

 

 

 

0.<--0001 is the same concept but it is mathematically possible, because the 0's run into the decimal point instead of an actual number.

 

But doesn't the phrase infinite indicate that it will never, ever, ever, ever.... ever, ever, stop? You would never get to the 1, and therefore be left with 0.999...

 

laura--how is .333... not a third of one? If I divide 1 by 3 manually I get .333...

 

10 cannot be divided into perfect thirds. Neither can one.

 

 

 

1/3=.333...; saying thats not perfect is a semantic debate.

 

 

 

edit

 

 

 

insertmynamehere--10x-x=9.999...-.999... leads to 9x=9 because 10 times anything minus the anything=9 times the anything; unless I am missing how that is being read your math teacher made some type of error.

[kazzin]

I showed this to my math teacher, specifically the second point.

 

 

 

x = .999...

 

10x = 9.999...

 

10x - x = 9.999... - 999...

 

9x = 9

 

x = 1

 

 

 

He looked at it for a bit, and found an error in 9x = 9.

 

Since x (at least at that moment) is not equal to one, .999... x 9 isn't equal to 9.

 

 

 

So I guess there's something impossible that happens?

 

 

 

No. Your math teacher doesn't understand math very well, and that's why he's teaching high school instead of getting his PhD.

[Zierro]

1/3=.333...; saying thats not perfect is a semantic debate.

 

 

 

Precision is important when we're dealing with the smallest possible thing in existence. :)

[mmmcannibalism]

[hide=]

2. 0.999... is always 0.000...1 less than 1, and therefore not 1. Saying 0.999... is equal to 1 is like saying 9 is 10... except on a scale with many more numbers.

 

 

 

I see where you're coming from, but that's the wrong way of showing that number because it implies that the 0's aren't infinite - they stop before the 1.

 

 

 

0.<--0001 is the same concept but it is mathematically possible, because the 0's run into the decimal point instead of an actual number.

 

But doesn't the phrase infinite indicate that it will never, ever, ever, ever.... ever, ever, stop? You would never get to the 1, and therefore be left with 0.999...

 

laura--how is .333... not a third of one? If I divide 1 by 3 manually I get .333...

 

10 cannot be divided into perfect thirds. Neither can one.

 

 

 

1/3=.333...; saying thats not perfect is a semantic debate.

 

Well, it's just that the fact that your arguments are based on a third of one, and one can't be divided into thirds... it just seems... stupid?

[/hide]

 

 

 

I have to be blunt now, if you cant accept thet .333...=1/3 then your wrong and its impossible to debate this with you.

[kazzin]

 

Well, it's just that the fact that your arguments are based on a third of one, and one can't be divided into thirds... it just seems... stupid?

 

 

 

It can't? 1/3 is a third of 1.

 

 

 

I assume you're suggesting that 1/3 != 0.333..., in which case I have to ask what exactly you think the decimal expansion of 1/3 is? And yes, every real has a decimal representation.

[Zierro]

It can't? 1/3 is a third of 1.

 

 

 

It is possible with fractions but not with decimals.

[Laura]

 

I have to be blunt now, if you cant accept thet .333...=1/3 then your wrong and its impossible to debate this with you.

 

Then why does it continue? It's a representation of 1/3 in decimal form because we cannot achieve 1/3 in this manner.

 

 

 

.999...=1 assumes that 1/3=.333...

 

And .999...=/=1 assumes that 1/3=/=.333...

[kazzin]

Look, the Reals form an ordered field, which most importantly has the property that given x and y from R, x is either less than y, equal to y, or greater than y, where x < y is taken to mean that y - x > 0.

 

 

 

Obviously no one will argue that 0.999... > 1, so let's assume 0.999... < 1.

 

 

 

Then 1 - 0.999... > 0

 

 

 

But what is 1 - 0.999...? Well, the greatest lower bound of the set {0.1, 0.01, 0.001,...} of course, which is clearly 0. That is, 1 - 0.999.. is SMALLER than any positive real number, and thus 1 - 0.999... = 0.

 

 

 

Thus we conclude that 0.999... = 1, and we are done.

[Laura]

Look, the Reals form an ordered field, which most importantly has the property that given x and y from R, x is either less than y, equal to y, or greater than y, where x < y is taken to mean that y - x > 0.

 

 

 

Obviously no one will argue that 0.999... > 1, so let's assume 0.999... < 1.

 

 

 

Then 1 - 0.999... > 0

 

 

 

But what is 1 - 0.999...? Well, the greatest lower bound of the set {0.1, 0.01, 0.001,...} of course, which is clearly 0. That is, 1 - 0.999.. is SMALLER than any positive real number, and thus 1 - 0.999... = 0.

 

 

 

Thus we conclude that 0.999... = 1, and we are done.

But it eventually doesn't stop at one. It rests when the number beings subtracted stops, which it doesn't. So then I don't see how that equals one, should both stop at any given point, its value will equal ...1.

[Zierro]

Why do people continue to plug and chug infinity in an equation and think it is just as legit as a normal equation? That being said, can anyone tell me what infinity plus 1 is?

[kazzin]

Look, the Reals form an ordered field, which most importantly has the property that given x and y from R, x is either less than y, equal to y, or greater than y, where x < y is taken to mean that y - x > 0.

 

 

 

Obviously no one will argue that 0.999... > 1, so let's assume 0.999... < 1.

 

 

 

Then 1 - 0.999... > 0

 

 

 

But what is 1 - 0.999...? Well, the greatest lower bound of the set {0.1, 0.01, 0.001,...} of course, which is clearly 0. That is, 1 - 0.999.. is SMALLER than any positive real number, and thus 1 - 0.999... = 0.

 

 

 

Thus we conclude that 0.999... = 1, and we are done.

But it eventually doesn't stop at one. It rests when the number beings subtracted stops, which it doesn't. So then I don't see how that equals one, should both stop at any given point, its value will equal ...1.

 

 

 

No idea what you're talking about. You may find it useful to attempt to define 1 - 0.9..., however. That is, define the difference between 0.9... and 1.