I cant belevie ppl cant put this through their heads

[steve jones]

that .999... and 1 are the same number.

I just spent 10 minutes areguin with someone -_- , and the things he said...

If everyone in the world gave me (100% - 99.999%) of their money, then I would be a rich man. 

So in that context I must disagree that 0.999 and 1 are not exactly the same number.

SJ

[johnny-boy]

well done. i like the bit about existing even though it cannot be written discussion closed. everyone kn ows a third is 0.333~, and that 3/3 = 1, so 0.999~ must be one! it only m,akes sense!
Tom

Yes, and "1" can be written so it can't be "1". 

.333 is 333/1000, .3333 is 3333/10000, etc. also, .999 is 999/1000, .9999 is 9999/10000 so where the heck did 3/3 come from?

John

[prometheus]

Yes, .333~ and .999~ are attempts to write down fractions as real numbers. So if you are going to see .999~ as the fraction it is trying to express then it does equal one.

If you see it as a folly attempt then it can be problematic. Do realise that there can't be a 0 in the line of these numbers. .99990 is not the right way to round it down. If you limit yourself to three decimals, like .999, then if the number is used in math it means .9990. So that's not right.

In physics .999 has a different meaning. It is a limit. If you stop writing down more 9's then  that doesn't mean there is a 0. It means the measurement isn't any more specific than two decimals. Whatever number comes then is unclear and irrelevant. This is because in physics numbers are measurements. In math they are a lot more 'holy'.

[prometheus]

If everyone in the world gave me (100% - 99.999%) of their money, then I would be a rich man. 

That's because .00009~ percent over a large number is a lot. But the point is that you also have to give them back all those 9's.

[prometheus]

Yes, and "1" can be written so it can't be "1". 

.333 is 333/1000, .3333 is 3333/10000, etc. also, .999 is 999/1000, .9999 is 9999/10000 so where the heck did 3/3 come from?

John

Try to write down the fraction 1/3 as a rational number. I am not talking about .333 as in 333/1000.

[johnny-boy]

I am not talking about .333 as in 333/1000.

Then what are you talking about? 1/3 of 9 = 3 (that sounds pretty rational to me).

John

[steve jones]

I do get what the original poster is talking about:

3/3 = 1

0.333~ = 1/3

0.999~ = 1


However, I insist that there must be some explanation for this, as 0.999 is clearly less than 1:

If I buy something for 99 pence and pay with a pound, then I get 1 pence change!


But obviously, not being a mathematician, I dont know the reason for this strangeness. Maybe someone who is still at school / college / university should ask a Maths bod?

SJ

[johnny-boy]

I don't know Steve; if I'm eating an apple it'll be extremely difficult to convince me I'm eating an orange.

John

[tompilk]

Yes, and "1" can be written so it can't be "1". 

.333 is 333/1000, .3333 is 3333/10000, etc. also, .999 is 999/1000, .9999 is 9999/10000 so where the heck did 3/3 come from?

John

i multiplied it by three...
Tom

[johnny-boy]

i multiplied it by three...
Tom

And what point did that make?  .333 multiplied by 3 equals .999 (not "1").

John

[henrah]

I don't know Steve; if I'm eating an apple it'll be extremely difficult to convince me I'm eating an orange.

John

Johnny boy caught it on the skin. 0.999 is clearly < 1. 0.999, 0.9999 and 0.99999999999999999999999999 are completely different than 0.999~ as they are rational numbers. 0.999~ can only be expressed through the mentioning of it being a recurring decimal, or through being written down as a fraction. In this case, 0.999~ is 1/1, which is the same as 1 I'm sure you know.

If I buy something for 99 pence and pay with a pound, then I get 1 pence change!

This is because £0.99 is a rational number, and can be written down without the use of fractions or recurring decimals.


0.9, 0.99, 0.9999 and 0.9999999999999999999 =/= 1.
                                                                  0.999~ == 1.

Simple.

EDIT:

And what point did that make? .333 multiplied by 3 equals .999 (not "1").

Of course it does. But he stated it as a recurring decimal, which - when multiplied by 3 - does = 1. As someone originally said: 1/3 = 0.333~ and 1/3 * 3 = 1, therefore 0.333~ * 3 = 1.

People are confusing 0.333 and 0.999 with 0.333~ and 0.999~.

[prometheus]

Then what are you talking about? 1/3 of 9 = 3 (that sounds pretty rational to me).

John

1/3rd, the fration written as .333~ as a irrational number.

As for the meaning of 'irrational number', use a dictionary.

However, I insist that there must be some explanation for this, as 0.999 is clearly less than 1:

1 -.999 is clearly .0001, yes.

But .999~ is something strange.

Imagine it as having a cake. First we take .9 part of the whole cake, 90%.
Then we take 90% of that what is left, the 10% percent. Then we have 99% of the cake. Then we take 90% of the 1% left. We get 99.9%.

Then we take that what is left .1% of the original cake. And again we take 90% of that. So we can add another 9 to get 99.99% Now for 99.999~ to be 1 we have to go on for ever. If we do not do that we will have something of the cake left. But if we stop we will have to round the number up. That would mean taking 100% of what is left instead of 90%. If we just stop our number just ends after so much nines and we do not have the number we are talking about.

So in the end .999~ is trying to express 1 but it will fail in practice because you cannot do something infinitely. .999~ is just a stupid flawed way of writing 1.

[monsieurrenard]

emmdoubleew, I can't agree with any of the proofs you use.  The first assumes you can multiply an abstraction like .999.... with a real number.

Hi,

This is a very interesting debate and I had to participate.
First, let's make it very clear that we are in the realm of math and absolutely must abide by its rules. Surely we can all agree on this point, since .999... and 1.000... are purely mathematical numbers.

If .999... is not a real number, then how come it's obviously less than 2 and greater than zero? According to mathematics, the number .999... is indeed part of the set of all real numbers. May I also remind you that the number 1 also has an infintely repeating amounts of 0 after it: 1.0000... and so on. This is true for absolutlely all numbers.

The second shows that the limit of the series used to give .999.. approaches 1, which I'll readily conceed.

Actually it doesn't, that would barely take two steps. It does use convergence, but I don't think you fully understood the proof, which is quite OK, but keep an open mind.

 

The third once again uses .1111... as a number instead of an abstraction.

.111... is mentioned nowhere in the third proof  . In any case, just because it has an infinite amount of digits after the decimal point doesnt not make it any less of a number. Are you denying that Pi is a number? Surely you wouldn't want to do that, as it's necessary to multiply it by the diameter to find the circumference of a circle. 

The  last one I think fundamentally misinterprets what a limit(or convergence) is about.  As you take the function f(x)=1/x to infinity, it gets infinitely close to f(x)=0, but I doubt you'd be silly enough to claim that f(infinity)=0.  As I said before, infinity is only a nice and convenient concept.  .9999... doesn't exist except as an abstraction. 

Yes you are right, but we are not talking about 1/x, 1/x has only one limit, while .9 + .09 + .009 ... has two limits (1 and .999...), which is impossible! Hence they must be the same.

I think emmdoubleew put it very nicely:
"All numbers are concepts. Some numbers, like 1, have stronger links to reality than others, but we are looking at mathematics here, not the real world. If you're going to throw away numbers which can't concretely exist, then you're throwing away pi, e, i, zero, and, frankly, almost all of mathematics."

So this whole thread is about 0.999... and 1 being the same number?

well, as you can see its clearly not. Claiming that they are the same is stating something that there is no answer to, because no matter how many decimals you put after the comma, you can always put one more...and then 1 more...and then a million more...so bascally you are just rounding off 0.99... to be 1, two different numbers. yes they are close, insanely unbelievably close, but they are still different. Im gonna get yelled at now  Sorry 

Hi nicco, I like that you are thinking about infinity! Such a hard concept to grasp. Yes, if you kept adding 9 after the decimal point, you would get closer and closer to 1, but there is already an infinite amount of 9s after the decimal point, so you are already there!

Here's the original post:
that .999... and 1 are the same number

They are not!  If you want to round it off to "1" that's fine with me.

.999........ no matter how far it's extended it's still going to come up short from "1".

John

Hi John,

I see your thinking but it is actually fallacious. You cannot extend .999... because it already extended to infinity, hence already "at" 1. .999... is static, not a series on which you keep adding 9s. Hope this helps you understand  Also, since we are talking about math, it would help if you could provide a mathematical proof that they are not the same instead of speaking on such abstract terms. Good luck though, because you won't find one.

So in the end .999~ is trying to express 1 but it will fail in practice because you cannot do something infinitely. .999~ is just a stupid flawed way of writing 1.

It's silly, but far from flawed. People simply cannot accept what confuses them. And that's okay, only they tend to keep a closed mind.

I hope I clarified a few things.

- Monsieur le Renard

[henrah]

So in the end .999~ is trying to express 1 but it will fail in practice because you cannot do something infinitely. .999~ is just a stupid flawed way of writing 1.

That's the beauty of fractions

[monsieurrenard]

That's the beauty of fractions

It's the beauty of mathematic.s 

-Monsieur le Renard

[prometheus]

It's silly, but far from flawed. People simply cannot accept what confuses them. And that's okay, only they tend to keep a closed mind.

I didn't comment on the fact that they are equal in size. I meant that it is strange to write .999~ if you want to write 1. Just like one would not write 9/9 when 1 is meant.

[johnny-boy]

I read this and I still don't buy it.
https://www.purplemath.com/modules/howcan1.htm

Also I don't accept that 1/3 is the same as .333.
9 times 1/3 = 3 whereas 9 times .333 = 2.997.   

John

[monsieurrenard]

I read this and I still don't buy it.
https://www.purplemath.com/modules/howcan1.htm

Also I don't accept that 1/3 is the same as .333.
9 times 1/3 = 3 whereas 9 times .333 = 2.997.   

John

John,

No one has ever said 1/3 = .333
1/3 = .3333333.....
9 * .333... = 2.999.... = 3

The main flaw in your thinking is your refusal to acknowledge that there is an infinite amount o 9s after the decimal point.

However, John, will you tell me what 1 - .999..... equals?
I would be happy to know the "difference" between the two numbers.

 

Hope this helps you understand.

-Monsieur le Renard

[prometheus]

However, John, will you tell me what 1 - .999..... equals?

It would take forever to calculate since there are an infinite amount of 9's.

[monsieurrenard]

It would take forever to calculate since there are an infinite amount of 9's.

If you think latterally it takes a split second 

-Monsieur le Renard

[johnny-boy]

Also, since we are talking about math, it would help if you could provide a mathematical proof that they are not the same instead of speaking on such abstract terms. Good luck though, because you won't find one - - Monsieur le Renard


Funny, I thought the proof being used in mathematical theory to prove .999 is “1” was abstract (and flawed).

Best, John

[johnny-boy]

However, John, will you tell me what 1 - .999..... equals?
I would be happy to know the "difference" between the two numbers. - monsieurrenard

.001, .0001, .00001, etc.

John

[johnny-boy]

If you think latterally it takes a split second 

-Monsieur le Renard

Yeah, just round it off to "1".

Best, John

[monsieurrenard]

However, John, will you tell me what 1 - .999..... equals?
I would be happy to know the "difference" between the two numbers. - monsieurrenard

.001, .0001, .00001, etc.

John

Multiple answers? I wanted the difference between the two numbers.

Also, since we are talking about math, it would help if you could provide a mathematical proof that they are not the same instead of speaking on such abstract terms. Good luck though, because you won't find one - - Monsieur le Renard


Funny, I thought the proof being used in mathematical theory to prove .999 is “1” was abstract (and flawed).

Best, John

John,

It wasn't flawed, and it certainly wasn't abstract, only musik_man did not understand it. See my first reply in this thread. If you like, point out where you think those proofs were flawed and I will show you that they are not.

John, let me tell you right now that I am not trying to prove that .999... = 1. Emmdoubleew has done it, millions of mathematicians around the world have done it, I don't need to do it. The fact that .999... = 1is accepted in the mathematics world, and whethere you like it or not, you will have to accept this as well if you plan on studying physics, engineering, electronics, computer sciences, or any subject which makes use of math.

I am only trying to help you understand. I'm trying to show you that you can't just trust your instincts. This is the beauty of mathematics, this is the beauty of intelligence. You are not restricted to what you see, you can dvelve deeper and come to seemingly unbeleivable results! Don't let your mind be your own prison, it should be your liberator .

It's good that you are facing this fact with much skepticism, because it means you are thinking, but don't prevent yourself from accepting such a cool tidbit of mathematics.

-Monsieur le Renard.

PS: When i was first told that .999... = 1, it took me a whole month of rejecting the proofs and going over them again and again until I finally figured out why it made sense. I was as stubborn as you .
PPS: This is the proof that eventually convinced me:
x = 0.9999...

Multiply both sides by ten:

    10x = 9.9999...

Subtract x from both sides:

10x - x = 9.9999... - 0.9999...
     9x = 9.0000...

Divide by nine:

      x = 1.0000...
Substitute:
   .999... = 1.000...

[prometheus]

Multiple answers? I wanted the difference between the two numbers.

Doesn't he mean .00111~ ? A one for every 9 that is incomplete? Don't ask me why he starts off with two zero's though.

But that isn't right. 1/900 is .00111~ and if you subtract that from 1 you get .00888~ which would be 8/900

[monsieurrenard]

Doesn't he mean .00111~ ? A one for every 9 that is incomplete? Don't ask me why he starts off with two zero's though.

But that isn't right. 1/900 is .00111~ and if you subtract that from 1 you get .00888~ which would be 8/900

Prometheus,

I see, I didn't realize he meant to combine them.

Thanks for clarifying,

-Monsieur le Renard.

[johnny-boy]

Monsieurrenard,

I understand it is accepted in theory. Oh course "theory" is just an unproved assumption - not necessarily true. I’m also aware that mathematicians and scientists depend on their precious theories for their livelihoods and will hinder any evidence to the contrary, like Carbon 14 Dating for instance.

Multiple questions require multiple answers. If you ask me what 1 minus .999 is - it’s .001.

John

[henrah]

If you ask me what 1 minus .999 is - it’s .001.

What is 1 - 0.999999999?

What is 1 - 1/3, expressed as a decimal?

What is 1 - (1/3 * 3), expressed as a decimal?

Hence, what is 1 - (0.333~[/i] * 3), expressed as a decimal? (I enlarged the ~ to be sure that you noticed it and that it meant the 0.333 was recurring)

[monsieurrenard]

Monsieurrenard,

I understand it is accepted in theory. Oh course "theory" is just an unproved assumption - not necessarily true. I’m also aware that mathematicians and scientists depend on their precious theories for their livelihoods and will hinder any evidence to the contrary, like Carbon 14 Dating for instance.

Multiple questions require multiple answers. If you ask me what 1 minus .999 is - it’s .001.

John

John,

I am curious why you keep ignoring the fact that we are talking about .999... and not .999, not only that but I asked you a single question: what is 1- .999... ?

Or another way to state it: prove to me there is a difference between the two number by naming a number between them.

It's not a theory John, it's a fact, and once you understand it it makes perfect sense.  As for your  unfair demonization of mathematicians and scientists: the most famous one to ever exist is the one who completely denied all theories previously accepted. You probably know him as "Einstein"  . I think it's rude and unjust of you to put down scientists like that, because without them you'd wouldn't even be connected to the internet. The job of a mathematician and a scientist is to come to his own conclusions. And if you stop and figure out this proof for a second, you will come to the conclusion that .999... = 1 all on your own:


x = 0.9999...

Multiply both sides by ten:

    10x = 9.9999...

Subtract x from both sides:

10x - x = 9.9999... - 0.9999...
     9x = 9.0000...

Divide by nine:

      x = 1.0000...
Substitute:
   .999... = 1.000...

-Monsieur le Renard

[johnny-boy]

Okay, your right Monsieur. Now relax.

John

[monsieurrenard]

Okay, your right Monsieur. Now relax.

John

Johnny,

It's nothing but a pleasure to argue with you , and if you are only saying that I am right so that you no longer have to deal with me, then shame on me  , i never meant it to be a bother and I apologize if it was.

It was never an argument about me being right.

Best, Monsieur le Renard.

[johnny-boy]

Johnny,

It's nothing but a pleasure to argue with you , and if you are only saying that I am right so that you no longer have to deal with me, then shame on me  , i never meant it to be a bother and I apologize if it was.

It was never an argument about me being right.

Best, Monsieur le Renard.

I was only joking with you Monsieur. I have work to do. We can continue this debate later.

BTW, my remark about scientists and mathematicians was directed towards the general scientific community, not the handful of great inventors like Edison, Bell, Franklin, etc.

Best, John

[prometheus]

Oh course "theory" is just an unproved assumption - not necessarily true.

First off, this is math, not science. It is based on axioms.

Second, scientific theories aren't unproven assumptions. It has nothing to do with the definition of a scientific theory. In fact, you can never prove a theory. You can only refute it. That is why it is required for a theory to be falsifiable to be accepted.

[steve jones]

1/3rd, the fration written as .333~ as a irrational number.

As for the meaning of 'irrational number', use a dictionary.

1 -.999 is clearly .0001, yes.

But .999~ is something strange.

Imagine it as having a cake. First we take .9 part of the whole cake, 90%.
Then we take 90% of that what is left, the 10% percent. Then we have 99% of the cake. Then we take 90% of the 1% left. We get 99.9%.

Then we take that what is left .1% of the original cake. And again we take 90% of that. So we can add another 9 to get 99.99% Now for 99.999~ to be 1 we have to go on for ever. If we do not do that we will have something of the cake left. But if we stop we will have to round the number up. That would mean taking 100% of what is left instead of 90%. If we just stop our number just ends after so much nines and we do not have the number we are talking about.

So in the end .999~ is trying to express 1 but it will fail in practice because you cannot do something infinitely. .999~ is just a stupid flawed way of writing 1.

Ahhh, Im with you now, yes.

It is the infinitesimal nature of the reoccuring third that leads to this 0.999~ = 1.0 scenario. So essential 0.999~ is just a highly misleading way of writing 1/1? And in reality, 0.9 can NEVER equal one.

Cool, so what is 2 then? 

SJ

[johnny-boy]

Ahhh, Im with you now, yes.

It is the infinitesimal nature of the reoccuring third that leads to this 0.999~ = 1.0 scenario. So essential 0.999~ is just a highly misleading way of writing 1/1? And in reality, 0.9 can NEVER equal one.

Cool, so what is 2 then? 

SJ

So I guess we were right from the beginning. Thanks for clearing that up Steve.

Jeez, the scientific/mathematical community makes things so unnecessarily complicated.  I can’t help but wonder why. I do know why governments make things complicated and confusing though. So it takes the general public a while to discover their devious intentions. Hmm…

Best, John

[steve jones]

Is that why its so difficult to open milk cartons these days?

SJ

[johnny-boy]

Is that why its so difficult to open milk cartons these days?

SJ

Hmm...

[steveie986]

Jeez, the scientific/mathematical community makes things so unnecessarily complicated.  I can’t help but wonder why. I do know why governments make things complicated and confusing though. So it takes the general public a while to discover their devious intentions. Hmm…

Anything that's "unnecessarily complicated" is just too difficult for the not as intellectually endowed to comprehend. You should cease your juvenile comments about serious intellectual matters, because they are quite outside your league, johnny boy.

[johnny-boy]

Anything that's "unnecessarily complicated" is just too difficult for the not as intellectually endowed to comprehend. You should cease your juvenile comments about serious intellectual matters, because they are quite outside your league, johnny boy.

I agree; you're definitely not in my league steveie.999.

[henrah]

steveie.999.

Or should you say steveie.9999999~? Haha, great return to the original topic

[Ruro]

Thanks everyone for this great topic

Simple, yet pretty darn fascinating! We need another thread like this I hope it doesn't turn into one of these though...

Friggin count me out, I won't understand it... or atleast, I won't devote the time to it ^_^;; This one was rather quaint & sweet, perhaps because I'm a noobie Mathametician though!

Or we should work on these: https://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics Splitting a Million Dollars, what... how many ways? Among a 100 members? $10k each

[steveie986]

I agree; you're definitely not in my league steveie.999.

No hard feelings. I still love you as much as I love everyone else here.

[monsieurrenard]

It is the infinitesimal nature of the reoccuring third that leads to this 0.999~ = 1.0 scenario. So essential 0.999~ is just a highly misleading way of writing 1/1?

That's precisely it Steve! If there was anything less than a static infinite amount of 9s it would not equal 1 .

-Monsieur le Renard.

[pianistimo]

it's called splitting hairs.  people kill over things like this.

what i don't understand is why when u add up % on the back of pepsi cans they don't add up to 100%.  so what's missing.  air bubbles take up the rest of the space?

total fat 0%
sodium 1%
total carbs 14% (sugar)

protein 0 grams
now where's the other part?  it's the carbonated water, right?

[johnny-boy]

No hard feelings. I still love you as much as I love everyone else here.

Definitely no hard feelings Steveie. It's always fun to do a little razzing in between the lines. I had to bail-out earlier (late for an appointment) and have a tooth pulled - ouch!

Best, John

[phil13]

it's called splitting hairs.  people kill over things like this.

what i don't understand is why when u add up % on the back of pepsi cans they don't add up to 100%.  so what's missing.  air bubbles take up the rest of the space?

total fat 0%
sodium 1%
total carbs 14% (sugar)

protein 0 grams
now where's the other part?  it's the carbonated water, right?

Um, that's the % of what you should ideally have in a day, not what's in the cola. 

Phil

[pianistimo]

oh.  i see.

[pianolearner]

If I buy something for 99 pence and pay with a pound, then I get 1 pence change!


But obviously, not being a mathematician, I dont know the reason for this strangeness. Maybe someone who is still at school / college / university should ask a Maths bod?

SJ

That's because 99 pence is NOT strictly 0.99 just like 99 people does not mean 0.99.

 (100 x 1) - (99 x 1) = 1

[pianistimo]

what about the lost sheep?  i guess i shouldn't bring the bible into this...but it says 'he left the other 99.'  in military terms, too, if u forget one - ur forgetting something very important.  that's the one that's minused from the 100.

[prometheus]

it's called splitting hairs.  people kill over things like this.

No, that's called religion. Mathematicians don't kill each other if they don't agree. They don't have to.