1 = .9999.....

[NeoCracker]

FOr some reason a guy I went to school with couldn't bring himself to believe this statement, though I shall show you two ways in which it is true, and you can try to disprove it. :p

1/3=.333.....
2/3=.666.....

1/3 + 2/3 = 3/3 = 1

.333... + .666... = .999...

There fore, 1=.999....


Next up, imagine a number line. No two different numbers can occupy the same space. So, mathametecly, there will always be space between numbers, therefore there if there is no space between two numbers, they must occupy the same point.

SO, can anyone here tell me what number exists between 1 and .999....?

If not, logically, they occupy the same space, and are therefore the same number.

For those of you who dont' get the .... proceeding the nines, it means the nine is infinitely repeating.

Discuss.

[demondude]

It was about time this topic came up again. D:

[NeoCracker]

When did it come up before?

[blackmage_nuke]

I believe 0.99... doesnt equal 1 but approaches, similar to how the function y=e^x approaches the line y=0 but never touches it.

[Levian]

It's a very long time ago, but I still hate the topic.

In my humble opinion, 0,999 does not equal 1. :mog:

[demondude]

Lots of times, lots of different websites. =p

http://upload.wikimedia.org/math/9/9/9/9999f0ace517cc0702d78a67675f14b0.png

[Timekeeper]

Yea, they are the same.
When you reach infinity, anything becomes possible :)

EDIT: The number itself is different to 1, as you can never reach infinity.

If you want to use the number and don't want to reach infinity then you round and it becomes 1 anyway.
If you decide you can reach infinity then my earlier statement goes.

[Darkswordofchaos]

.999.... is a smaller number than 1 althought the diffrence is infinitly small so small in fact you couldent measure it. no matter what the space between them can be always get smaller but never disapear.

[Aerith's Knight]

And that's why you learn in the first year that you have to round up to the least accurate number in your calculation.

3 * 0.3333... = 1 * 10^1

3.0 * 0.3333... = 1.0 * 10^1

[Captain Maxx Power]

The major problem here is that an infinite series isn't and can never been a true representation of a fraction as given here. Take for example the fraction 1/4. Decimalised it is 0.25. The number 0.25 is a complete decimal representation of the fraction 1/4. However, the decimalised number for the fraction of 1/3 is of course 0.333... Because of this it becomes impossible for us to use it as the true fraction of 1/3. You would have to use a finite decimal in calculations such as 0.3333, or 0.3333333, but neither of these is the correct value. It would be like using 0.24 to represent 1/4; close but no cigar. It's from this misunderstanding that the equation 1/3 + 2/3 = 3/3 / 0.333... + 0.666... = 0.999.... can arise, because you're essentially adding together decimals that aren't a correct representation of the fraction being shown. You can't add infinite series of decimals together even if logically they make sense because they are infinitely long. Better to try and add together the distance of two infinitely long pieces of string.


Next up, imagine a number line. No two different numbers can occupy the same space. So, mathametecly, there will always be space between numbers, therefore there if there is no space between two numbers, they must occupy the same point.

SO, can anyone here tell me what number exists between 1 and .999....?

If not, logically, they occupy the same space, and are therefore the same number.

This is a falsity because it only applies to finite decimal values. If a given value is an infinite series then it becomes impossible to locate it within a number line because of this as the inherent value of that number is impossible to place. For example, 0.999 would not be in the same place as 0.9999. Because of this attempting to place any reoccuring decimal, such as 0.999..., into a theoretical number line is a pointless task because no matter where you choose to place that number, chances are someone could come along, add another 9 on the end and cause it to have to move.

For those with attention deficiency disorders, here is a summarized snippet of the general ramblings for you to ponder over - version: Infinite series and decimal values do not a good combination make.

[Rantz]

Doesn't make a practical difference to me whether they're the same or not. The fact that the decimals go on infinitely is just a limitation of the decimal system anyway, so the discussion is vain to me. A third is a third and the fact decimals can't account for that doesn't make it some big mysterious number that goes on for infinity. It's still just a third. Three thirds together make a whole, so the number .999... is a human construct that doesn't exist in the practical world. .333... symbolises a third and .666 symbolises two thirds, so .333... + .666... equals 1, not .999...

Edit: Yeah, CMP said it first

[Dignified Pauper]

Actualy, 1/3 stands in for the exact decimal number .33333... because you can multiply and divide by it and get an answer. If you take 9 * 1/3 you get a definite 3. If you take 9 * .333 you get a slightly different answer.

[Captain Maxx Power]

Actualy, 1/3 stands in for the exact decimal number .33333... because you can multiply and divide by it and get an answer. If you take 9 * 1/3 you get a definite 3. If you take 9 * .333 you get a slightly different answer.

Exactly. It's the unnecessary conversion to decimal numbers that creates the issue.

[oddler]

1 != 0.9... as 0.9... approaches 1 but never equals the limit. That's my 2 cents.

[Timekeeper]

.333... symbolises a third and .666 symbolises two thirds, so .333... + .666... equals 1, not .999...

Raps it up nicely methinks.

[Luara]

...ah...erm...ugh...


...



you're funny

[Madame Adequate]

How's it going, thread? Been awhile since you rolled around!

[Peegee]

I believe 0.99... doesnt equal 1 but approaches, similar to how the function y=e^x approaches the line y=0 but never touches it.

An expression is not a function. Good try though.

Think about it this way: graph the equation y=0.9999...

It's the same as if you were to graph the equation y = pi - it would be a horizontal line

The expression 0.99999... = 1 is an equality and is also true/valid/whatever

[Rye]

... :mad2:

[Peegee]

Well what if i worked it out as a series 0.9999... is like y = (1/9)*(10^-n) where n = 0,1,2,3,4,5...

edit:wait let me rewrite that

But it's not a fuction it's an equality.

It's
Not
A
Function

[blackmage_nuke]

I believe 0.99... doesnt equal 1 but approaches, similar to how the function y=e^x approaches the line y=0 but never touches it.

An expression is not a function. Good try though.

Think about it this way: graph the equation y=0.9999...

It's the same as if you were to graph the equation y = pi - it would be a horizontal line

The expression 0.99999... = 1 is an equality and is also true/valid/whatever
I was using it as a comparison not an example

[Madame Adequate]

Listen to peegee, he's azn, he knows. :colbert:

[blackmage_nuke]

Bah it doesnt matter, the next time this topic comes around my opinion will change again anyway.

[Peegee]

I believe 0.99... doesnt equal 1 but approaches, similar to how the function y=e^x approaches the line y=0 but never touches it.

An expression is not a function. Good try though.

Think about it this way: graph the equation y=0.9999...

It's the same as if you were to graph the equation y = pi - it would be a horizontal line

The expression 0.99999... = 1 is an equality and is also true/valid/whatever
I was using it as a comparison not an example

Listen to peegee, he's azn, he knows.

Well no, I'm quite terrible at math, but if you looked up old eoff threads about this I used to think 0.999... was akin to a function that was as close to 1 as you could get.

Now in practice no number is 'the smallest', as you can always divide it by another real number, thus getting a smaller number.

The expression x = y is just that - an expression. When you see x you sub in y.

So in the example x = 0.999... all you get to do with it is sub 0.999... into x. Since there's no associated function (like y = x^2) there's nothing more you can do with it.

It's I dunno, number theory.

[Madame Adequate]

:p I dunno how good you are at math, but I know you're right about this.

[blackmage_nuke]

The real question we are asking is does 1/infinity truly equal to zero? becaus 0.999... is really 1 - 1/infinity

[Unbreakable Will]

ow...
my head hurts...

[Peegee]

The real question we are asking is does 1/infinity truly equal to zero? becaus 0.999... is really 1 - 1/infinity

Infinity is not a finite, real number.

I dunno how good you are at math, but I know you're right about this.

I was reading a pre-calculus notebook yesterday. Though formally I've gone about as far as integrals before my brain hurt very much.

[blackmage_nuke]

The real question we are asking is does 1/infinity truly equal to zero? becaus 0.999... is really 1 - 1/infinity

Infinity is not a finite, real number.


Yet we constantly use it in other proofs

And no amount of evidence or reasoning will make me see that the infinitely small difference between 0.999... and 1 does not exist

[Rodarian]

... I sucked at Math....

[Aydin]

............ I failed Algebra II..

You guys are really smart... >_<

[Madame Adequate]

And no amount of evidence or reasoning will make me see that the infinitely small difference between 0.999... and 1 does not exist

:eyebrow: "Evidence and proof will not convince me!"

[blackmage_nuke]

Yes Im like a hardcore christian except in a non religous way

[Peegee]

It's okay man. I'll just be relentlessly logical until you grudgingly accept.

It's how I was taught.

Take a really small real number. Say one divided by the amount of prions in the universe. In practical reality something that small does not exist, but in mathematics it does.

Now with that number you divide it by two. Oh crap you have an even smaller real number! Divide it again! and again! Divide it by the inverse of itself! Oh crap it became even more smaller! Do it again and again!

You can do this "infinitely" and it would always be a smaller real number. There is no such thing as a smallest real number greater than zero.

Once I realized that, 0.999... ceased to be an irrational number

[blackmage_nuke]

Programing has lead me to believe that whoever wrote the universe has included some version of

#include<limits>
#define fltmin numeric_limits<float>::min()

for the universe. Except rather than float we use all numbers

[Peegee]

Programing has lead me to believe that whoever wrote the universe has included a

#include<limits>
#define fltmin numeric_limits<float>::min()

for the universe

English.

[blackmage_nuke]

Well in programming there is a smallest number greater than 0 and I believe the universe is a giant program. When I learnt about Planck lengths that blew my mind

[blackmage_nuke]

Yes mathematics commonly has sexual intercourse with other sciences. No matter how many large english's breasts get, maths just doesnt swing that way

[Quindiana Jones]

Decimals and fractions are different. 1 does not equal 0.999.....

[qwertysaur]

In Math to prove something fully, you also have to disprove it is false. You can not disprove that 0.999999... =/= 1, therefore the statement is not true. (when you disprove falsehood you use indirect proofs usually)

Infinity can never be reached, only approached, and also does not actually exist in math. The full proven statement here is that as the number of digits after the decimal place approaches infinity, the value of the number (which is still rational) approaches 1, but never quite reaches it.

[Raistlin]

Let ".999..." equal a decimal followed by an infinite series of nines.

Ok, all of you who are claiming that .999... does not equal one are wrong. The number between them is not "infinitely small," because there is no number between them. They are not "basically equal with rounding," they are in fact exactly equal.


Three thirds together make a whole, so the number .999... is a human construct that doesn't exist in the practical world.

Yeah it does. I'm pretty sure you can identify a single object in the "practical world". ".999..." is just a different representation of it, but just as valid as "1."


It's from this misunderstanding that the equation 1/3 + 2/3 = 3/3 / 0.333... + 0.666... = 0.999.... can arise, because you're essentially adding together decimals that aren't a correct representation of the fraction being shown. You can't add infinite series of decimals together even if logically they make sense because they are infinitely long.

This makes no sense. It's not a "misunderstanding," it's adding together rational numbers.


This is a falsity because it only applies to finite decimal values. If a given value is an infinite series then it becomes impossible to locate it within a number line because of this as the inherent value of that number is impossible to place.

This is also wrong, and I'm not even sure where you got this. It is a basic rule of algebra that any two, distinct real numbers have an infinite number of other distinct, real numbers between them. 0.999... is a real number.

Someone who remembers more calculus than me will have to post the actual proof. It was posted a couple of years ago in the last thread but I'm too lazy to look it up. But yes, it is actually, definitively proven that .999... = 1.

EDIT: Ok, previous threads on this: First one (http://forums.eyesonff.com/general-archive/45989-math-makes-me-happy.html). Second one (http://forums.eyesonff.com/general-chat/82621-hi-explain-why-0-999-1-please.html). Third one (http://forums.eyesonff.com/general-chat/105259-1-3333-does-not-equal-4-3-a.html).

[Rantz]

Three thirds together make a whole, so the number .999... is a human construct that doesn't exist in the practical world.

Yeah it does. I'm pretty sure you can identify a single object in the "practical world". ".999..." is just a different representation of it, but just as valid as "1."

I'm not talking about what the rules about mathematics are, I'm debating the actual merit of ever using the representation .999... when all it does is cause confusion. Some would undoubtedly say that the .999... representation exists to increase clarity, but as proven by this discussion and countless others, it clearly does just the opposite. The simple fact that they are equal renders the representation .999... obsolete as far as I'm concerned.

[Denmark]

Well in programming there is a smallest number greater than 0 and I believe the universe is a giant program.

Programming is not mathematics. Programming is defined, restricted, by the limits of the computer. Computers cannot comprehend infinity. Most humans can't comprehend infinity, why should a commercially-available machine MADE BY HUMANS be able to do it? When you see "infinity" on a computer, that's just a pre-defined bitstring that the computer interprets as "infinity", just like all other numbers are pre-defined bitstrings interpreted as the computer as explicit numbers. If you desire, I would be glad to have a sitdown discussion with you about computer architecture.

The universe, on the other hand, is not limited. This is the reason why we have the <b>concept</b> (did you get that) of infinity, because the human mind can comprehend this concept and it is a useful concept in mathematics.

Your "belief" that the universe is a giant program does not make it so. Reject your beliefs and accept science.

tl;dr - stop trollin and lrn2math

[blackmage_nuke]

The universe has it's limits, or atleast you cant prove it doesnt

[Vyk]

If you use it in something its going to come out so close it won't really matter. But technically and logically they are different

If you placed the lines on a graph, they'd be close. You might argue they'd even be touching. But they wouldn't be the exact same

[Jessweeee♪]

1 = 2

[NeoTifa]

SO, can anyone here tell me what number exists between 1 and .999....?



Okay, do you remember infinity? There are an infinite number of numbers after the decimal. True, the number approaches 1 as it goes further into infinity, but to say they are equal is just ignorant. We round and assume for simplicity's sake, but they are not equal!

0.999999999999 !=
0.99999999999999999999999 !=
0.999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999999999999999999999999999 !=
1

Those are all different numbers.

[Aerith's Knight]

This is actually all irrelevant seeing as 1/3 and 0.3333... are two different numbers.

[Captain Maxx Power]

It's from this misunderstanding that the equation 1/3 + 2/3 = 3/3 / 0.333... + 0.666... = 0.999.... can arise, because you're essentially adding together decimals that aren't a correct representation of the fraction being shown. You can't add infinite series of decimals together even if logically they make sense because they are infinitely long.

This makes no sense. It's not a "misunderstanding," it's adding together rational numbers.

Rational number shows what happens when you add 1/3 and 2/3 together;

1 / 3 + 2 / 3 = (1 * 3) + (3 * 2) / (3 * 3) = 3 + 6 / 9 = 9 / 9 = 1

In case I wasn't clear this is what I mean; in any given situation involving an infinite series of decimal numbers, it's not possible to give an accurate answer to any kind of function performed on the number, especially when doing so with another infinite series, because the series are infinite. It's possible to solve this expression via the above fractions because we are dealing with a finite set of values.



This is a falsity because it only applies to finite decimal values. If a given value is an infinite series then it becomes impossible to locate it within a number line because of this as the inherent value of that number is impossible to place.

This is also wrong, and I'm not even sure where you got this. It is a basic rule of algebra that any two, distinct real numbers have an infinite number of other distinct, real numbers between them. 0.999... is a real number.

Probably a bit of cognitive dissonance on my part, especially since I was working around the analogy of a number line. Technically speaking within the realms of real numbers, 0.999... is equal to 1. But this is done on the basis that Infinite isn't just a concept but a tangible, mathematical reality. I was attempting to rationalize why for example in the real world you can't cut a pie into three pieces and lose an infinitely small fraction of the pie. Then again we are talking about a profession that could probably prove the Universe doesn't exist technically, and which I only have a bare bones understanding thereof.

[rubah]

The real question we are asking is does 1/infinity truly equal to zero? becaus 0.999... is really 1 - 1/infinity

Infinity is not a finite, real number.


Yet we constantly use it in other proofs
you use the limit definition. You don't just use 'okay, 1/0=∞' or if you do, I'm sure your teacher marks you off :p

[Tavrobel]

0.999999999999 !=
0.99999999999999999999999 !=
0.999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999999999999999999999999999 !=
1

Those are all different numbers.

Because you are giving us finite numbers, numbers that have a distinct end. .9... is infinite; it goes on forever, and thus, you can't determine its end. Think of addition, which side to you start from when you want to add numbers? The right? But there is no right end in an infinite number.

Has anyone here (who is wrong) ever considered that we can have different looking numbers that represent the same thing? 1/2 == 4/8. No one has a problem with those once they have the knowledge. .9... and 1 are just another part of the stable of identical numbers; it's no different, you just need higher level math to prove it. Ohh, and there's plenty of proofs.

As I see it, all of this can be blamed on the awesome that is Calculus. Not only does it screw with minds, but it screws with lives and creates unnecessary DORAMA. .9... == 1, though. Just use the "can you find a number in between them" proof. There's also a proof involving limits and Taylor series, but I cba to find my math notes.

[NeoTifa]

I was using finite numbers to show that they are not the same, that there are always infinately more 9's to add.

[NeoCracker]

Hell, I learned this In my trigonometry class. :p

And a number that goes on infinately is not irattional, just impractical to use.

There is no real practical use to knowing 1 = 0.999..., but that doesn't change the fact that it does. :p

And PG made 2 of those threads? WTF?


Edit: Neotifa, you can't add a 9 to the end of something that has no end. So in this case, there are not always 9's to add to the end, proving your argument false.

[Tavrobel]

I was using finite numbers to show that they are not the same, that there are always infinately more 9's to add.

You do know that the "..." already makes that assumption, right? There's no need for you to do it for us, because it would be impossible to show by spamming 9s.

Do you even know what the concept of infinite means?

[oddler]

0 < 1

Just use the ones place. :monster:

[Madame Adequate]

I was using finite numbers to show that they are not the same, that there are always infinately more 9's to add.

But being infinite means it isn't something you can parallel with finite numbers. 0.999... doesn't mean a lot of 9's and then you can add a few more to make a number between 0.999... and 1. It means that go on forever and there is no end, ever. At all. There is nowhere you can put another 9 on the end of because it has no end.

I know that 0.999... = 1 looks counterintuitive. I get that. The first time I saw it was scoffed at the obvious nonsense. But it is true despite the fact that it looks weird to the uninitiated.

[rubah]

Let's do this thread again, but in base 3.

[NeoTifa]

Ughh, what the fuck ever. You're not understanding what I'm saying. Whatever. Suck it.

[Denmark]

I love arguments like this.
A: *something*
B: *logical contradiction to thing*
A: WHATEVA

@rubah: 0.222... = 1

EDITGA²: Being able to argue in this thread first requires a firm background in Calculus and what 0.999... ACTUALLY MEANS. If you don't have this, you're doing it wrong and should probably just stop trying now.

[qwertysaur]

Like I said before, disprove it's false. That is the only way to fully prove it is true.

[Ouch!]

Like I said before, disprove it's false. That is the only way to fully prove it is true.

Burden of proof lies on the positive claim and all that junk?

[NeoCracker]

SHow me a couple examples of how to disprove somethings false, we can give it a try. :p

And isn't all the evidence that points to it being true disproving it's false? Isn't that, by definition, HOW you disprove something is false, by showing how it is true?

[Tavrobel]

If something is false, that doesn't make its opposite necessarily true.

[NeoCracker]

In this case though, it kind of does? Does it or does it not equal 1? If No is false, Yes is the only logical answer. Though yes, in most situations, I agree with Tav.

But that also means in this situation, proving it true will prove the opposite false, simply because there are only two viable options for answers, yes or no. And either one or the other is true.

And if all the evidence points to yes being true, no is false. Unless someone can explain how it can be both equal to, and not equal to 1 at the same time. :p

[Jowy]

shut up, that's why.

[rubah]

watch out for converse error!

P -> Q
~Q -> ~P
Q -/> P
~P -/> ~Q

[qwertysaur]

Math has some slight changes to certain definitions. True means "never false", not the opposite of false.

The simplest method to disprove is through indirect proofs, where you force a contradiction. But that is only usefull to disprove truth, because you only have to do that once to finish.

However usually you want to disprove that something is true, wher you have to do it once and be done with it. Disproving it is false is exponentially harder, as you need to check every case imaginable and make sure that no slight variation in the execution of the theory will result in a false result or a contradiction in what is aready known. It is really tedious and annoying to do, but it is possible to disprove something is false. Another method is an implicit proof where you prove it for any number n and then n + 1, then test for any exclisions that would create a bad result to exclude.

In short, there is no way to actually disprovce 0.999 = 1 is false. There is a good reason you get to name a discovery after yourself, the math community is very picky in accepting new ideas and tests the life out of it to try and break it.

[NeoCracker]

From what I'm gathering, you only need to disprove it's false if it's going to have some form of application or another, just in case there is a single situation out there it won't be true. This is a process that is likely used just to be on the safe side.

IN short, if you can't disprove something to be false, it hardly means it isn't true. It just means the community wont' risk using on the off chance it's wrong.

Given the impractical use of .999... and the fact I see no way it could ever come up, having to disprove its false is an entirely pointless excersize, regardless of if it was possible or not. :p

Either that, or the Idea of disproving falseness has nothing to do with eliminating risks, and all mathemticians are just Elitists assholes. And if thats the case, it still has no real relevance to this conversation. :p

[blackmage_nuke]

If there was an omnipotent being similar to 'God' who looked at infinity, what would it see?

[Tavrobel]

It's a cosmetic adjustment. Apparently, a single digit is too bland for the likes of hardcore mathematicians, and thus we have to use bigger numbers and involve concepts into our work.

Mathematicians are elitist assholes because their work is the purest (that is, unfettered by a need of knowledge in any other discipline). It takes a certain personality to stomach higher level math. After the necessary knowledge that we need to prove .9... == 1, math starts to involve far fewer numbers, and more involved concepts.

[rubah]

It really has less to do with covering your ass than you're thinking.

take a discrete math course!

[NeoCracker]

On the note of math..

YouTube - I Will Derive! (http://www.youtube.com/watch?v=P9dpTTpjymE&feature=rec-HM-r2)

[Raistlin]

0.999999999999 !=
0.99999999999999999999999 !=
0.999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999 !=
0.9999999999999999999999999999999999999999999999999999999999999999999999999999999 !=
1

Those are all different numbers.

Um, yeah? So? But when the amount of nines after the decimal is infinite, it equals 1. This is a fact. It's not opinion or debate, it's been proven.

[Darkswordofchaos]

.999... is not the same number as 1 if you use the two in application the diffrence in the answers youd get would be so miniscule you wouldent even be able to see it so you would apear to have the same answer. but there is a space between 1 and .999.. its just infinitly small. so in a practical and mathmatical sense they are the same number but if you wanna get down to the nity gritty technicaly no they are not the same number

[Tavrobel]

No, they are the same number. The difference is not infinitely small; it's non existent.

If you can find me a number between .9... and 1, then I'll admit that they have a difference. But since you can't, because you can't reach the end of .9..., then they are the same number. There is no difference; it is neither a practical nor technical difference. It is identity.

[NeoCracker]

Mathmatically you are getting to the nity grity technicalities. If it is getting infinately closer, you would be able to list an infinite number of numbers between the two, yet it is impossible to find but one.

And no, the is no miniscule difference in the answer, there is no difference.

Edit: That part I deleated meant absolutely nothing close to what I had intended, so for those who read it, my bad. :p

[oddler]

EDITGA²: Being able to argue in this thread first requires a firm background in Calculus and what 0.999... ACTUALLY MEANS. If you don't have this, you're doing it wrong and should probably just stop trying now.

Peace! See you guys later. :strut:

[NeoTifa]

Everybody just shut up and dance! (http://www.youtube.com/watch?v=HcOZ6xFxJqg)

[Darkswordofchaos]

i wish i was still in school so i could ask my calculus teacher she would no

edit: you cant use a number that goes on for ever so for practicalitys sake you would have to eventualy stop it and that number would be diffrent than one but i suppose .999... would = 1 however i refuse to belive they are the same number

[Raistlin]

No, your calculus teacher would, if she knew what she was talking about, say yes. The proof is based on calculus, in fact.


.999... is not the same number as 1 if you use the two in application the diffrence in the answers youd get would be so miniscule you wouldent even be able to see it so you would apear to have the same answer. but there is a space between 1 and .999.. its just infinitly small. so in a practical and mathmatical sense they are the same number but if you wanna get down to the nity gritty technicaly no they are not the same number

That makes absolutely no sense. Between any two distinct, real numbers there exists an infinite number of other, distinct real numbers. 0.999... is clearly a real number because it is a rational number, and since you're claiming that it is distinct from 1 there should be an infinite number of other numbers between them. How about you list some?

[Tavrobel]

i wish i was still in school so i could ask my calculus teacher she would no

edit: you cant use a number that goes on for ever so for practicalitys sake you would have to eventualy stop it and that number would be diffrent than one but i suppose .999... would = 1 however i refuse to belive they are the same number

Your Calculus teacher would say yes, actually, if she were qualified to teach it. Whether or not you can use a number that is infinitely long is not the discussion here. It's about if .9... == 1, which it does. I suppose that you are willing to accept that 2/4 == 4/8 == 16/32 == 1/2, but they don't look the same, either. What's wrong with this one? There are several proofs to the situation already mentioned. It's no logical quandary.

EDIT: Man, Raist, don't ninja me.

[Darkswordofchaos]

i retract all previous statments upon studying this i have come to the conclusion that .999..... is the same as 1 my confusion comes from the fact that most mathmaticians wouldent put 1 in the place of .9999..... the would put say .9999999999999999999999999999 and end it. please forgive my ignorance

[Raistlin]

I found the calculus proof posted in an old thread. Here it is:


lim (n->infinity) Σ(i=1->n) 9/(10^i)=lim (n->infinity) Σ(i=1->n) (9/10)(1/10^(n-1))

which sets up a property of Riemann sums, so this equation becomes:

lim (n->infinity) (9/10)(1-(1/10^n))/(1-(1/10))

As n->infinity, the denominator of 1/10^n become larger and larger, and the quotient gets closer and closer to zero, and thus lim (n->infinity) 1/10^n=0. So we have:

(9/10)(1-0)/(9/10)=1

[Rye]

This entire thread = 50 patoots.

[NeoCracker]

Welcome to math Rye. :p

[rubah]

quick, integrate her before she escapes our bounds!

[Aerith's Knight]

If we convert her to a Taylor polynomal, she can't run due to her infinite nature!

[Ultima Shadow]

I was using finite numbers to show that they are not the same, that there are always infinately more 9's to add.Infinity/infinity = infinity

infinity x infinity = infinity

infinity + infinity = infinity

infinity - infinity = infinity

No matter how much you add or take away from infinity, it won't change the slightest bit.



As for regarding .999 = 1. Well, the difference is infinitely small, thus it basically don't exist. Or rather, it's not allowed to exist, since it's impossible to define. Something like that. >_>

[Rye]

Noooooo, not math! :cry:

I just finished my last math requirement of my life in my first semester of college! I can't go back now! My math skills will stay and die in the Pre-Calc era of my life, because that was the highest math I got to. xD

Now that I don't have to take math anymore, I've not gotten anything lower than an A- in any of my classes! G-G-GPA boost!

(Incidentally, I did like really simple polynomials though. And by simple, I mean the really simple ones you learn in 8 or 9th grade. They were kind of fun, though I'm not sure I'd still be able to do them. :greenie:)

[Tavrobel]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.

There is no difference; it's not infinitely small, it's non existent.

[Aerith's Knight]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.

There is no difference; it's not infinitely small, it's non existent.

Infinity/infinity = infinity

This one demands the theory of L'opital. You have to differentiate both formulas until infinity is on only one side of the divider.

[Denmark]

SOMEONE SAVE US FROM THE WRETCHED FUTILE ARGUMENT

Guys, guys, guys. If there is a proof (<a href = "http://en.wikipedia.org/wiki/0.999#Proofs">and there are several</a>) of something, THEN IT IS PROVEN TO BE TRUE AND THERE IS NO REASON TO DEBATE ABOUT IT.

How many people here are mathematics professors? How many? Raise your hands high. If your hand isn't raised, STOP FUCKING DEBATING THIS.

*closes thread*

[Aerith's Knight]

Because wikipedia has a foundation in truthful topics? :p

It's all debateable, my friend.

[Denmark]

Show me your mathematics PhD.

Also, all of the proofs are referenced.

BLIND ACCEPTANCE. MATH = SCIENCE = RELIGION?

[Raistlin]

As for regarding .999 = 1. Well, the difference is infinitely small, thus it basically don't exist. Or rather, it's not allowed to exist, since it's impossible to define. Something like that. >_>

No, wrong. There is no difference at all. They are just as equal as 1/3 and 0.333.... "Infinitely small" is a meaningless term. The only useable definition would perhaps be lim (x->infinity) of (1/x), which equals 0 anyway.

[Peegee]

Because wikipedia has a foundation in truthful topics? :p

It's all debateable, my friend.

Agreed. You can debate the merits of the claim, but often with mathematical theorems, any attempt to debate it usually requires showing an error in the proof. This requires an understanding of what the proof is attempting to show.

Just because it's debatable doesn't mean it's not futile though. For example arguing against the claim that 2 > 1 is silly. Or not understanding that 2 = 1 + 1 and demanding a proof shows an elementary misunderstanding of the relationship between numbers.

I am personally shocked that the concept of infinitesimals have not been brought up.

DARN IT PG SHUSH! Oh crap, here goes

[Darkswordofchaos]

well for for the dif between .9999.... and 1 to be infinitesimal, .999... would eventually have to end witch it dosent so there is no need to bring that up in this discussion

[Madame Adequate]

Because wikipedia has a foundation in truthful topics? :p

It's all debateable, my friend.

As much as the Encyclopedia Brittanica does :monster:

[Dignified Pauper]

the only number that can be between .999... and 1 is .00...01 where there are an infinite amount of zeroes. However, eventually, we reach a stop at the 01 where the 9s continue forever, and when that is the case you reach a number where n > 1.

.999... for all intensive purposes has no mathematical difference to 1

[Tavrobel]

Infinity/infinity = infinity

This one demands the theory of L'opital. You have to differentiate both formulas until infinity is on only one side of the divider.

Well, we know that but not the people who were wrong know that and it would stand to reason that if you don't understand why .9... == 1, then you wouldn't understand limits and differentiation, anyways, which means it effectively couldn't be evaluated. Besides, it could equal zero too.

Denmark, it takes way less than a PhD to argue mathematics. Just ask a computer science major.

[Darkswordofchaos]

Denmark, it takes way less than a PhD to argue mathematics. Just ask a computer science major.

yeah you only need a bachelor

[Raistlin]

I understood and accepted that .999... = 1 when I was 13 when it was first explained to me. Not the calculus proof, of course, which involves limits, but the algebraic ones are easy enough to understand.

[Dignified Pauper]

According to Arithmetic: 0/0 = 1, 0, undefined (infinity) too...

[Raistlin]

Marick, I will smack you. :mad2:


the only number that can be between .999... and 1 is .00...01 where there are an infinite amount of zeroes.

How can you even have a 1 at the END of an "infinite amount of zeroes?" That's a contradiction. If the zeroes are infinite, there is no end.


.999... for all intensive purposes has no mathematical difference to 1

1. It's "for all intents and purposes." :p
2. Not only is it true for all intents and purposes, it doesn't even need that qualification. It's EXACTLY equal.

EDIT: haha, I thought you'd edited your post but I realized it was on the end of my last page that I had missed earlier.

EDIT 2:
According to Arithmetic: 0/0 = 1, 0, undefined (infinity) too...

Only one of those is right (undefined). And undefined does not equate to infinity, it means just what it says: undefined.

[Dignified Pauper]

I should have explained that better.

The arithmetic part was a joke.

And, in the .00...01 example, I even disproved it.

That said, it's still funny why anyone argues over it.

I could say infinity/infinity = 1, infinity.

[yuno]

you dont even need calculus to prove this.i teach this in my algebra classes.we have to first agree that "0.999..." is in fact a real number (this makes sense,since 0.999... is indeed a repeating decimal),and then all we need are the axioms of equality.

proof:let x=0.999...
=> 10(x)=10(0.999...) (by multiplication property of equality or MPE)
=> 10x=9.999...
=> 10x-0.999...=9.999...-0.999... (by addition property of equality or APE)
=> 10x-x=9.999...-0.999... (here we replaced the 0.999... on the left side by x,since they are equal)
=> 9x=9
=> x=1 (again,by MPE)
now,by transitivity of equality,since x is equal to both 0.999... and 1,then 0.999...=1.

QUOD ERAT DEMONSTRANDUM.

a few remarks:in this proof,please note that the addition and multiplication properties of equality only require that we add/multiply real numbers,regardless of whether these real numbers are terminating,infinitely repeating or non-terminating,non-repeating decimals.

and i have made this thread just a tiny bit boring.

[Raistlin]

Math is fun, not boring. Also, yeah, there are algebraic proofs which were discussed earlier this thread (and all the previous threads on this), and have the benefit of being much easier to understand for a layperson. The calculus proof just allows you to demonstrate how an infinite string of 9's after a decimal does, in fact, equal 1.

This debate was apparently so ubiquitous on the WoW forums that Blizzard even made an announcement a few years ago on its website that .999... does, in fact, equal 1.

[Tavrobel]

QUOD ERAT DEMONSTRANDUM.

By multiplying .9... by 10, that would give it an end with 0, even though you can't technically get there, the resulting number is no longer properly infinite.


This debate was apparently so ubiquitous on the WoW forums that Blizzard even made an announcement a few years ago on its website that .999... does, in fact, equal 1.

It didn't even take WoW for Blizzard fans to go nuts with it; it's been around on the WCIII and DII boards long before WoW was released in 2004. Sometime shortly after the release of Frozen Throne, one of their April Fools jokes was replacing the whole of the website with a giant plaster stating that .999... = 1.

[I Took the Red Pill]

EDIT 2:
According to Arithmetic: 0/0 = 1, 0, undefined (infinity) too...

Only one of those is right (undefined). And undefined does not equate to infinity, it means just what it says: undefined.No, it's indeterminate. Quite different from being undefined. If you define division in terms of multiplication (18/3=6 because 6x3=18), and then look at something like 3/0, you can clearly see that we cannot define an answer for this because there is absolutely no number that, when multiplied by zero, will result in the number three. Hence the term undefined. 0/0, however, is different. Let's look at it in terms of multiplication then. Is the answer zero? Well, 0x0=0, so that would certainly work. But what about the identity that anything divided by itself is 1? Would 1 work here? Well, 1x0=0, so yes. It now becomes clear that the answer works for all real numbers. 0/0 could be equal to sqrt(2), because sqrt(2)x0=0. Whereas an expression being undefined means we literally cannot define an answer with our current definition of multiplication, the indeterminate form is quite in its own category and becomes integral (pun intended) in laying the foundations for differential calculus.

Yeah I'm not here to jump into the whole .999...=1 argument, because anyone denying the fact is, well, thickheaded and wrong. Just here to clear up some mathematical jargon.

[Raistlin]

By multiplying .9... by 10, that would give it an end with 0, even though you can't technically get there, the resulting number is no longer properly infinite.

No, the algebraic proof is valid. If the nines are infinite, where's the end that you'd stick the 0? Infinity minus one is still infinity. The end result would simply be moving the decimal point over so it'd be 9.999... (still goes onto infinity).

[Tavrobel]

No, the algebraic proof is valid. If the nines are infinite, where's the end that you'd stick the 0? Infinity minus one is still infinity. The end result would simply be moving the decimal point over so it'd be 9.999... (still goes onto infinity).

Well, as it turns out, Blizzard said it to be true, so therefore, I must believe to also be true.

In case you cared, their .999... = 1 spiel was 2004, several months before WoW was released.

[Raistlin]

Yeah, I saw it in your previous post. I just didn't see it until after WoW was released. Regardless of when it was, that action shows it was a hot topic of debate.

Also, you all fail for not noticing my calculus error earlier in the thread, which I just caught. I had lim (x->0) of (1/x) = 0 (when it is actually infinity), when it should be lim (x->infinity) of (1/x) = 0.

[Tavrobel]

Woah man, math and computer notation don't mix, okay?

And I figured that you had seen my comment, but not known the precise time when it occurred. I've been trying to find a screenshot of the whole spiel, but to no avail. I believe that it had Kel'Thuzad's picture as a background for the text, but I can't be precisely sure.

[Darkswordofchaos]

not in math but in real life there is no such thing as .999.... cause you cant have a never ending part of somthing in the real world so it has to be that .999... equals 1 .666666..... =2/3 and .3333333 equalls 1/3 simple as that

[eestlinc]

we want to believe that .9999... < 1 because there is that receding tiny difference that goes on and on. But there is only a difference if we stop and assess at a specific number of 9s. Once we go on infinitely, there is no difference and .9999... = 1 because you never stop at any set of 9s.

[Ultima Shadow]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.
Well, let's say you have an infinite line of red and yellow sticks (yes, sticks. Because sticks are cool). Now, take away all the yellow ones. Since the amount of coloured sticks total is infinite, there's an infinite amount of yellow sticks and also an infinite amount of red sticks. The amount you take away (all the yellow ones), is infinite, yet what remains (the red ones) is also infinite. There, just took away an infinite from an infinity, yet infinity remains. In other words, infinity - infinity = infinity is 100% possible.

As for dividing, it's exactly the same. There's just an infinite amount of "different coloured sticks" that you divide into an infinite amount of infinite lines.

If you do it that way, it's possible.:greenie:

[I Took the Red Pill]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.
Well, let's say you have an infinite line of red and yellow sticks (yes, sticks. Because sticks are cool). Now, take away all the yellow ones. Since the amount of coloured sticks total is infinite, there's an infinite amount of yellow sticks and also an infinite amount of red sticks. The amount you take away (all the yellow ones), is infinite, yet what remains (the red ones) is also infinite. There, just took away an infinite from an infinity, yet infinity remains. In other words, infinity - infinity = infinity is 100% possible.

As for dividing, it's exactly the same. There's just an infinite amount of "different coloured sticks" that you divide into an infinite amount of infinite lines.

If you do it that way, it's possible.:greenie:
No. It's undefined. Lrn2Cardinality. The problem arises because there can be different "kinds" of infinities. For example, are there more whole numbers or perfect squares? The answer, to both, is infinite, even though that would defy intuition; there has to more whole numbers than perfect squares, right? Nah, they're just different infinities. That's one reason why ∞ - ∞ is undefined. Not to mention the fact that infinity is not a number and doing operations on it is silly.

If you want something more concrete and with an actual example, check this out. (http://www.philforhumanity.com/Infinity_Minus_Infinity.html)

edit: what you're saying is basically the same as the example in the link I posted.

Assume ∞ - ∞ = 0

Add infinity to both sides: (∞ + ∞) - ∞ = 0+ ∞

Since infinity plus infinity is infinity, and 0 plus infinity is infinity, you get

∞ - ∞ = ∞

In the same way that you can assume that ∞ - ∞ = 0, add pi to both sides, getting (∞ + pi) -∞ = pi, and since ∞ + pi = infinity, you get ∞ - ∞ = pi.

It all has to do with the Cardinality of infinites.

[KentaRawr!]

Hey guys, what do you call a bird that doesn't eat?

A polynomial.

Hahahaha.

[Ultima Shadow]

No. It's undefined. Lrn2Cardinality. The problem arises because there can be different "kinds" of infinities. For example, are there more whole numbers or perfect squares? The answer, to both, is infinite, even though that would defy intuition; there has to more whole numbers than perfect squares, right? Nah, they're just different infinities. That's one reason why ∞ - ∞ is undefined. Not to mention the fact that infinity is not a number and doing operations on it is silly.

If you want something more concrete and with an actual example, check this out. (http://www.philforhumanity.com/Infinity_Minus_Infinity.html)

edit: what you're saying is basically the same as the example in the link I posted.

Assume ∞ - ∞ = 0

Add infinity to both sides: (∞ + ∞) - ∞ = 0+ ∞

Since infinity plus infinity is infinity, and 0 plus infinity is infinity, you get

∞ - ∞ = ∞

In the same way that you can assume that ∞ - ∞ = 0, add pi to both sides, getting (∞ + pi) -∞ = pi, and since ∞ + pi = infinity, you get ∞ - ∞ = pi.

It all has to do with the Cardinality of infinites.
Oh, what the heck. Fine then. >_>

[yuno]

infinity minus infinity is not undefined,it is indeterminate.more specifically,it is what we mathematicians call an indeterminate form (http://en.wikipedia.org/wiki/Indeterminate_form).indeterminate is very much different from undefined.undefined means we cannot assign any real number to be the answer.on the other hand,indeterminate means the answer may still be any number;we just are not sure yet.

consider the following:the expression x+1 blows up to infinity as x goes to infinity.the expression x+2 blows up to infinity as well if we let x go to infinity.therefore,the expression (x+2)-(x+1) represents infinity minus infinity as x approaches infinity.but here,we know that (x+2)-(x+1)=1,so here,even if x approaches infinity,infinity minus infinity is equal to 1.

next,consider the following two expressions x+1 and x+3.again,both blow up to infinity as x approaches infinity.so (x+3)-(x+1) represents infinity minus infinity again.but now,infinity minus infinity is 2.

another!how about 2x+1 and x+1?again,both approach infinity as x approaches infinity,so this is also infinity minus infinity.but (2x+1)-(x+1) is x,so here,infinity minus infinity is infinity!zomgwtfbbq!

to put it simply,indeterminate means the answer cannot be determined yet.as i tell my students,indeterminate means we are not yet done.


Math is fun, not boring.

i absolutely (not conditionally I cried aloud with mirth and merriment) agree.i cant say much about other people though.

[Jiro]

I wish I was smart enough to take all this higher level maths. It is quite brilliant, I just fail at grasping it.

I'll try to explain the concept for everyone who, like me, has no idea what is going on and you smarter people tell me if I'm correct.

Even though .9... is not a rational number, it is still equal to the number 1, as there is no number between them. Y/N? Also, does this mean that 1 == 1.1..., and therefore .9...==1.1...?

[Big D]

Even though .9... is not a rational number, it is still equal to the number 1, as there is no number between them. Y/N?Y
Also, does this mean that 1 == 1.1..., and therefore .9...==1.1...?Nope, 'cause 1.1 recurring is higher than 1.1, which is higher than 1.09, and so on. You can get numbers that are lower than 1.1..., but higher than 1.

[Jiro]

Yeah, I thought about that, but wasn't sure. :)

[yuno]

hey,0.999... is in fact rational,because it is a repeating decimal.

[Raistlin]

Even though .9... is not a rational number, it is still equal to the number 1, as there is no number between them. Y/N? Also, does this mean that 1 == 1.1..., and therefore .9...==1.1...?

No. 0.999... is a rational number because it is a repeating decimal that can be represented as a relationship between two integers (9/9, or 1).

And how do you conclude that 1 = 1.1...? 1.1... is 10/9, not 1. And that's easy to see, because there's an infinite amount of numbers between 1 and 1.1... (i.e., 1.01, 1.001, 1.0001, etc.).

[Jiro]

I don't conclude anything, I was trying to make sense of it because my brain is smaller than everyone else's on here ^__^

[Raistlin]

This is rather counter-intuitive. At least you're accepting it instead of vehemently arguing against it. Ignorance by itself and be forgiven and rectified.

[rubah]

jiro, try 1.099999999999999999999999999999999999etc = 1.1

[Peegee]

Considering how many people tend to argue against this equality, is the equality very easily understood by mathematicians / math students of a certain level?

I am just imagining a room full of professors arguing that 0.9999... is not equal to 1

[Raistlin]

Nah, I doubt any serious math professors argue about this ever, because it's been very conclusively demonstrated. It's only counter-intuitive until you understand limits, and then it all fits. So basically, only lay people tend to argue about it, as shown by the biggest arguments about this are found on video game message boards. :p

[NeoCracker]

I didn't even think this would be argument worthy. I was just really bored and, for some reason or another, remember that one guy in my Trig Class getting all uppity about it not being true. XD

[blackmage_nuke]

0.999 is definitely equal to 1, anyone who said otherwise in this thread is a raving lunatic

[Tavrobel]

Nah, I doubt any serious math professors argue about this ever, because it's been very conclusively demonstrated. It's only counter-intuitive until you understand limits, and then it all fits. So basically, only lay people tend to argue about it, as shown by the biggest arguments about this are found on video game message boards. :p

Professors only ever talk about this on the first day of introducing series and sequences (if that). Otherwise, students go to their Discussion classes, the topic is brought up, and the TA simply goes on and proves it, to the amazement of all of the math students in the classroom. Frequently, the only expression to explain it is "oo," "cool," and "awesome." There's no objection, because it's trivial.

More people in this thread just need their Calculus II.

[Jiro]

This is rather counter-intuitive. At least you're accepting it instead of vehemently arguing against it. Ignorance by itself and be forgiven and rectified.

I'm not going to argue against something that requires knowledge outside of my own to understand. One day I hope to fully understand why, rather than just knowing it's true.

Still, I find it funny that so many people argue the point.

[NeoCracker]

0.999 is definitely equal to 1, anyone who said otherwise in this thread is a raving lunatic

Incorrect sir. 0.999 equals no such thing.

0.999... however equals one. :p

[blackmage_nuke]

It was implied by context!

[NeoCracker]

Maybe thats not good enough for me?

[Jiro]

NeoCracker has a good point - there is no room for error in maths!

[Timekeeper]

[QUOTE=Raistlin;2697896]I'm not going to argue against something that requires knowledge outside of my own to understand. One day I hope to fully understand why, rather than just knowing it's true.
This is the difference between knowledge and wisdom :)

[Jiro]

How did you end up quoting Raist? And yes I understand what you're saying, you explained that thing to me once.