Did you know that 0.999999999999999=1?

I have another one!!!!!

At a party last weekend, a math student taught me about different kinds of infinities. It seems like infinity should just be infinity, right? But some infinities are bigger than others.

For instance, between 1 and 2 there are infinity decimals.

And then between 2 and 3 there are also infinity decimals.

And then between 1 and 3 there are ALSO infinity decimals.

But the infinity decimals between 1 and 3 is, obviously, bigger than the infinity decimals between 1 and 2 because it includes the infinity between 1 and 2 AND the infinity between 2 and 3!

WILD, huh?!

No wonder the time between the first Margarita and the second seems so short compared to the first to the third. Lordy. Thank god I don’t start with the first and then go to the third…I’d never make it.

And, then there are orgasms pancake never mind.

how bored were you when you calculated this?

Haha I had to check, and it works :P That’s cooooooooooooooooooool.

Where is finkelitis when we need him?

Fun question, but it doesn’t work. You’re putting a 0.999… on the right side of the equation whilst leaving the left-side X untouched. In essence, you’re faking your way through a two variable inequality.

The premise of the question is incorrect.

0.999999999999999999999 is not equal to one.
0.999999999999999999999… is equal to one.

That is, 0.9 followed by an infinite sequence of 9’s is equal to one.

It is, of course, not actually possible to write an infinite number of 9s.

And the concept of degrees of infinity is well defined and well explained in mathematics.

The wiki link does a better job of explaining the infinity principle.
I was about to say something when I noticed the link.

I learned this in my calculus class the other day. literally blew my mind!!

@Robmandu

Look it up. The math checks out.

Yes.

@ MrItty

It was implied (to the reader) that the 9’s were infinite. I just don’t know how to do a 9 with a bar over it on a keyboard.

the link showed 0.(9) as an alternate display.
But i doubt that anyone would have known what that meant…

I vaguely remembering learning about this, but I absolutely forget why mathematically it was not true. Rob, the math does work out though.

Something else quite interesting is if you take a look at fractions and there equivalent decimal you get, 6/9 = .66666, 7/9 = .77777, 8/9 = .88888, 9/9 = 1.

Rob; read the link article. There is more than one proof.

Another example of this is that 0.33333 (let me point out that the 3’s are repeated to all you MrItty’s out there) is exactly 1/3. Most people have no problem accepting that 0.333333 is exactly 1/3 but when you say that 0.999999=1 they refuse to believe it.

I have trouble believing in the repeating decimal reaching some number rather than approaching it.

I have trouble believing that there are an infinite number of steps between here and the door.
My feet are much bigger than that.

@fireside; check out Zeno’s paradox.

I don’t believe that 0.33333 == 1/3. It’s a decimal approximation. Same goes 3.1415926535897932384626433832795 being a decimal approximation for π.

That said, I’m freakin’ out just a little doing the math for myself with 0.99999… It does work out to equal 1, dammit. Weird though that when attempted with 0.111111… or 0.22222… or 0.88888… you always end up back with the original identity.

This trick seems only to work for 0.9999…

Math is stupid. ツ

Rob; read some of the other proofs in El’s link.

yah, I did. Just didn’t want to bog down with “nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999….”

@R; you are almost as funny as Milo.

I was just about to say that, Robmandu.

Trippy stuff, huh?

YES! I did know that. A few months ago, we a had a longgg argument about this in the chat room. pete(thepothead) and Allie disagreed with me, uberbatman and lefteh about it. They still won’t agree that 0.9999…..=1 . So I am gladdddddd you asked this. Now I just need to link them to this thread :D

@PnL

haha!

Well I want a math prof. to get on here and validate it once and for all.
Most people don’t believe it but it’s freaking true!

I hate this thread. (Nothing personal.)

@Gail – so the race is already won!
or do i still have to take the steps to get across the room?

@fireside

Great link!

@robmandu

By the same logic, 0.333… = 1/3. If X = 0.333… then 10X = 3.333… then 9X = 3.333… – X then 9X = 3.0 then X = 1/3. Not really a decimal approximation since (as Mritty said previously) 0.(3) is not a decimal, but a decimal representation of a fraction. Put another way (as the wikipedia page proves), think of a limit function f(x) as x approaches infinity.

Oh, JesusfuckingChrist. You have got to be kidding me.
I can’t believe this question has come back to haunt me—on Fluther!
I’ve already spent nearly a week debating this very question already. Gah.

For all real world purposes, I can see why 0.(9)=1. But if you rewrite it as a limit,
by definition shouldn’t x be approaching 1, (but never actually reaching it)?

Apparently though, I am the lone dissenter.
Nearly the entire Math graduate department seems to agree that 0.(9)=1.

Fucking math.

Oy! Keep the change already!!

Nimis: I agreed with you about the limit of x is almost one (in an answer ^^ somewhere above.) Is your math department familiar with Tom Lehrer’s New Math? Of course it is.

Ah, yes. You…and I suspect Petethepothead and Allie too.
I meant lone dissenter the first time ‘round. =)

I don’t care what anyone says.

.9999… and 1 are not the same number, and never will be.

How about this: If X = 0.(9) then it’s already solved. Just leave it alone and don’t ask any questions. You might end up making the universe implode. And no universe means no Fluther.

Simple Math.

GO MATH!!

Ok.

I think I figured it out. At least for me.

The “trick” to the so-called proofs is that they obfuscate the importance of the decimal place. By having an “infinite” repetition of nines, the 10x – x = 9.999… -0.999… appears to yield 9x = 9. But it does not. Or it does, if you’re willing to ignore the decimal.

Here’s why. Try this without the repetitive nines.

x = 0.99
10x = 9.9 <== here’s your first clue.
10x – x = 9.9 – 0.99
9x = 8.91 <=== see that, how by respecting the decimal place earlier, you get a non-magical value?
x = 0.99
x ≠ 1

Now I can imagine some responses.

1) But look at all the other proofs!—I don’t care about other proofs. I need only disprove it once.

2) But this is about repetitive nines!—Yah, I know. But as I mentioned earlier with pi, there are many cases where a decimal number can only approximate a value, never define it completely.

3) But really, this is about repetitive nines!—If I put you on the 10 yard line, and with each play you get half-way to the goal line, how many plays will it take for you to get a touchdown? Infinity!

4) But, why are you hung up on the decimal place?—Look at your paycheck. Move the decimal one place to the left. Tell me you don’t care.

5) Last time, this is about the repetitive nines, for cryin’ out loud!—Yes, it is. Because of the supposed infinite list of nines, it allows us to be sloppy. That’s the point of the proof. But when you must account for the decimal places by using a finite, easy to comprehend number of digits, then you can understand that the same must be done for the repetitive nines, too. In the wiki proof, that’s not being done.

—

Don’t hate me because I’m beautiful.

@robmandu its not 9.9 -.99 though, thats where your wrong. Its 9.99999999999999(for ever) – .999999999999999999999999(forever) therefore your indeed left with 9. Proof works, sorry. Ive gone over it with quite a few college profs, it checks out.

I still say .99999…. can never equal 1. It’s a never-ending number. A finite number can’t be equal to a infinite number. It’s just an approximation.

I want my .000000…...000000…000001 back on every 1 dollar I have ever paid, plus interest.

@allie

” Students are often “mentally committed to the notion that a number can be represented in one and only one way by a decimal.” Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.

Some students interpret “0.999…” (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 “at infinity”

Uber: First of all, I don’t appreciate the phrase “mentally committed” at all. Check your tone, mister. :p Secondly, my point wasn’t how the number is represented. I could care less about the decimals. My point is that one is finite and one is not. As for your last paragraph, if it is indeed finite, then how can it equal 1?

1/3 does not equal .333333333… either

and here is no “last nine” at the end of .99999…

.999999… and 1 are not different representations of the same number; they represent different quantities.

Proof doesn’t work. It’s a clever trick, but it’s wrong in the basic representation of an infinitely repeating decimal. Check out this short, sweet article.

In fact, uber, your argument was one AGAINST your own.

PS: I suppose that the proof is acceptable, it’s just that decimals can’t really represent infinity properly.

0.9999… approaches infinity.

Seriously, where’s finkelitis?

your face

Hahaha.. what the fuck? ^^

your mum’s face

You have to be very careful when dealing with infinity. Those of you who have had calculus should have an understanding of the concept of limit. The reason why this “proof” works is that the series .9999… has 1 as a limit, that is, the more 9’s you put on the end, the closer you get to 1. I do not have any examples that I can think of now, but there are many cases of series that do not have limits where you can perform operations similar to what was done with .9999… and get meaningless and contradictory results.

0.9999999999 (ON TO INFINITY) EQUALS ONE

GET OVER IT PEOPLE.

GOODNESS GRACIOUS.

Shouting don’t make it so, Moe.

Yeah, I was just being stupid.

But on the serious:
-The math checks out.
-You can ask a mathematician.
-If you fail to accept it then it is your own intellectual inability to accept things that don’t agree with your perspective of the world that should be questioned; not the math.

@el; I have emailed the problem to a relative (who is also one of the most intelligent people I know and presently getting a PhD in geometric algebra.) ((or is it algebraic geometry. An abstruse topic, in any case, and understood by only 21 people in the word.))

@gail

everytime you call me el, I think about the Hebrew god

http://en.wikipedia.org/wiki/Ēl

lol

Lurve to robmandu just because I don’t hate you and you are beautiful. :^>

Is elchoop better? “What does el choopanebre mean, anyway.”

Feel free to call me “la.”

That was one heavily researched and documented article. There are giant discussions about the same of God in the OT. (I ended up just now reading about Hebrew grammar; the three letter roots and the many prefixes and suffixes that change the root.)

Elchoopanebre was a monster in a Futurama episode- the one where they went underground into the sewers. I assume it’s a play off of ‘el chupa cabra.”

It’s supposed to be “el chupa nebre” but I spelled it wrong the first time I used it as a username and then just kept it every time since.

I guess the only logical explanation for me that makes the equation correct is by attempting to subtract 1 – 0.(9), where in essence you get 0.(0).

The issue I have with adding, subtracting, multiplying, dividing by infinity is that you are assuming that infinity is an integer when in reality it is a mathematical concept which cannot truly be found. Now I know we are not subtracting infinity but we are using the conept of infinity. We have limits which attempt to find some logical explanation to what happens when things approach a point but never really reach them, but when attempting to use basic mathematical functions using infinity I think all basic math functions break down.

I guess what makes the equation above true is that we assume that 0 is equal to 0.(0). I know my education doesn’t allow me to argue this point but I think it could be argued. I still don’t think 0.(9) = 1

Does 0.(9)+0.(9)=2?

@breedmitch no, but 0.9999999999999999999999999999….+0.999999999999999999999999999….=2

I thought that the parenthesis implied that the 9 went on repeating forever.

didnt know that, do now :)

haha pwnd

really, pwnd? i just call it mild stupidity on my part. :P

i call it pwnd cause you got busted for not reading the other responses :P

I think the two of you are missing some vowels. What’s a pwnd?

http://www.urbandictionary.com/define.php?term=pwnd

URGH!

0.(9) + 0.(9) =/= 2

little frustrated there petey? =P

0.(9) + 0.(P) = almost 2.

0.(9) + 0.(9) = (really) close to 2

^^ Dancing cheek to cheek?

Is it like this concept?

When we’re taught in High school that some Greek guy was able to calculate the circumference of the Earth by not seeing a shadow in a well at two different points and doin’ some trig and stuff

but this is assuming that because the earth is so very far away from the sun that the rays are almost (but not really) parallel

So what it means is we should just usually round, up?

One more time.
The terms .9, .99, .999, ...
form a sequence. This sequence has a limit of 1. So what does that mean? It means you can do the following. Choose any small number, say 1/1,000,000. If you go far enough in the sequence you will get and stay within 1/1,000,000 of 1 for all members of the sequence beyond that point, in this case, after the 6th term of the sequence.

Limits of sequences have some very nice properties. They act in many ways like ordinary numbers. If you multiply all the terms of a squence by some number, the effect is to mulitply the limit by the same number. That is why the so-called proof works.

So are we justified in treating a sequence with a limit as if it is a number whose value is the limit? This is a philosophical question. From a practical point of view there is not any harm done. However, this only works for sequences with limits. In this case, it easy to formalize what I said above to show that the limit is 1, at which point there is nothing more to do.

Sorry I’ve been missing from this link. Let’s see if I can shed some light. LostinParadise has been doing a good job of highlighting an important issue, though. (lurve)

What’s going on here is that we’re considering two different ways of thinking of numbers. One is as points on a number line. None of us has a problem with 1.

The second, more subtle way of thinking of numbers is as the limit of infinite sequences. 0.999… is really a process: we’re taking a series
9/10 = .9
9/10 + 9/100 = .99
9/10+ 9/100 + 9/1000 =.999
etc.

This is what we represent by 0.99999…

The reason that these things seem different is that they are different ways of conceiving of points on a number line. The repeating decimal is a sequence of points getting closer and closer to something but never passing (or reaching!) it. The number one is just a single point.

However, the number one is precisely what the infinite sequence is drawing closer to without passing. The technical name for this is the limit of the sequence. As Lost pointed out, not all sequences have limits. But all infinite decimal expansions do (they’re a constrained kind of sequence).

Once you get used to the idea that the limit must be the only suitable number that the sequence could “mean,” if we want to consider it as a number, then figuring out what it is not usually too hard. The proofs above work. Indeed, 0.9999… = 1 is true, once you accept the idea of letting a single point represent the sequence. For infinite decimals, that’s generally the convention, since it turns out that they contain all the same numerical information. You can also consider the differences:

1 – 0.9 = 0.1
1 – 0.99 = 0.01
1 – 0.999 = 0.001
etc.

Clearly the difference is getting closer and closer to zero. We say that it “vanishes in the limit.”

This means that 0.(9) + 0.(9) = 2. That’s fine.

It gets a little tricky to consider a sequence, which we imagine as building one piece at a time (and hence never finishing) of somehow being finished. But this is the least of the subtlety in dealing with infinity.

@robmandu—your discomfort with shifting the decimal is sensible in our normal experience (with finite situations), but this is one of the weird things about infinity: you can shift stuff off to the side, and never have to deal with it. An interesting gambling paradox: if you don’t have a maximum bet, and have any amount of credit, you can always win. Here’s how:

1. Bet a dollar. If you win, repeat.
2. If you lose, double your bet.
3. Keep doubling as long as you lose. The moment you win, you win back all your money, plus a dollar.
4. After you win, reset, and begin betting a dollar again.

Essentially, having an infinite line of credit allows you to shift your debt off, and you never have to contend with it. So that’s an example (there are many) of paradoxes that have to do with infinity.

But here’s the real weird thing: as nikipedia mentioned (first answer) there are different sizes of infinity. However, the ones she mentioned are all the same size! One of the hallmarks of an infinite set is that it contains subsets that are the same size as it. So the set of numbers between 1 and 3 is the same (infinite) size as the numbers between 1 and 2, even though one is contained in the other.

Coming up with infinite sets that are actually different sizes requires a little more doing. But I’ll leave that for another conversation.

I hope that clears up the first matter, at least, (and makes you curious about the rest).

0.999… or 0.999999999999999, which is the X?

Stumbled across this today. Numbers are cool, man.

X = 0.999…

@robmandu: See @critter1982‘s 1st comment above.
Numbers are cool.

9X in not 9.
Maybe it’s 8.99··· ··· ( super amount of 9 ) ··· ···991

Oh my god! This is why I hate math. So why do I drive myself crazy make some sense out of it?

Questions:
Does 1.(9) = 2?
Why is it automatically 10X? Why not 5X or 20X?

The other way to look at this is name one single number between the two. You can’t because they are the same. Distinct numbers have an infinite number of points between them.

I just re-read this thread and wanted to point out how much I love it.

.99999999999999999999999999999… does not equal 1.

My brain hurts

Team ≠One stands strong. I support you, Petey! <4

I’m not even going to bother arguing. It just made me remember our long conversation about it in CF and that reminded me of other awesome conversations in there. Good times. <4

It reminded me what its like to argue with a brick wall and how frustrating it is when you can explain something a millions ways logically and still be denied :P

@benuberbatmandrew Neh.

@uberbatman, isn’t that why we fluther in the first place?

@robmandu touche.