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        riddles >> general problem-solving / chatting / whatever >> .999 =1: True or False? or Both? 
        
(Message started by: KeyBlader01 on Oct 8^th, 2008, 7:03am)

Title: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 7:03am

Yesterday, my class and professor ridiculed me because I said that there are mathematical proofs that state .999 =1 and I have seen them on the web. But can we really state that .999=1?Yes or no? Is it true or false? If it's true, prove it! Name me titles of books and who invented that theory, formula, etc.. If it's false, prove why it's false and name some titles of books and theories,formulas that disprove it.

If it's both, state why it could be both and in what cases and etc...

Thanks!

Also, would you consider math to be discrete or concrete?

Title: Re: .999 =1: True or False? or Both?
Post by SMQ on Oct 8^th, 2008, 7:16am

Well, not ".999", which is 1/1000 less than one, but ".999..." with a never-ending string of nines is indeed equal to 1. For an advanced treatment you can see this famous thread (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1083464351) on this message board (I suggest installing the Math Symbols support script (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_suggestions;action=display;num=1171644142) first). You can also check out the article on Wikipedia (http://en.wikipedia.org/wiki/0.999...).

My personal favorite "simplified proof" goes like this: given any two distinct (not equal) numbers, a and b you can find at least one new number between them (for example, by taking the average: (a+b)/2), right? So name a number between 0.999... and 1. When they answer 0.999...5, remind them that putting digits after the ... is nonsensical since the string of nines never ends. Since there aren't any numbers between 0.999... and 1, they must be equal.

--SMQ

Title: Re: .999 =1: True or False? or Both?
Post by Grimbal on Oct 8^th, 2008, 7:17am

It is 0.999... = 1. Don't forget the '...'!.

It was one of the longest thread on this forum.
See http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1083464351;start=0

Sometimes, even teachers are wrong. I remember a physics teacher who claimed that if you try to jump from a boat to the riverside, and the boat is lighter than you, you will actually jump backwards!

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 8:08am

on 10/08/08 at 07:16:19, SMQ wrote:

Thanks but the last proof. Couldn't it be just 0.9999.... be the last number on the number line toward 1 so they are not necessarily equal.

Also, I downloaded the script but when viewing the page, I just see html for the smileys. How do I fix it so I see the smileys?

Title: Re: .999 =1: True or False? or Both?
Post by towr on Oct 8^th, 2008, 8:13am

on 10/08/08 at 08:08:12, KeyBlader01 wrote:

Thanks but the last proof. Couldn't it be just 0.9999.... be the last number on the number line toward 1 so they are not necessarily equal.

But how could it be the last number? The average of two distinct numbers is by definition another number; and by construction it is distinct from both and falls in between them. So for any number less than one, there's at least one other number between it and 1, the average.
It's like how there isn't a largest integer, you can always add 1 to get an even greater number.

Quote:

Also, I downloaded the script but when viewing the page, I just see html for the smileys. How do I fix it so I see the smileys?

Do you also have greasemonkey installed (or the IE equivalent)?

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 8:27am

on 10/08/08 at 08:13:18, towr wrote:

1) So you are saying that the average of .9999... and 1 is 1.

2) Yes I installed greasemonkey and it's still not showing up.

Title: Re: .999 =1: True or False? or Both?
Post by SMQ on Oct 8^th, 2008, 8:44am

on 10/08/08 at 08:27:33, KeyBlader01 wrote:

2) Yes I installed greasemonkey and it's still not showing up.

If you look at the Greasemonkey submenu in the Tools menu is Enabled checked? If you go to Manage User Scripts from that submenu is wu::forums Math Symbol Support in the list? When you click on it is Enabled checked at the bottom? If it's not in the list, but Greasemonkey is enabled, try clicking the user script link again: is should ask if you want to install it, not just download it.

--SMQ

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 8:49am

on 10/08/08 at 08:44:11, SMQ wrote:

Thank you so much! I see what I did wrong. I forgot to download the math script >.< haha... thanks again!

Title: Re: .999 =1: True or False? or Both?
Post by towr on Oct 8^th, 2008, 8:58am

on 10/08/08 at 08:27:33, KeyBlader01 wrote:

1) So you are saying that the average of .9999... and 1 is 1.

The average is 1, but that's because 0.999...=1.
The point is rather that were 0.999... and 1 distinct, then their average would lie between them. And since it can't lie between them (because you'd get 0.999... again), they cannot be distinct; and so they are all three the same.
It's a proof by contradiction.

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 9:27am

on 10/08/08 at 08:58:29, towr wrote:

Oh I'm starting to understand it better now. Thanks. So, what you are saying is that .999..... =1 because of that proof by contradiction. What if you look at it in terms of limits because that's what we're doing in class (I'm taking Calculus 1) by professor says that if .999...=1 then why does plugging in 1 where the lim h->1 x^2+1/x-1 make the denominator zero and 0.99 (well it's going to have to finite to calculate) doesn't? Is it because he's using a finite number there when making the calculation but if he used 0.999.... it would also make the denominator zero? Is that a good reason?

Also one of my classmates said, you can't bound infinity or (0.99999999.....) as a number. How can I prove him wrong?

Also, is it possible to print out that thread with all the proofs with the math symbols on it? I went to print preview and it symbols don't show and I tried copying and pasting in Word and the math symbols didn't show again. I want to show my professor the proof.

Thanks a lot for respecting my thoughts on this matter.

Title: Re: .999 =1: True or False? or Both?
Post by SMQ on Oct 8^th, 2008, 10:05am

:o Wait, time out, your calculus professor who is teaching you limits doesn't believe 0.999... = 1?!

0.999... = (by definition) lim as nhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif from k=1 to n of 9http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif10^-k
= (by rearrangement) lim as nhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif from k = 0 to n-1 of (9/10)(1/10)^k
= (by sum of geometric progression (http://en.wikipedia.org/wiki/Geometric_progression#Geometric_series)) lim as nhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif of (9/10)[1 - (1/10)ⁿ]/(1 - 1/10)
= (by simplification) lim as nhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif of 1 - 10^-n
= 1

Q.E.D.

As for printing, I'd never tried that; huh. Stranger yet, IE does the same thing. I have no idea why that doesn't work. I suppose you could take screenshots (Alt-Print Screen) and print that, but...eww. :(

--SMQ

edited for preciseness, typos

Title: Re: .999 =1: True or False? or Both?
Post by Michael Dagg on Oct 8^th, 2008, 11:00am

Trivially: suppose n=.999... . Then 10n=9.999...
but 10n-n=9 ==> 9n=9 ==> n=1.

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 11:07am

on 10/08/08 at 10:05:42, SMQ wrote:

That proof you gave me is kind of confusing for me but I'll look it over and show it to him. Maybe he didn't know that because I proved it by (1/3 + 1/3 +1/3 =3/3 and .999 =n 10n=9.999 - n = 9n=9 where there was no ... to prove it was an infinite sequence although if he knew what I was talking about he would have just said you mean .999...=1 in which it does?)

Michael, I tried to prove it that way to him but I forgot the ... maybe if I include it on the next lecture, he'll understand?

Title: Re: .999 =1: True or False? or Both?
Post by rmsgrey on Oct 8^th, 2008, 11:50am

Sometimes teachers pretend ignorance to test their students' understanding. Sometimes teachers are aware of some subtle yet significant point that makes the question the student thought he was answering subtly different from the question the student was answering. Sometimes teachers actually are ignorant.

Whichever may be the case here, under any reasonable, rigorous definition of the notation, "0.999...=1.000..." is a true statement

Quote:

Also one of my classmates said, you can't bound infinity or (0.99999999.....) as a number. How can I prove him wrong?

It depends what he means by his statement. I don't understand what he's saying, so I can't figure out any way of convincing him he's wrong...

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 8^th, 2008, 11:57am

on 10/08/08 at 11:50:01, rmsgrey wrote:

I meant to say, he says 0.99999.... goes to infinity as we can't comprehend the number. It's infinitely .99999999..... So, you can't say it's equal to 1 because 1 is a number, it is defined as 1. where as infinity isn't, you can't bound infinity to one number. As you can't say 0.9 =0.999

Also, my professor said if 0.999... is equal to 1, where are there two different ways of writing it? Why can't they exist as one concept/number? But I think I know the answer to that which is 3/3 is another way of saying 1?

Title: Re: .999 =1: True or False? or Both?
Post by 1337b4k4 on Oct 8^th, 2008, 12:19pm

I really really hope I didn't understand you correctly...

Title: Re: .999 =1: True or False? or Both?
Post by towr on Oct 8^th, 2008, 12:21pm

on 10/08/08 at 11:57:04, KeyBlader01 wrote:

I meant to say, he says 0.99999.... goes to infinity as we can't comprehend the number.

Well, perhaps he can't comprehend it. But it is quite well defined. As SMQ wrote above you can see it as an infinite sum which converges on 1.

A decimal ...d₃d₂d₁d₀.d_-1d_-2... (where each d_i is a digit) is http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif d_i10ⁱ over all i
For 0.999... all digits d[i]=9 for i < 0 and 0 otherwise. And the sum converges.

Quote:

It's infinitely .99999999..... So, you can't say it's equal to 1 because 1 is a number, it is defined as 1. where as infinity isn't, you can't bound infinity to one number. As you can't say 0.9 =0.999

Take a piece of paper; cut of 9/10th off and put it to one side. Then take 9/10th of the remainder, put it with the first piece. Repeat ad infinitum (that'll keep you busy awhile.)
The pieces you lay to the side will never total more than the piece of paper you started with. It is bound by 1, and approaches 1 the more steps you take.
(At the first step you've laid aside 0.9 of the paper, at step 2 it's 0.99, at step 3 it's 0.999 etc)

Quote:

Also, my professor said if 0.999... is equal to 1, where are there two different ways of writing it? Why can't they exist as one concept/number? But I think I know the answer to that which is 3/3 is another way of saying 1?

Yup. There are an infinite number of ways to write 1. x/x for any x other than 0; or (a+1)-a for any a; or an infinity of other ways.
As far as decimal notation goes (rather than using composite expressions), there's always 1; 1.0; 1.00; 1.000 etc.

Title: Re: .999 =1: True or False? or Both?
Post by rmsgrey on Oct 8^th, 2008, 12:41pm

on 10/08/08 at 11:57:04, KeyBlader01 wrote:

How does he handle the problem of writing a number that's equal to 1/3, or pi, or...? Or is it only decimal strings that end with an infinite number of nines that he doesn't consider to represent numbers?

Allowing infinite sequences of digits to represent numbers is a useful convention, but you could insist that only finite sequences be allowed - in which case various numbers have no decimal representation, rather than a very small (but still infinite) number of them.

Quote:

A better answer is to point out that the representations of numbers very slightly less than one look very different from the representations of numbers very slightly more than one - 0.99999999 looks different from 1.00000001 despite the two differing by only 0.00000002 - while 0.333332 looks a lot more like 0.333334 despite having a much larger difference in absolute value (0.000002)

As the number at the transition between numbers that look like 0.9999... and numbers that look like 1.0000..., it's not too surprising that 1 should have representations that look like each.

Title: Re: .999 =1: True or False? or Both?
Post by Sir Col on Oct 8^th, 2008, 3:28pm

When comparing 0.999... to 1 there are three mutually exclusive possibilities:
(1) 0.999... < 1
(2) 0.999... = 1
(3) 0.999... > 1

In trying to decide which one is true:
(1) is true in the sense that it appeals to intuition.
(3) is true in the sense that 0.999... contains more digits than 1.
So on average (2) must be true.

As the above argument is flawed it follows that 0.999... cannot equal 1. ::)

Title: Re: .999 =1: True or False? or Both?
Post by 1337b4k4 on Oct 8^th, 2008, 5:01pm

on 10/08/08 at 15:28:09, Sir Col wrote:

very nice :)

Title: Re: .999 =1: True or False? or Both?
Post by iono on Oct 8^th, 2008, 10:40pm

1/3=.333.......
1/3+1/3=.333...+.333...=.666...=2/3
1/3+1/3+1/3=.333...+.333...+.333...=.999...=3/3=1
Simplest explanation I can think of

Title: Re: .999 =1: True or False? or Both?
Post by towr on Oct 9^th, 2008, 1:26am

on 10/08/08 at 09:27:55, KeyBlader01 wrote:

Also, is it possible to print out that thread with all the proofs with the math symbols on it? I went to print preview and it symbols don't show and I tried copying and pasting in Word and the math symbols didn't show again. I want to show my professor the proof.

I've made a png image from the first 3 posts (which are probably all you want). You can try and see if you can print that. (See zipped attachment)
You might want to cut it up in pieces before printing, because it won't fit legibly on one page, and sentences might get cut in half. Or you can have it printed as a banner at your local print shop. ;)

Title: Re: .999 =1: True or False? or Both?
Post by Grimbal on Oct 9^th, 2008, 1:42am

I think the confusion comes from the mistaken assumption that the decimal representation of a number is the number and calculations are nothing more than the manipulation of the symbols that represent the numbers.
You can work like that with integers, that is why people have this intuition I guess, but it doesn't hold for real numbers.
"0.999...", "1.000" and "1" are all different representations of the same number, just as well as "0.333..." and "1/3" are the same number under another name.
People object to 0.999... = 1 because it is not written the same. People don't object to 1.000... = 1 because they know that the zeroes don't count. If they understood that an infinite series of 9's counts the same as a unit in the previous digit, they might see that "0.999..." and "1" are just the same number under a different name.

Title: Re: .999 =1: True or False? or Both?
Post by mikedagr8 on Oct 9^th, 2008, 4:16am

.999...=9/10+9/100+9/1000+...

S_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif}=a/(1-r)
a=9/10
r=1/10

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

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 9^th, 2008, 6:24am

on 10/09/08 at 01:26:58, towr wrote:

Thank you so much! I'll try to cut it up, thanks a lot! Unfortunately my next math lecture is a test. So, the excitement will kill me until after the test is over >.<

Grimbal, thanks for your thoughts. I'm going to share that info with my classmate.

Mikedagr8, Interesting way of putting it. Thanks for sharing.

Title: Re: .999 =1: True or False? or Both?
Post by mikedagr8 on Oct 10^th, 2008, 1:58am

on 10/09/08 at 06:24:27, KeyBlader01 wrote:

Mikedagr8, Interesting way of putting it. Thanks for sharing.

No problem. Everyone in one of my maths classes had to find a unique way to show that .999... was equal to 1. This was mine. Hope it was helpful.

Title: Re: .999 =1: True or False? or Both?
Post by KeyBlader01 on Oct 21^st, 2008, 6:24pm

Thanks everyone. I told my Calculus professor about .999...=1 is equal to one and he laughed and said, yeah that's right. He's knows already!! And he made me suffer trying to find all these proofs for this lol. It's because I said .9999=1 first I suppose. But hey, now I know =D

Thanks again!

Title: Re: .999 =1: True or False? or Both?
Post by rmsgrey on Oct 22^nd, 2008, 12:38pm

on 10/21/08 at 18:24:08, KeyBlader01 wrote:

His goal in doing so would probably have been:

1) to determine whether you understood the correct statement or were merely parroting something you'd read and not really understood.

2) to force you to consolidate your understanding of it.

After all this, you're not likely to forget that 0.999...=1, nor to think that .999=1

There's a difference between "knowing" something because someone told you it's true, and knowing it because you have proved it to be true.

Title: Re: .999 =1: True or False? or Both?
Post by Grimbal on Oct 22^nd, 2008, 2:40pm

on 10/22/08 at 12:38:56, rmsgrey wrote:

There's a difference between "knowing" something because someone told you it's true, and knowing it because you have proved it to be true.

That's true, I read it somewhere on Internet. ;)

Title: Re: .999 =1: True or False? or Both?
Post by hamzak on Nov 25^th, 2008, 5:48pm

well for .999 to equal one there must be 11 trailing 9s including the three already there the reason for this is that at that point they are so incredibly close they say .99999999999 = 1 but from my view it is still not 1 it is .99999999999.
hope i could help

Title: Re: .999 =1: True or False? or Both?
Post by towr on Nov 26^th, 2008, 12:11am

on 11/25/08 at 17:48:46, hamzak wrote:

There must be more than just 11 trailing 9's for it to equal 1, there must be an infinity of them.
Then they're not just close to equal, but are equal. Because there is no difference between them at all.

Title: Re: .999 =1: True or False? or Both?
Post by Mickey1 on Mar 11^th, 2010, 5:16am

There is only one way to challenge that 0.999... is not equal to 1, namely to deny that 0.999... is a number. It cannot be written down, and such a statement is not illegal.

It is a process to design numbers arbitrarily close to one, so if both 0.999... and 1 exist (and 1 must exist for legal reasons) they must be equal.

0.999... can either be described a limit of a sum where n goes to infinity, or as an endless collection of partial sums, 0.9, 0.99, 0.999 etc.

You have very little choice regarding the limit concept but you could argure that only rational numbers exist (but I don't know why you would do that).

Title: Re: .999 =1: True or False? or Both?
Post by Grimbal on Mar 12^th, 2010, 4:43am

But then, 1/3 = 0.333... doesn't exist either? Or does it?