How exactly does 0.999~ = 1?

[QuantumTheory]

According to this site, it does:
http://www.straightdope.com/columns/030711.html"

Is this really true? I don't see how, because it seems like a limit problem to me.

 

[matt grime]

It is true because of the way we have defined what decimal representations* of real numbers are, and exactly because of what limits are. 0.9recurring is the limit of a cauchy sequence of the partial sums of the obvious geometric progession

9/10+9/100+9/1000+...

which is 1.

End of story.

Shall I await the "but you never get to the end of an infinite sum so it never truly equals 1" arguments?

* I presume you have no problem with the fact that 1/2 and 2/4 (and an infinite number of other fractions) all represent the same rational number? If so why is it so surprising that *two* (and only 2) decimal representations are of the same real number?

 

[QuantumTheory]

Incredible. Well, I never fully finished algebra, geometry, or pre calc or calculus. It's like you konw bits and pieces and not the whole thing, which makes thigns like this very difficult.

I won't doubt you ,but I still don't understand. However, my math teacher did tell me that there is an infinite numbers between 1 and 2.

I thought the whole point of a limit was that you can get close and not reach it? Wait a second. Will 0.9 always repeat itself or eventually get to 1?

 

[QuantumTheory]

Well it doesn't matter, but it could be 0.99999999999999999999999 and isn't that still 0.000000000001 close from 1?

 

[QuantumTheory]

I guess it only equals one if ou let it go for infinity. But if you stop the 'process' then it does not equal one.

 

[matt grime]

The reals are something that obeys a set of rules, one of those rules directly implies that in the decimal representations *we must identify those two numbers if they are to behave properly*. Decimals are just representations of the real numbers. If you want to understand those definitions then you need to do significantly more maths (analysis), but it is just a formal consequence of hte definitions. It is not mysterious.

Of course no finite truncation of the infinite sum is 1, but I don't see what that has to do with the infinite sum. You just appear to have some misconception about what a limit is. In general I'd say you're reading far too much into what after all are just strings of digits that obey certain rules. If you don't know what the rules of the game are you cannot play it. You're 'let it go to infinity' thing is one such indication. This isn't something that adds up things one at a time as if it were a machine, this isn't some process, you don't let things 'run forever'. It is just a symbol with certain properties.

One of which is it is the smallest number larger than any of the finite truncations.

 

[Hurkyl]

I guess it only equals one if ou let it go for infinity. But if you stop the 'process' then it does not equal one.

Right -- 0.999~ does not mean "a lot of 9's" -- it means the 9's keep going without ending.

More precisely, for every positive integer n, there is a 9 in 0.999~ located exactly n places to the right of the decimal point.

(There are infinitely many 9's in 0.999~ -- nothing is "going" to infinity)

 

[nazgjunk]

1/3 is 0.333~, right?

3 * 1/3 is 1, right?

All you then need to accept is that 3 * 1/3 is "also" 0.999~.

To make it somewhat clearer, 2*2=4, 2 * 0.222~ is 0.444~. So if 3*3=9, 3*1/3=0.999~.

When I first stumbled across this, my main problem was accepting the seemingly illogical truth.

 

[TD]

1/3 is 0.333~, right?
3 * 1/3 is 1, right?
All you then need to accept is that 1/3 is "also" 0.999~.
When I first stumbled across this, my main problem was accepting the seemingly illogical truth.

I've seen this very often but what I don't see is why someone would accept 1/3 = 0.3~ and not accept 0.9~ = 1, yet this is often used as 'proof'.

 

[HallsofIvy]

1/3 is 0.333~, right?
3 * 1/3 is 1, right?
All you then need to accept is that 1/3 is "also" 0.999~.
When I first stumbled across this, my main problem was accepting the seemingly illogical truth.

??Typo? I presume you meant "All you then need to accept is that 1/3 is "also" 0.333~".

 

[TD]

I think he meant that people would already accept that 1/3 = 0.3~ (don't ask me why though), then they'd need to accept that 3*1/3 is not only 1 but also 3*1/3 = 3*0.3~ = 0.9~.

 

[Mindscrape]

I've seen this very often but what I don't see is why someone would accept 1/3 = 0.3~ and not accept 0.9~ = 1, yet this is often used as 'proof'.

Yeah, I always thought this was strange.

 

[vaishakh]

There is a lot of difference between something tending to 1 and something being 1. we approximate it to 1, but need not be that it is actually 1. The explanation given by Nagzun was really fantastic. Something tending to 1 being written as 1 when we don't know the actual value is a different case. This is what happens in GP. We don't know the actual value and so the formula given is a/1-x wherew x<1. Here a Mathematician is sue that no one tommorow will give a unique perfect answer to this question about summation about GP to infinite values, So he approximated the formula. We write root-2 as 1.414, But 1.414 is a unique rational number. However root-2 is not really 1.414, it only tends to that value. We have no use of finding root-2 to hundred decimals to solve numericals. So we neglect the remaining. We can be sure about the fact that 1 is something very near to the number 0.999~. However we cannot tell, that both are equal.

 

[TD]

There is a lot of difference between something tending to 1 and something being 1.
[...]
So he approximated the formula. We write root-2 as 1.414, But 1.414 is a unique rational number. However root-2 is not really 1.414, it only tends to that value. We have no use of finding root-2 to hundred decimals to solve numericals. So we neglect the remaining. We can be sure about the fact that 1 is something very near to the number 0.999~. However we cannot tell, that both are equal.

There is a big difference between an approximation, which 1.414 is for sqrt(2), and another representation for exactly the same number, which 0.9~ is for 1. There is no "choice" for us, to say whether or not those two are equal. If we're working in the real numbers, there is no other possiblity than for those two to be exactly the same real number - it's just another way of writing it.

 

[matt grime]

There is a lot of difference between something tending to 1 and something being 1.

Excuse me, but such displays of ignorance tend to rile me. Who said 0.9... tends to one? It is one, it is not tending anywhere; its partial sums tend to one. One is the smallest real number greater than any of the strictly increasing partial sums, hence is the limit **by the definition of the space in which the argument takes place**.

 

[russ_watters]

If nazgjunk's simple and elegant proof (minus the typo, of course )isn't enough to quench all doubt, perhaps a question:

If 0.999... is not equal to 1, then there must be a number between the two: what is it?

And don't say 0.000...1, because there is no such thing.

 

[Integral]

I cannot believe that we are having https://www.physicsforums.com/showthread.php?t=5513" conversation again. To bad there is not someway to make it necessary to read the entirety of the linked thread before posting here.

 

[Maxwell]

Like a sticky with the topic:

"BEFORE YOU POST 0.999 = 1, READ THIS!" with a link to that post?

 

[Hurkyl]

I have been getting the feeling for a while now that each 0.999~ = 1 thread has been "better" than the previous one... I'm not sure what that really means, though.

 

[Sir_Deenicus]

We simply happen to use a system where 1 and 0.999... are equal. On the hyperreals, R*, a *logically vaild* superset of our current system, 0.9999... and 1 are not equal.

 

[nazgjunk]

Sorry for my little typo. What I did mean was "all you then have to accept is that 3 * 1/3 "also" equals 0.999~". I will edit that one now.

Ow, how I love to force people into accepting what they just proved themselves.

 

[matt grime]

I have been getting the feeling for a while now that each 0.999~ = 1 thread has been "better" than the previous one... I'm not sure what that really means, though.

eventually the series of threads might converge, and if we let it go to infinity it might equal one explanation...

 

[Mindscrape]

The large infinity or the small infinity?

I still don't see why people so readily accept (1/3)=.333~

 

[Sir_Deenicus]

eventually the series of threads might converge, and if we let it go to infinity it might equal one explanation...

Only if each following thread got shorter and shorter... :D

 

[Hurkyl]

On the hyperreals, R*, a *logically vaild* superset of our current system, 0.9999... and 1 are not equal.

That's incorrect -- every true internal statement about the reals is also true about the hyperreals, and that includes 0.999~ = 1.

(I can explain further in private message, or a new thread -- I just don't want to muddle this one up)

 

[matt grime]

I still don't see why people so readily accept (1/3)=.333~

My take on this is that they think of decimals of representing the answer of doing a division, thus dividing 3 into 1 by long division gives them the sequence 0.3, 0.33, 0.333,... etc, and they somehow think that "thus the result is 0.3..." they do not see such a division for 0.999...

Further, to compound the problem, i suspect that since they do not know what convergence is they only accept 0.333recurring is 1/3 as some kind of decimal 'best guess', thus subconsciously they do not truly think of them as being exactly the same; this is the 'but you never reach the end' argument.

Of course this notional proof of 1/3 times 3 requires that we prove that the algebraic operations on decimals are well defined, which typically is not done.

 

[Sir_Deenicus]

That's incorrect -- every true internal statement about the reals is also true about the hyperreals, and that includes 0.999~ = 1.
(I can explain further in private message, or a new thread -- I just don't want to muddle this one up)

Is that even possible? To muddle this, that is. I can't believe I said R*. I meant *R. Me is embarrassed.

Anyway, I think you might be misapplying the transfer principle or extension axioms or something. I think I can prove this as well. The basic ideas is that the limit never converges to 1 since there are an infinite number of numbers between.

 

[matt grime]

It converges to 1 in the hyperreals too.

What's the difference between 1 and 0.9...? It is a real number (since the reals are a subring) that is also an infinitesimal, thus it is zero since there is exactly one infintesimal. (If it were otherwise how could it be a superset? (the injection would fail to be even a function))

 

[Sir_Deenicus]

It converges to 1 in the hyperreals too.

What's the difference between 1 and 0.9...? It is a real number (since the reals are a subring) that is also an infinitesimal, thus it is zero since there is exactly one infintesimal. (If it were otherwise how could it be a superset? (the injection would fail to be even a function))

hmm.. I see, although i still disgree. I need to work this out myself. Perhaps I was wrong to call them superset then, i dunno. Ill shall henceforth be posting either my thanks for you correcting me or a proof as to why i am correct.

 

[shmoe]

I still don't see why people so readily accept (1/3)=.333~

I think part of the acceptance of 1/3=.33~ is there's no alternate decimal expansion. The unfounded notion that every number must have a unique decimal expansion is probably fuel for many of the .99~ complainers.


These threads always make me try to remember what I actually thought the real numbers were in high school, but I can't seem to remember doing any math before my first analysis class.

 

[QuantumTheory]

If nazgjunk's simple and elegant proof (minus the typo, of course )isn't enough to quench all doubt, perhaps a question:
If 0.999... is not equal to 1, then there must be a number between the two: what is it?
And don't say 0.000...1, because there is no such thing.

I searched for this thread before I posted one but couldn't find one, forgive me.

It seems to defy logic sense. Why? Because of limits. I thought the whole point is of a limit THAT YOU APPROACH SOMETHING but never actually reach it. You're still a very very closet o it, and as you do so it effects other varibles. (I think f(x). I've read about limits although my math behind it is obviously fuzzy.

What I don't undestand is what 'matt grimes' says that actually reaching the limit is the whole part of it. Huh? I thought you can't reach it but can only get very close, and as you get closer, f(x) gets closer an closer to c, right?

There is nothing after 0.99~. Because it goes on forever. I'm 17, still, even though it equals 1 using fractions (which is obvious) the decimals are not so obvious.

 

[nate808]

Quick question, if you had line .999~ cm long vs a line 1 cm long would they be exactly the same or would the .999~ be missing the point at 1

 

[Curious3141]

Wow, this again. But since this thread seems to have been allowed to go on a little longer this time, let me try a pet argument I've been nursing to convince the unbelievers.

I'm sure I've seen the erroneous objection from some that they can find a 'number' between 0.9999~ and 1, viz. 0.999...5 (an infinite string of nines and a 5 miraculously tacked on to the end).

Well the next time they make that argument, I'm going to spring this one on them :

In binary, 0.1111~ = 1 This is the exact analogue of the problem in decimal. Now find me a binary number between 0.111~ and 1

 

[Sir_Deenicus]

The problem with confusion can be traced back to the notion of decimal, irrational, rational not being taught properly in the elementary levels and the finitude of the human mind. Hence the formation of crystallized misconceptions which more often than not, have hardened beyond the point of remoldability and must be broken and rebuilt.

As for approaching, think not of it that way but rather as becoming complete or more formally, converging [to a final limit value]. Also, try to remove all aspects of time in your reasoning when considering limits.

 

[Hurkyl]

It seems to defy logic sense. Why? Because of limits. I thought the whole point is of a limit THAT YOU APPROACH SOMETHING but never actually reach it.

The sequence:

0, 1/2, 2/3, 3/4, 4/5, 5/6, ...

has limit 1. Does that mean that "1" is approaching something, but never quite reaching it?


Of course not, that's silly.

The sequence:

0.9, 0.99, 0.999, ...

has limit 0.999~. It is similarly silly to talk about "0.999~" approaching something.

(Of course, this sequence also has limit 1)