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 Ricky
 Moderator
Re: I disagree with 0.999... = 1
It's also a good lesson as to why we can't use inductive logic in science. Math however, is a different story.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
but 0 really isnt a number  the romans never had any concept of a zero in their system the Greeks managed fine without a zero for quite a while too
infact you will see what I am saying in a second
the number 1378 is merely a kind of shorthand for the sum 1000.000... + 300.000.... + 70.000... + 8.000... notice all the zero's after the point stretch to infinity
the zero merely denotes the complete abscence of a number at a certain point in the sum so 1305 = 1000.000.... + 300.000... +5.000... I rest my case  zero is the abscence of a number  not a number in itself therefore it is true that numbers range on the real number scale as 3,2,1,1,2,3 and since .999... is neither 1 or +1 then the difference is between 1 and 1 that has to be the case it just cannot be otherwise
I havent got time to continue this right now  I'll be back tomorrow. 
Last edited by cray (20061006 15:01:10)
 MathsIsFun
 Administrator
Re: I disagree with 0.999... = 1
Zero is indeed a strange bird.
It is true that zero is a "placeholder" when writing down numerals. 302 is not 32.
But I believe it also has a role as a number. Imagine the descending number series 5,4,3,2,... What happens next? 1 (a number), 0 (a what?), 1 (a number), 2,3,... etc.
To me a number is a count or measurement. It is debatable if you can count "0", but you can measure it. 0 degrees, 0 meters above sea level, $0 bank balance. And if two bags have the same number of apples each, then the difference is 0.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
Like I said before cray, if zero isn't a number, then you throw entire branches of math into complete chaos.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
And it's strange. In kindergarten, they teach you the number 110, then 1020, but never 0. What gives?
I believe 0 doesn't exist  It is just a term given to nothing.
1, 2, 3...however, is different, since those numbers represent debts.
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
Mary has 5 apples. Joe has 5 apples. How many more apples does Joe have than Mary?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
Again  Even though we say 0 to represent the distance between the two amounts of apples, it is still nothing, so if 0 = nothing, then the distance between two of the same numbers must be nothing.
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
You're right, zero is nothing. And because of that, it's the most important number:
x + 0 = 0 + x = x
No other number has that special property. And without it, all of algebra is lost.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
And James Bond would also be ruined. He wouldn't be 007  He would just be 7.
Yeah, I was thinking of the algebra factors of the problem. 0 is a crucial part to play, and we cannot do without it. But is there a better way to define 0 than 'nothing' or 'the lack of something'?
Something similar  About 0^0: http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
Re: I disagree with 0.999... = 1
Nice definitions  Although there are other forms of zero.
Let's say, in graphs. To draw a straight line from 3 to 3, you would need the zero to help you.
If zero told us the amount, I think we could say 'nothing'  No dogs, no dollars, etc. I think zero is just another word for nothing or the lack of something  I know I'm not 100% right, though.
Re: I disagree with 0.999... = 1
Ok then I accept that but I just feel the solution is something to do with the idea of zero. except incase it isnt I'll put that on hold , though I have another question ! Is the reason why this is wrong 1/3 = .333... 2/3 = .666... etc etc because 1/3 is not a number but an idealisation of a perfect fraction ? after all 3/3 would literaly mean 3 divided by 3 equals 1 but a third of 1 is a lot more complex in reality than we think? so it acts more like a label than a actual number whereas .333... is an actual number?
maybe its to do with that ?
Last edited by cray (20061006 21:59:57)
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
All numbers are labels. 0 is the label for the identity with respect to addition. 1 is the label for the identity with respect to multiplication. 2, 3, 4, 5... are the labels for adding a certain number of 1's. The negatives of all of these are the labels for the additive inverses of them. 1/x is the label for the multiplicative inverse. a/x is just the same as saying a * 1/x.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 Kurre
 Power Member
Re: I disagree with 0.999... = 1
if not 0.999... = 1 , then 1.000... isnt = 1 since there could be a 1 or a 5 or any number at the end of these infinite zeroes...
and also, as said, 0.999... = 9/10+9/10^2+9/10^3+...+9/10^n then the difference (h) between 0.999... and 1 must be 1/10^n but since n = infinity, h=1/infinity=0 (since 1/0=infinity, 1/infinty=0) so the difference between 0.999... and 1 is 0, and therefor 0.999...=1
 George,Y
 Super Member
Re: I disagree with 0.999... = 1
zero is a useful concept, which means "nothing", "not any". It is particularly useful when it comes to negative numbers. I guess the first time a negative number occurred was the time when people had property and trade with debts. Hence zero played an important role as the cancelledout. I owe you 5 pigs, I have 5 from you, but then I give you 5, I have 5 from you, thus due.
cray, decimal system isn't just symbols, and every decimal represents some exponent of 10. 123.45 represents 1*10^{2}+2*10^{1}+3*10^{0}+4*(1/10)+5*(1/10)^{2} so decimals are fractions by nature. Infinite decimals add even one more idealization than rationals.
infinitesimal isn't zero, at least in classic defination made more than 120 years ago.
That's the trick. Actually there could be two types of functions or series having the same limit C.
One is constantly C, within some domain or since some step (which could be 1 to be general)
The other one is approaching C, and never reach C in the field of functions . However, some people believe it can reach C when it is a series and at Entry Infinity. Here another piece of strong evidence of objection  inconsistency.
1.0=1 1.00=1 1.000=1 ... though 1.000... may not exist, it can lead to some 1.00000000 in practice and caus little trouble. It's fortunate enough to get a clear result to put it other words.
But 0.999... do cause some trouble. Perhaps 0.9999999999 in practice. For example, NASA can reduce the chance of accident by having one more check. Are you confident that after many checks no accident? Infinite checks??? Come on...
X'(yXβ)=0
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
infinitesimal isn't zero, at least in classic defination made more than 120 years ago.
Since infinitesimals don't exist in the reals, they are 0. But in more, uh, "complex" number systems, there are nonzero infinitesimals.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
zero is a useful concept, which means "nothing", "not any". It is particularly useful when it comes to negative numbers. I guess the first time a negative number occurred was the time when people had property and trade with debts. Hence zero played an important role as the cancelledout. I owe you 5 pigs, I have 5 from you, but then I give you 5, I have 5 from you, thus due.
Interesting  I must read up on the history of math more ! I mean I was always under the impression the Romans didnt have a symbol for 0, and I heard they were also fantastic at bartering domestic animals etc. It maybe a mistake I have made in thinking that some cultures did not understand either the concept of zero (at least I thought they didnt have a term for it)
But 0.999... do cause some trouble. Perhaps 0.9999999999 in practice. For example, NASA can reduce the chance of accident by having one more check. Are you confident that after many checks no accident? Infinite checks??? Come on...
This comment blew me away, I just realised that a mathematical problem like this seems trivial and like a minor puzzle, but in actual fact it could have very serious consequences. My immediate response was to realise that the NASA mathematicians and programmers must surely "engineer" problems like this out of their systems by using a more precise definition of terms, its safe to say my knowledge of mathematics runs out at being awe inspired by the simple qualities of a sum like 3/3 = .999... it shook me up to realise that there must be programmers out there who dont realise this crucial simple peice of mathematics might have serious implications unless they understand the problem properly  in the aviation business for example. This problem has done my head in ! I thought maths was totally absolute and could be defined always in absolute terms Infinity has a lot to answer for and we should ban it ! LOL ha ha ha
Just one more line of questioning from me and then I really have to stop thinking about infinity  mainly because it makes my poor little brain hurt!
Does the concept of infinity only produce problems ? Is infinity a major problem for mathematics?
Last edited by cray (20061010 02:29:18)
 Dross
 Power Member
Re: I disagree with 0.999... = 1
cray wrote:This problem has done my head in ! I thought maths was totally absolute and could be defined always in absolute terms
It is and it can  there is no opportunity for discussion about it, the simple fact of the matter is that 0.999... = 1.
What this brings out is simply the fact that one number can have more than one decimal expansion. So what? The number 1 is unique, even though we can write down more than one way to represent it. This is simply what you could call a (minor) "flaw" in the way we represent numbers  the problem, as ever, is with the humans, not the mathematics.
There is a lot that could be written about reccuring numbers. What exactly do we mean when we write a number such as 0.333...(recuring), for example? There are clear definitions of this that should not be ignored.
cray wrote:Does the concept of infinity only produce problems ? Is infinity a major problem for mathematics?
Infinity, as with anything else, is entirely well defined and poses no problems. It is, as it happens, a more complicated subject than you might think, though.
Bad speling makes me [sic]
Re: I disagree with 0.999... = 1
It is and it can  there is no opportunity for discussion about it, the simple fact of the matter is that 0.999... = 1
But that leaves open the possibility of ambiguity doesnt it? Surely the computer that takes astronauts to mars is going to be programmed to define 1 as 1, isnt it ? Else if there is a calculation where the sum turns out to be .999... it just might conflict with a backup calculation that gathers the answer 1 to be more appropriate  I mean you can imagine a situation where no matter how infinitely small the difference between .999... and 1  it could have a consequence in the physical world. Or is that just not possible?
 George,Y
 Super Member
Re: I disagree with 0.999... = 1
Ricky wrote:infinitesimal isn't zero, at least in classic defination made more than 120 years ago.
Since infinitesimals don't exist in the reals, they are 0. But in more, uh, "complex" number systems, there are nonzero infinitesimals.
Hey, since Ricky don't exist in the reals, he is 0just kidding.
An infinitesimal isn't even a number at all, at least highlighted in most caculus books.
X'(yXβ)=0
Re: I disagree with 0.999... = 1
cray wrote:I mean you can imagine a situation where no matter how infinitely small the difference between .999... and 1  it could have a consequence in the physical world. Or is that just not possible?
infinitely small, can never make a difference, for example
if you have a number a, no matter how many times you add 1/inf to it, its still 'a' and it does not change. Dont think of it as adding a one to the end of an infinate chain of 0's. Thats trying to think of infinity and infintisemals as numbers. there not.
The Beginning Of All Things To End. The End Of All Things To Come.
 Dross
 Power Member
Re: I disagree with 0.999... = 1
cray wrote:I mean you can imagine a situation where no matter how infinitely small the difference between .999... and 1  it could have a consequence in the physical world. Or is that just not possible?
You seem to be trying to say that a computer would not be able to tell the difference between 0.999...(recuring) and 1. A computer would never have to deal with 0.999...(recuring) as a decimal expansion, since it does not have enough memory. It only has a certain number of floating points. It may, if the programmer chooses, round off to 1 if it gets close enough, but there again lies the human factor. The maths is all good.
Bad speling makes me [sic]
 Ricky
 Moderator
Re: I disagree with 0.999... = 1
A computer would never have to deal with 0.999...(recuring) as a decimal expansion, since it does not have enough memory.
"All [computer scientists] ask for is that you engineers fit an infinite amount of transistors on a finite sized chip"  A professor of mine.
An infinitesimal isn't even a number at all, at least highlighted in most caculus books.
Did I not just say that infinitesimals don't exist in the reals? And what is calculus? The study of real functions.
But you need to go to other number systems: superreals, hyperreals.
Think of .999... this way. How much would you have to subtract from 1 to get to 0.999.....?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: I disagree with 0.999... = 1
It may, if the programmer chooses, round off to 1 if it gets close enough, but there again lies the human factor.
Exactly, so within say the 300 Million lines of code it will take to put a manned space craft on mars  well there is enough of a gamble to call it a risk.
However I have at last come to accept that they are one and the same, or 1 and .999..., to be less precise.
 Kurre
 Power Member
Re: I disagree with 0.999... = 1
lucadeltodesco wrote:if you have a number a, no matter how many times you add 1/inf to it, its still 'a' and it does not change. Dont think of it as adding a one to the end of an infinate chain of 0's. Thats trying to think of infinity and infintisemals as numbers. there not.
just noticed a funny thing here. 1+n(1/infinty)=1 and 1+n*0=1 but if n= infinity, you add an infinty amount of 1/infinity, wouldnt this look like: 1+infinty(1/infinity)=1+infinity/infinity=1+1=2 but 1+infinity*0=1+0=1 1<2 1/infinity > 0
thats if infinty can be used as a number, which i guess it cant, but still...funny
Last edited by Kurre (20061011 05:00:45)
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