February 23rd, 2006, 12:50 AM
today we were shown a method of converting recuring decimal places to common fractions.
eg. 0.3 recuring=n=0.33333...
then 10n=3.33...
10n-n=9n
9n=3
n=3/9=1/3 (first time i've put maths into the internet so i hope you can all read the mess i just put up)
so if thats correct then for the case n=0.9 recuring...
n=0.99999...
10n=9.99999...
10n-n=9n
9n=9
n=9/9=1
0.9 recuring equal to 1????
You got it! So every number with a finite number of digits can be written as a number with infinite digits ending in repeating nines. Or to put it another way every number ending in repeating zeros can also be written as a number ending in repeating nines.
for example: 1.36323 = 1.36323000000... = 1.3632299999999...
This rewriting can be represented by subtracting 1.0*10^n and adding 0.99999...*10^n.
Notice that if you take any number with infinite digits like pi or 1/3 = .3333333... or 3/11=.2727272727.... and perform this same operation of subtracting 1.0*10^n and adding 0.99999...*10^n, you get the same number completely unchanged.
for example: .2727272727.... = .1727272727.... + .09999999.... = .2727272727.... because 7+9 = 16 and 2+9=11
.1
.16
.011
.0016
.00011
.000016
.0000011
.00000016
+.00000001....
---------------
.27272727....
February 23rd, 2006, 09:31 AM
Hah! I was going to say that you've gone a long way around it, because all you have to say is 1/3 = 0.333333.... 3 x 0.3333333 = 0.9999999.... and 3 x 1/3 = 1 therefore 1 = 0.999999........, then I realised that by saying 10n = 3.33333 and 9n = 3, what you had done was first prove that 1/3 = 0.33333...... which is better maths than I could ever have done!Originally Posted by wallaby
1 = 0.99999....... is a mathematical identity. We have argued elsewhere mainly with very obstinate people who don't believe in infinite expansions, or who don't understand the definition of "rational number".
Yes, you argued it with me.
My conclusion was in the limited space we have 0.999999 = 1. Only because we don't have room in the universe to place all the extra 9's. With unlimited space it never rolls over to a true 1. For all useful purposes I'll concede that it equals 1, even if it really is illogical.
so when we add 0.9999... to itself we get 2, intead of say 1.888....
Actually we get 1.9999999999999... that is 2.
Your conclusion appears to be based on trying to visualise an infinite amount of space for all the 9s, and to declare that if there was infinite space to put them, they would indeed be two different numbers. Whereas what the mathematics clearly shows (without the need to see all the nines) is that the two numbers are identical which ever way you cut it. They are identical through rigorous logic, not merely a consideration of a notional infinitesimal 0.00000.....0001 difference.Yes, you argued it with me.
My conclusion was in the limited space we have 0.999999 = 1. Only because we don't have room in the universe to place all the extra 9's. With unlimited space it never rolls over to a true 1. For all useful purposes I'll concede that it equals 1, even if it really is illogical.
Care to share this logic with us ?They are identical through rigorous logic, not merely a consideration of a notional infinitesimal 0.00000.....0001 difference.
COME ON!!!!!! walaby did it in the very first post!!!!
I think the problem is we have no way to really solve the problem, we always end up rounding as we have no real choice in the matter, once rounding occurs our logic shows 0.999... to equal to 1. Basic logic and common sense will tell you that it doesn't really equal 1. It's a paradox that can not be resolved. I can argue that the 9's can continue to infinity and one can also argue that the longer it continues the closer it gets to being a true perfect 1.
The other problem is the notation itself, we are not writing out all the digits. If we did so (impossible) we would find that 0.999... can not equal 1. The reason being is that we would never finish writing out the first part.
The math world has no choice but to treat this as a perfect "1", if not problems could never be solved and would never end.
Sorry but if I have a score of 10.0 and you have a score of 9.999... I still win
Insanity...did you miss the OP? Or when Mitchell told you to go back and read the OP?
Nope,
This problem can't be done without notation...
n=0.99999... (Keep going...you'll never finish)
10n=9.99999... (This is correct..but keep going...forever, wait you should not be here yet)
10n-n=9n (This right here is compromise and rounding, n does not equal 1)
9n=9 (Nope sorry, you still can't be here yet as your still on the first part)
n=9/9=1 (This is of course right if the first part was)
No matter how you stack it unless you round up 0.999... can never equal 1. Again I think people miss my point, I'll agree that it's a 1 only to satisfy our limitations. Logic however says it's not. Compromise says it is.
Limitations do not present truths.
we know n is not equal to 1, if n were equal to 1 then we could not get our answer.
n=0.999... to infinity.
10n=9.999... to infinity.
subtract the all the 9's after the decimal place, which is an infinite number, in n=0.9999... from all the 9's after the decimal in 10n=9.9999..., which is again an infinite number.
in both cases there are an infinite number of 9's after the decimal place.
infinity-infinity=0
just as 1-1=0
this leaves us with a grand total of 9n=9.
from there
n=9/9=1.
ta daaaa.
it may look like magic but it isn't.
subtract the all the 9's after the decimal place, which is an infinite number, in n=0.9999... from all the 9's after the decimal in 10n=9.9999..., which is again an infinite number.
Well see now that actually makes sense
Look, are you telling me that you don't accept that 9.99999... = 9 + 0.99999...? Because it follows that 9.99999... - 0.99999... = 9. It is almost as if you are saying that unless we write all the digits down one of digits might not be a 9! We are taking about numbers here not measurements so the numbers are what we say they are and the ... means that they are all nines. It is all about the consistency of mathematics which is not something that can be proven but must be taken on faith. So when we perform two mathematically equivalent operation we must say that the results are equivelent.Nope,
This problem can't be done without notation...
n=0.99999... (Keep going...you'll never finish)
10n=9.99999... (This is correct..but keep going...forever, wait you should not be here yet)
10n-n=9n (This right here is compromise and rounding, n does not equal 1)
9n=9 (Nope sorry, you still can't be here yet as your still on the first part)
n=9/9=1 (This is of course right if the first part was)
No matter how you stack it unless you round up 0.999... can never equal 1. Again I think people miss my point, I'll agree that it's a 1 only to satisfy our limitations. Logic however says it's not. Compromise says it is.
Limitations do not present truths.
9/9 = 1 right?
but 9/9 = 5/9 + 4/9 right?
but 5/9 = .55555... and 4/9 = .44444...
so 9/9 = .55555... + .44444... = .99999...
9/9 cannot be equal to two different numbers therefore 1 and .99999... must be the same number. Or perhaps you think that sometimes 5+4 does not equal 9 but is sometimes equal to some other number?
Of course sometime people are just stubborn, and it is anyones perogative to believe that mathematics is not consistent since that is something which cannot be proven.
Well I hope what you liked did not include this one because this is in fact incorrect or flawed in any case.
If we are going to treat infinity as a number we get bizzare conclusions.
what for example is infinity+1? Well we usually say that
infinity + 1 = infinity
but this implies that infinity - infinity = 1
we also say that for any number x
infinity + x = infinity
which implies that infinity - infinity = x
so infinity - infinity can be shown to be equal to any number you like
So what we usually say about infinity - infinity is that it is undefined.
so would any element in an infinite sequence be distingishable from the other elements.
Well I hope what you liked did not include this one because this is in fact incorrect or flawed in any case.
If we are going to treat infinity as a number we get bizzare conclusions.
what for example is infinity+1? Well we usually say that
infinity + 1 = infinity
but this implies that infinity - infinity = 1
we also say that for any number x
infinity + x = infinity
which implies that infinity - infinity = x
so infinity - infinity can be shown to be equal to any number you like
So what we usually say about infinity - infinity is that it is undefined.
pi - pi = 0 as well, now try to tell me what pi + pi equals
Sorry I don't think I really understand your question. Without further explanation and taking your question literally I would say why not?
Sure, any way you like. I'll tell you what, you give me pi as an example of the way you would like pi+pi and I will give pi+pi in the same form.
i'm not sure if i could explain it any better.Sorry I don't think I really understand your question. Without further explanation and taking your question literally I would say why not?
could we issolate one specific element of an infinite set or sequence to see that it is unique from all other elements by attaining a specific value for it?
Why not just say 2pi
Sure, any way you like. I'll tell you what, you give me pi as an example of the way you would like pi+pi and I will give pi+pi in the same form.
You mean like the sequence of even numbers? This can be represented by 2, 4, 6, ... 2n, .... where the nth term is 2n. So take the kth term and the jth term which are 2k and 2j. If we suppose that these terms are equal (not unique), that is 2k = 2j then we can divide by 2 to get k = j which means they are exactly the same term in the sequence. Thus I have proven that two terms in the sequence can only be equal if they are exactly the same term in the sequence, or in other words that every term in the infinite sequence of even numbers are all different or unique as you say.i'm not sure if i could explain it any better.Sorry I don't think I really understand your question. Without further explanation and taking your question literally I would say why not?
could we issolate one specific element of an infinite set or sequence to see that it is unique from all other elements by attaining a specific value for it?
Is that the kind of thing you are talking about? I would not be surprised if this is not what you are talking about which is why I thought I needed you to explain what you meant.
But here is another sequence 16, 9, 4, ... n(n-10)+25, ... The same kind of proof will not work of course and if you continue the sequence you quickly find that not all the members of this infinite sequence are unique:
16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, ...
that answers my question.
ExactlyWhy not just say 2pi
Sure, any way you like. I'll tell you what, you give me pi as an example of the way you would like pi+pi and I will give pi+pi in the same form.
If you say pi then I say 2pi.
If you say 3.14... then I say 6.28...
If you say (circumference of a circle)/(diameter of same circle),
then I say 2(circumference of a circle)/(diameter of same circle).
If you say 4 * inverse tangent of 1 then I say 8 times inverse tangent of 1.
If you say ln(-1)/i then I say 2 ln(-1)/i.
etc. etc. etc.....
Well, instead let's change the notation to base 3 instead of the (incredibly poorly chosen) decimal notation. one divided by three in that notation is 0.1. Three times 0.1 = 1.0
You see, the reason that 0.99999.... is equivalent to and precisely equal to 1 is that in reality 0.9999..... does not exist. It's a consequence of the decimal numbering notation we all use, not a number that actually exists in its own right which is different to 1.
Take a base numbering system, base k. 1/(k-1) = 0.1111111... in all bases (because k divided by (k-1) always leaves a remainder of 1). Therefore (k-1) x 0.111111.... = 0.kkkkkk...., but (k-1) x 1/(k-1) also = 1, by straight algebra.
Great point. At least this demystifies the whole thing to some exent by showing that it is an artifact of base representation. However your math requires a couple corrections, so I will do that if you dont mind.
The problem is that "(k-1) x 0.111111... = 0.kkkkkk..." is clearly incorrect, but this can be fixed by using base n + 1, and replacing k everywhere in your statement with n+1 or k-1 with n, but replacing 0.kkkkk... with 0.nnnnnn...
"Take a base numbering system, base n+1. 1/n = 0.1111111... in all bases (because the base n+1 divided by n always leaves a remainder of 1). Therefore n x 0.111111.... = 0.nnnnnnn...., but n x 1/n also = 1, by straight algebra."
D'oh!
how can 0.999999999999....... get 1.99999999999.....? do you just round the first 0.9999999999..... to 1 but just add 0.9999999999? i dont understand the logic.
not a math person... just a part-time realist making observations.
I think the discrepency (illness) is a flawed mathmatical system. The system is full of valid and good uses but, of course, isn't perfect. The differences, discrepencies, (I love the 4/9 + 5/9 proof) or symptoms are proof that the system is errant, not that .99999999999999999999999999 etc = 1. By definition 2 differnt values cannot be truly =.
hugs.
you can fight the symptons all your life but you wont make real progress till you Identify the disease and take steps to correct the root of the problems.
Sure, which is we conclude that .999... and 1 represent the same damn number.
Why people have so much trouble accepting that a real number can have two decimal expansions and yet have no trouble with 1/2=2/4 is beyond me. I mean 1/2 and 2/4 'look' different right? How can they be the same thing? Nonsense of course.
The problem is people who haven't bothered to actually study mathematics beyond the partial and incomplete picture given in their high school classes. They have no idea what these symbols they are complaining about actually mean. How they believe they can pass judgement without knowing what they are passing judgement on is also beyond me. Step 1- realize ignorance. Step 2- take steps to correct ignorance.
can 0.9999...9 be written as a fraction of integers.what i want to know is is 0.9999...9 a rational number.
If you mean the ellipsis (the "...") to mean a finite number of 9's, then yes it is rational. All decimal expansions that terminate are rational.
If you mean the ellipsis to mean an infinite number of "9"s followed by a final 9, then this is total rubbish. There is no "last" decimal position after an infinite number of terms.
If you were instead asking about 0.999... and were just sloppy with your notation, I have to ask if you read this thread at all? 0.999...=1, and 1 is a rational number. So yes 0.999... is a rational number.
1/2 and 2/4 are the same value
if you have .999999(inf...) there will always be an infinitely small seperation from the value of 1. Its inconvenient for math's sake but the difference is still there. You can consider it to be = to 1 all day long but it doesn't change the fact that there is a portion missing
The usual questions:
1) Can you even define what 0.999... means? No waffling, there will be some epsilons and deltas involved here or something equivalent in this case (e.g. least upper bounds).
2) Can you even define what you mean by "infinitely small seperation"? Again, no waffling. Give me a precise definition and how it makes sense within the axioms of the real numbers.
3) Do you know the difference between a limit of a sequence and the terms of the sequence?
4) If 1 and .999... do not represent the same real number, then there will be a real number between them. Can you give me a decimal representation of this number?
Until you can answer (1) I will assume you have no clue. So start there if you care to discuss this furthur.
1. .999=999/1000, .9999=9999/10000, you could continue adding a 9 to the numerator and a 0 to the denominator in equal numbers forever and the value would never = 1. Every decimal has a corrisponding fraction. The fractions for very large or small numbers are very cumbersome to write. If you change the value of 1 by the smallest of amounts in either direction you have changed the value it is no longer = to one. If x is < 1 then x is not = to 1. no matter how far you go out with the repeating 9's because there is a .9 at the beginning that shows the value to be <1. It might be pretty close to 1, but it is still < 1.
2. you answered your own question... adding epsilon subtracting delta... changes the value from 1 to another value entirely.
3. nope sure don't but I dont have to for this discussion. (I'll look it up though)
4. 1> 9999/1000 > .999. or 1 > .9999 >.999 > .99 > .9
What is next to the decimal is more important (significant) than what is farther away
1=1 period
p.s. relax : P
This is waffley and does not define .999... in any way. It looks like a vague attempt at describing the sequence .9, .99, .999, ... in terms of rational numbers though I don't know why? Try again, here's another hint-do you know what a limit is is?
I hope you don't think "epsilon" refers to some "infinitely small seperation"? (whatever that would be)
I'm going to take this to mean you can't define "infinitely small seperation". I'll make the suggestion that you stick to actual mathematics rather than making up your own terms that you can't define.
Yes you do need to know. There is a difference between the terms of the sequence .9, .99, ... and their limit.
I don't see anything here that's strictly greater than .999.... Go back to looking up the definition of .999...
I have no idea what the point of this is?
I'm practically comatose.
Claiming that there is a difference between 0.999... and 1 is the equivalent of saying that (1/3) x 3 < 1, but only if you use the decimal expansion. Mathematics does not work by allowing two different and logically inconsistent results to the same operation. That is, in fact, how it is determined that certain operations - like dividing by zero or treating negative square roots like real numbers, are disallowed.
April 22nd, 2006, 03:24 AM
After looking at the discussions I came to fing out that the problem is infinity. Lets look at infinity for a while.
Can any one define infinity? Probably not. The reason being so is the fact the the very notion on infinity is beyond the reach of our intellect.
Let me place it to you simply
infinity + (anything) = infinity !! correct?
if yes then
from our mathimatical reasoning we can subract infinity from itself to get the required result. But that is not the case as placed by my other friend as well.
The problem is that if we take infinity it is beyond the reach of understanding therefore playing with is mathimatically is also beyond understanding.
In simple mathimatics we call this the "undefined" why? Because we can not define it!!!. And for that matter remember one thing mathimatically we can work only on defined quantities.
If any one of you has studied advanced mathematics as in group theory, linear algebra, fuzzy logic and probability, this all deals in the approximation of understanding of "INFINITY".
Believe me using calculus and arthmetic and algebric principles will not solve the problem. These will help, but you will end up getting to the same points asdiscussed in the theorems of the above metioned fields.
Infinity is can only be abstractly understood. and mathematics works of deterministics not abstractions.
Believe my words. I have gone through all this and I know what happens.
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