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why would .999 repeating be an approx.? That's like saying "professional environmental safety and management engineer" is an approximation for janitor. Infinity is a concept, math is a concept, red is a concept, etc. Why don't we just get rid of fractions? I never understood their importance:D
We have numbers like pi and e, which can't be written as a fraction, but then we have fractions which can't be accurately written as decimals (like 1/7 I think). [edit] Well, they might be able to be written accurately, but not exact [/edit]Quote:
Originally posted by snakeeyes1000
Why don't we just get rid of fractions? I never understood their importance:D
I think we must agree to disagree. No, I am not a professional mathematician, and neither have I looked for any kind of professional proofs for what I'm talking about. I don't intend to simply because this debate really isn't that important to me, and I'd rather do some work on a programming project I'm working on;). Al of what I have said on the topic is based on my personal opinion and logic. That doesn't necessarily make it invalid though:).Quote:
Ok, obviously you disagree with the proofs because you think infinitely small doesn't equal to 0. or in other words:
lim (x->0) x !=0 (now you see, it is x aproaches 0, not x=0 -- in your words, there is an infinitely small difference between x->0 and x=0)
and you also don't believe that 1/3=.333...
because of some wierd reason thats to me is completely could not believe (like 1/3 cannot be translated into a fraction because it is an approximation even though .333... stands for .3 repeating).
and also you believe that infinity exists (which i do so too, but i believe that it is a concept and not a value/number)
now show me a proof of 1 and 2. does anyone ELSE believe you? give me a professional source (hopefully a university site) where it clearly indicates that either lim (x->0) x != 0 or that
1/3=.3333... I want this because I don't believe you are a professional mathematician that specializes in number theory. well you don't have to be one, unless you show me another person's proof or your own proof that may have been published in a professionally recognized publication then I'll believe you.
in short, you offered me no proof while you dismissed my proofs (which has been largely recognized as a valid proof) as rubbish without a proper reason.
By the way the incompleteness theorem just indicated that such a case may exist. as I said, no one has offered me any proper reason as to why .999... != 1, other than .999... is only an approximation even though it clearly is not.
I am no longer going to debate the issue, although I would like to clarify my stance on several of the issues that you misinterpreted.
I think that infinity is a concept, but not one that cannot be used in such terms as "infinitely large" or "infinitely smaller". You are correct in that I think that evenly an "infinitely small" value is distinct and separate from 0, and that 1/3 != (0.3). IMO, (0.3) is an approximation we use for 1/3, mainly because 3/3 != (0.9). (0.3) is infinitely close to the true value of 1/3 however.
I was not attempting to dismiss your proofs, but simply state that I find them to be faulty. That is my opinion.
I think that the case of these numbers that have an infinite number of decimal places, such as (0.3) or (0.9), is an example of Godel's Incompleteness Theorem.
I think that this last part is the part of my viewpoint that has been most confusing. With numbers, such as pi, (0.9), or (0.3), that have an infinite number of decimal places, I believe that there are only two ways in which they can be used: conceptual/algebraically or approximately. You can use these numbers in ratios or equations where they cancel out, or in cases where you want a result like 6*pi, without error. You could say (9*(0.9)) / (3*(0.9)) = 3, and that is fine. You could not, however, evaluate 6*pi to a single real value without the second usage of such numbers, approximation. In this case, you take an approximation of the infinite number you're going to use. The number of decimal places of the approximation depends on how much accuracy you need. You could evaluate 6*pi by saying that pi = 3.1415926, because you don't need anymore accuracy than that. Alternatively, if you needed LOTS of accuracy, you could approximate pi to 1000 decimal places, and get enormous accuracy. But even if you used 100 trillion decimal places, it still would not be exactly correct, because the actual value of pi is not a real number. Exactly what it is I do not know, but it cannot be contained in a real number.
Approximating numbers such as pi or (0.9) in nearly all cases is certainly sufficient. You could use 1000 decimal places or 1 million, and have whatever level of accuracy you need. The only problem occurs when you need infinite accuracy. As an example, take the error estimates used in scientific results. You could have 5.78+/- 0.05 as a number you were working with, do math with it, and carry through the error estimates to get the final error estimate. If you had 3 eggs or something, though, then there is no error, because 3 is an absolute counted value. pi, being a mathematical constant, has no error estimate; it is devoid of error. This, however, is only applicable to the true non-real value of pi. The value that Windows calculator gives for pi is 3.1415926535897932384626433832795. If you approximate this to 3.1415926, you are essentially introducing an error of +/- 0.0000001. If you approximate pi to 3.141592653589793238462643383279, you are introducing an error of only +/-0.000000000000000000000000000001. If you approximate pi to its first 1 million decimal places, any error would be nearly non-existant, and it would probably be accurate enough to use in astronomical calculations on stuff millions of light years away.
The key word, however, is nearly. In the extremely few cases where infinite accuracy is desired, even the tiniest, most infintessimal amount or error is infinitely greater than the desired accuracy of the result. If your error is +/- 1/100 trillion, even that is far, far, far greater than 1/infinity. The true, non-real value of (0.9) is (1 - 1/infinity). If you need accuracy to within 1/infinity, then 1/100 trillion is a huge, vast amount of error.
It is because of this that I find (1+(0.9))/2 to be a faulty proof, because the error you introduce when approximating (0.9) is far greater than the desired accuracy of the result. This proof is only valid when the absolute value of (0.9) is used. Unfortunately, (0.9) is not a real number, and it is simply not possible for us to evaluate such a question. Trying to do so would be the same as trying to count to infinity. The most powerful computer creater 1 million years from now would work at the task forever and never succeed.
I think that (1/infinity) != 0. You may think otherwise. If you do agree that 0 != (1/infinity), then 1 != (0.9). In any case, I hope that we can at least agree to disagree with understanding of and respect for each others' viewpoints:).
BTW, I've found this debate to be very interesting and enjoyable. I apoligize if in some posts it seemed as though I was being rude or angry. It was not intentional:(. I was simply too interested in the debate:).
Lots of interesting points brought up. A lot of them show people trying to use concepts they only have a wooly grasp of, but it's a start.
On the subject question: Can a = b when a and b are different integers?
Depends on your definition of different integers. I would define it as something like:
a element of N
b element of N
a != b
From this definition clearly a = b is a contradiction.
However, people saying that in computer memory a = b when a = -1 and b = 6--- is NOT contradicted by this proof, as a, b are not elements of N in this case.
If we use modulo arithmetic (similar to the computer memory approach), 1 = 5 (mod 4) or more generally a = xn + a (mod x) where n is some integer.
The main argument seems to be over whether 1 = 0.(9) [Nice way to notate it on a computer]
Many arguments to counter this have used the idea of infinity. However, they all seem to have got it wrong. If you are saying that there is a difference (significant or not) between 0.(9) and 1, then you are wrong. Talking about the "infinitieth" place is absolute rubbish, since infinity is not a number. However, it is more than just a concept. It is well defined. Because it doesn't appear on the real number line does not make it just a concept that can be ignored.
Infinity is used in this case as a limit. And this is how.
i.e., take 0.999.... with a finite sequence of n 9's, the answer is 0.0000...0001 with n-1 0's after the decimal point.Code:1 - 0.9 = 0.1
1 - 0.99 = 0.01
...
1 - 0.99999999 = 0.00000001
...
n
1 - E 9/(10^n) = 1/(10^n)
1
Now as n tends to infinity (note, you can't say when n is equal to infinity, that doesn't make sense), the answer tends to zero. Now if you know how limits work, we can see that we have a continuous differentiable function here, which means the limits are the values at all points and from both directions essentially (loosely worded, I know, but it gets the point across). Therefore, if you have an infinite sequence of 9's (i.e. 0.(9)), then the answer is an infinite sequence of 0's with a 1 in the last place. Now, where is the last place in an infinitie sequence? There isn't one. Therefore there isn't a place for the 1. Therefore 1 - 0.(9) = 0, and since we're using real numbers, we can use standard arithmetic rules to find that 1 = 0.(9).
Of course, this proof will be no more valid than any other offered to you so far.
One final point to take exception to: Pi is not a real number.
Do you actually know what a real number is? I think you're either using "real" when you don't realise there is a mathematical definition of a real number, or you're grossly mistaken about what the definition is. It lies on the number line, which makes it real. It's not rational, certainly. It's not complex, for sure. But it most definitely is Real.
And what makes you so sure that 1 is any more valid that Pi as an "accurate" number? 1 doesn't exist any more than Pi does. "1" is defined to stand for the value of x, such that a.x = a, where a is any number. "0" is defined to stand for the value of x where a.x = x for any a. If you can point me to 1 in the real world (not one thing, but to just one - the value, not the digit) then I'll concede the point.
I reckon 0 and 1 are the two most annoying numbers. They have so many similar properties (and probably even more unique properties).
a.x = x and a.x = a surely work for 0 and 1, but how do you go on from there? Say for two, would you say: a.x = a + a? That of course simplifies, but then you are using the number 2 to 'define' the number 2 (it's like trying to explain the word 'do' to a foreigner without using the word 'do'). Really, for two, you say a.x = 2.a. If that's the case, then for 1, it is really a.x = 1.a. We don't show the coefficient (when it is 1) because there's no need too. The number is still there, just not shown.
Ummm, yes, but that isn't the way you'd define 2, 3, 4, ...
It's only 0 and 1 that would need to be defined like that. 2 would be 1+1, 3 would be 2+1 etc. No need to get yourself in a mess like that method would surely get you into.
what the heck is a.x=a? don't understand it! :confused: someone explain?
It's an equation with two variables, x and a.
a is defined to be any number.
x is a specific value that we call "1".
the "." is a multiplication symbol.
I'll assume you know what the "=" means ;)
So, I guess if we say PI = 3.14, then PI is 42!
Cool!
:D
Usually the dot (or period) is placed in the middle of the line, not the bottom, so sort of like a bullet point. It's just too lengthy to look up the character code (or unicode value) for the dot that means times. I assume it will be under the math section of the unicode values.
In that case, could we say PI is the meaning of life? LOL :D.Quote:
Originally posted by NotLKH
So, I guess if we say PI = 3.14, then PI is 42!
Cool!
:D
lol i'd use a*x or ax, but never a.x lol:p
...and certainly not axx - that would definitely be confusing!
If 2=3 then 1=8 and 493=6984. In effect if any of this was correct then there would only be one true number in the entire universe :D Let's face it, if 2=3 then there would be no need for the number 3 in the first place as it would already have an equivalent (2). So every number would = 2 and so every number would be 2 and so 2 would be the only unique number in existence. I guess this argument has far reaching implications. If 2=3 then maths wouldn't exist because every number would be 2 and so manipulation of this number would always result in the equivalent...we wouldn't need calculators, we wouldn't need maths - we wouldn't need this forum. I guess what I'm trying to say here is that; if 2=3, I'm a donkey’s hairy behind :cool:
OK, just one confusing thing used:
many people are writing the C++ "!=" for 'does not equal'
please can you use the VB "<>" , for one main reason:
"0.(9) != 1" can be read as
"0.(9)! = 1" which is also true, but it's using the Factorial sign. This gets confusing when people write "a != b", where at first glance it seems they might mean "a! = b"
(N.B, 0! = 1 as well)
Anyway, an interesting question:
1 - 0.(9) = what???
1/infinity. I said that already:).
The flaw with Evil Giraffe's attempt to prove that they are equal is that he is approximating the answer. When you begin subtracting (0.9) from 1, yes, you get an endless number of 0s. But the fact that you cannot actually reach the theoretical 1 in the 'last place' does not mean that the theoretical 1 is meaningless. It is theoretical, not real. Not all meaningful numbers must be real.
By stopping because you realize you will never be able to finish the question, you are getting the answer for a different question! You are instead getting the result of 1 - 0.9999999999999999999999999999999999999999 (which BTW is NOT zero, so you're approximating that too). (0.9) is not a real number, but a conceptual value, and attempts to use it in real math will always fail. This does not mean you can pretend to use an approximation as its true value with the reasoning that its true value does not make sense!
1 - (0.9) = 1/infinity. 0 is, of course, the closest real approximation of that value, but is not the correct answer.
BTW, yes I can use 1/infinity, because I'm not doing anything remotely related to limits here. This is a single, non-changing conceptual value. If infinity was NOT the denominator, then (0.9) would have a finite number of digits.
BTW #2: The next time anyone asks you what pi is, say that it is the result of 6 * 9;).
One thing comes to mind:
"what a couple of dumbasses"
hahahaha
Oh yeah, and for those of you who say 1=2(or the like). You need to take 3rd grade math again. :rolleyes:
Back in 6th Form Physics, we had to write an essay on anything to do with what we'd covered so far. I did Nuclear Fusion and Fission, but that's besides the point. My friend did "How do bumble bees fly?". It may sound like a dumb subject, but a while ago, some physician (or is it physicist) proved bumble bees can't fly, which is absurd because, well, they DO fly. He got full marks for his essay, using the 'proof' that they don't fly to prove they do.
The moral of the story is, just because you think it's dumb, doesn't mean it is. So you shouldn't exactly call these people dumbasses.
Obviously to us all, 1 <> 2, it's getting to that point which is clever.
It's not actually theoretical even. It just doesn't exist. Please go and read up on the idea of infinity.Quote:
Originally posted by Alphanos
The flaw with Evil Giraffe's attempt to prove that they are equal is that he is approximating the answer. When you begin subtracting (0.9) from 1, yes, you get an endless number of 0s. But the fact that you cannot actually reach the theoretical 1 in the 'last place' does not mean that the theoretical 1 is meaningless. It is theoretical, not real. Not all meaningful numbers must be real.
I never said we were stopping before finishing. My point was that this pattern continued forever. I wasn't stopping, I had already calculated it!Quote:
Originally posted by Alphanos
By stopping because you realize you will never be able to finish the question, you are getting the answer for a different question! You are instead getting the result of 1 - 0.9999999999999999999999999999999999999999 (which BTW is NOT zero, so you're approximating that too).
Um, I'll say it again. Go and look up what is meant by a "real" number, and explain why you think 0.(9) isn't one. (a real number that is. Not why you think it isn't "1" :D)Quote:
Originally posted by Alphanos
(0.9) is not a real number, but a conceptual value, and attempts to use it in real math will always fail. This does not mean you can pretend to use an approximation as its true value with the reasoning that its true value does not make sense!
I'm not sure about your point of "real" maths. And, as I mentioned before, it's not an approximation, it is a real number, and the true value does make sense
So 0.(9) isn't a "real" number but 1/infinity is? 1/infinity is undefined. Using that as a value doesn't make sense.Quote:
Originally posted by Alphanos
1 - (0.9) = 1/infinity. 0 is, of course, the closest real approximation of that value, but is not the correct answer.
You're not dealing with limits, so you think you can use infinity as a number? It doesn't work like that. Infinity is the cardinality of the Natural Numbers, not an actual number, therefore you can't use it in arithmetic like you have there. You can only throw it in when you are dealing with limits.Quote:
Originally posted by Alphanos
BTW, yes I can use 1/infinity, because I'm not doing anything remotely related to limits here. This is a single, non-changing conceptual value. If infinity was NOT the denominator, then (0.9) would have a finite number of digits.
Okay, another argument:
0.(9) is the geometric progression with a=0.9 and r=0.1
Sum of an infinite G.P. is a/(1-r) = 0.9/(1-0.1) = 0.9/0.9 = 1
A geometric progression is aka a geometric series right? Otherwise I'm completely lost! :)
Yep. Geometric Series and Geometric Progression are just different names for the same thing
Actually, I have proven that .999... ==1, it does not suffer from round off error at all, and it has yet to be disproven.Quote:
originally posted by Gandalf_Grey_
i am not saying that the 8 is at any defined decimal place it is just at infinity. Maybe you aren't getting this specific reference however when i say .999 ... 8 i mean a number infitesimally smaller than .999 .... An infitesimally small number is possible (the difference between .99 ... and 1).
snakeeyes1000, it certainly does suffer from rounding error, that's the point:)!
Evil_Giraffe, I believe I have already explained my viewpoints clearly, so I will respond only that I certainly do think that infinity does exist, and that 1/infinity is not a real number. Infinity is not a real number, but a concept. It obviously makes no sense to use it in real math. However, 1/infinity is NOT an attempt to use infinity in real math, but rather the simplest and most accurate method of defining, literally, a value which is infinitely small. (0.9) = 1 - 1/infinity not due to real math, but conceptual math.
When I say that a number is not real, what I mean is that its true value is conceptual, and when we use it as a real we are only approximating it. I have personally come to the conclusion that all numbers with an infinite number of digits are not truly real, simply because it is impossible to USE them. You may disagree with me on that and many other points; that is my opinion. (Yes I know it is not exactly adhering to the definition of real numbers, I think the definition is flawed:)).
Sorry Alphanos, there isn't room for personal opinion here, since we're working within a common framework. Just because you disagree with the definition of a "real" number, doesn't mean you can disregard it.
And you're also spouting rubbish about "conceptual" values again. Mind telling me what's wrong with talking about numbers that can't be represented exactly in a non repeating decimal format? 1/7. Seems a fairly "real" number (using your definition of the term real number) to me. Then suddenly we look at it in decimal and it's a recurring decimal. How about we look at it in base 7? Oh look, it's 0.1 Seems fairly "real" to me.
Your concept of "real" numbers shouldn't depend on how you write them down. The notation used should have no bearing on the underlying maths. That's the whole point.
I guess you also disagree (personal opinion of course) with the summation to infinity argument?
Basically: i agree with Evil Giraffe
1) I think we all believe infinity "exists"Quote:
I certainly do think that infinity does exist
2) What, so we can't use Pi or e or 20.5 any more???Quote:
all numbers with an infinite number of digits are not truly real, simply because it is impossible to USE them
3) Oh, so 3^(0.9) = 3 and now you CAN use (0.9) to evaluate a real value, with (0.9) =1Quote:
You could say (9*(0.9)) / (3*(0.9)) = 3, and that is fine. You could not, however, evaluate 6*pi to a single real value without the second usage of such numbers, approximation
4) Sorry, i think i missed you proof, and can't find it. (There have been a lot of posts)Quote:
1 - (0.9) = 1/infinity
BTW: don't worry about sounding "rude or angry" ( it think you said that a few posts ago) Anyway, it is good to see someone standing up for what they think is right.
Famous quote of Einstein:
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.
I just thought I'd pop that quote in while we are debating about infinity and so on.
the concept exists, but number which has a value of infinity does not exist. but you can have infinitely many nines trailing (a concept here, not a value).Quote:
Originally posted by sql_lall
Basically: i agree with Evil Giraffe
1) I think we all believe infinity "exists"
actually, if we are going to disregard real and imaginary number definitions, is Sqrt(-10) really imaginary? I can read it just fine.
There is more to this than meets the eye!
When using the "=" sign it can be evaluated differently than what you normally would assume. The question you have to ask yourself is "what are the rules deciding that two numbers are equal". Mathematical rules like this are studied in subjects such as Group and Field theory and is normaly studied my MSc students.
Under the "normal" rules 1=2 or 1=4 is always false. This is so since the "counting rules" (the "identity object property" and reflexive property) imply that if a=b <=> (if and only if) a-b=0 (where a,b are real numbers), wich is not true in the case above.
However, to give an example when this can be true we turn to what is called "Modulo counting".
Assume that the "counting rules" belong to the field Z3:="integers modulo 3", then, as an example; 0=3,1=4, 5=11.
To see this you have to know that two integers are equal, modulo 3, iff the differece between them are divisible by 3. Ex:
a=b mod 3 <=> (a-b)/3 is an integer.
This "modul 3" counting rules stuff may seem like useless academia, however, the study of "counting rules" has important implications for advanced mathematics.
kimmy
Hey wow, if that's true I'm gonna start ripping people off by putting a "prices are in integer modulus 3 format" in the disclaimer.
Actually, you don't use the equal sign, nor do you say that they're equal when using modular arithmetic. When referring to numbers in this sense, you say that they're congruent, not equal. The symbol for this is the one similar to equal, but in that it has 3 horizontal lines, not 2.Quote:
Originally posted by kas
Assume that the "counting rules" belong to the field Z3:="integers modulo 3", then, as an example; 0=3,1=4, 5=11.
To see this you have to know that two integers are equal, modulo 3, ...
If I use -=- as the congruent sign (work with me :P), then your example would be:
0 -=- 3, 1-=- 4, 5 -=- 11 (modulo 3)
However, it can sometimes be thought of as being equal, since it's a very close definition. Modular arithmetic is definitely a very powerful tool.
Destined
It is true that when introducing modular arithmetics you use the "three line" equal sign and call it congruent.
However you can redefine the properties governing how the "=" can be used (this was ultimatly my point). If these properties follow certain rules you end up with a new type of arithmetics. (These rules are called the Groups, Rings or Fields properties, are a part of the study of Abstract Algebra.) "Normal" arithmetics or modular arithmetics are jus special cases. Another, common, arithmetic is matrix arithmetics where it is generally not true that AB=BA.
It is perfectly valid, in maths, to use the ordinary relation and operator signs even when using other (such as modular)than "normal" arithmetics . However, for the sake of clarity, you should clearly state wich space your numbers belong to (e.g. the "matrix" space).
kimmy
ok i think you can NOT have 1 = -1 or 1 = 2 becuase he said for just 1^2 = -1^2 that is not the same thing as 1 = -1 that is something totaly different.