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Post by StuartG on Jun 30, 2011 22:53:03 GMT 1
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Post by rjpageuk on Jul 1, 2011 10:54:30 GMT 1
I dunno really -- the closer you look, the continuum eludes us somewhat -- just that its such a SIMPLE concept in some sense, that it is very natural that it is what we have based so much of our physics on, both in terms of WHERE or WHEN things can happen, but also what values other variables can take. But maybe the continuum is just a deception, a consequence of our limited point of view -- just as even em waves turned out to be made of distinct things (photons), and as space and time themselves may be discrete rather than continous, so maybe the maths of the continuum isn't the real maths of the uiniverse, just an approximation to it. This was basically what I was getting at when I said that irrational numbers arent really there in any real sense. I suppose the real answer is we just dont know yet. The continuum is just what happens to discrete as the size tends to zero. I dont think it is fair to describe this as a deception or consequence of a limited view - it is essentially a catch all model that can describe discrete items no matter how small. The reason I was always drawn to maths is the beauty and perfection of it, the real world getting in the way is always an annoyance to me.
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Post by speakertoanimals on Jul 1, 2011 13:15:49 GMT 1
No it's not. The number of points in discrete steps along an infinite line is infinite, but a SMALLER infinity than the number of points on the continuum between 0 and 1. You can't go from one to the other by taking a limit (although we do often do that in practical, computational cases, but the nature of the underlying space isn't the same!).
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Post by rjpageuk on Jul 1, 2011 15:07:15 GMT 1
It is true there is no direct correspondence between a countable set (discrete) and uncountable (continuum), however I was trying to say that we can think intuitively about the continuum as what results from a discrete set of numbers where the size between the items tends toward zero holds. EDIT: You are right though this isnt true. In any case I dont think its adds anything to the discussion, so ignore.
What I really wanted to say is that the continuum is a catch all solution that can deal with discrete items no matter how small or how many. In this sense it is useful but in no way requires the continuum actually exists in any real sense - the errors such a model introduces would just be proportional to the "granularity" which may well be very small indeed.
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Post by abacus9900 on Jul 5, 2011 15:58:18 GMT 1
Pi doesn't come out exactly because you are trying to express a rational number (any number that can be expressed as the quotient or fraction a/b of two integers) in terms of the decimal system. Sometimes the two systems are not compatible so that you sometimes get a recurring fraction that never ends or in the case of Pi seems to go on indefinitely. It's like trying to express 1/3 in terms of a decimal number; all you get is 0.33333333... without ever achieving an exact match.
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Post by speakertoanimals on Jul 5, 2011 16:03:04 GMT 1
Wrong. Pi ISN'T rational, that's the thing.
Rational: finite or recurring decimals
Irrational, infinite, never-recurring decimals
10 isn't special, distinction holds WHATEVER integer basis you choose for your number system.
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Post by abacus9900 on Jul 5, 2011 20:23:49 GMT 1
Wrong. Pi ISN'T rational, that's the thing. Rational: finite or recurring decimals Irrational, infinite, never-recurring decimals 10 isn't special, distinction holds WHATEVER integer basis you choose for your number system. The problem here is that you are in effect defining Pi as irrational without any actual mathematical proof. Just because it has not been exactly defined to date does not prove it never will be - or do you know something the rest of us do not? Also, 10 may not be special but the point I was making is that by decimalising 22/7 you are attempting to make a 'translation' from one system to another, which sometimes cannot be completely accomplished. Whether you use the decimal system, octal, binary, etc,. you are going to encounter the same problem. It's like trying to make a perfect translation from say, French into English. Often, all you can do is to approximate. (You don't like analogies but it's the best I can do). STA, you are once again guilty of using labels such as 'rational' and 'irrational' as absolutes but nothing in maths is absolute - just place holders for current mathematical concepts. (Have you forgotten your Godel?)
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Post by eamonnshute on Jul 5, 2011 20:29:33 GMT 1
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Post by abacus9900 on Jul 5, 2011 20:54:33 GMT 1
This does not alter the original point that 22/7 (rational number) does not directly translate using another number system because it turns out to be irrational in this case. This answers the question of the OP.
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Post by eamonnshute on Jul 5, 2011 21:03:29 GMT 1
This does not alter the original point that 22/7 (rational number) does not directly translate using another number system because it turns out to be irrational in this case. This answers the question of the OP. It doesn't matter what number base you use, if a number is rational in base ten then it is rational in any base. ditto irrational numbers.
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Post by abacus9900 on Jul 5, 2011 21:25:24 GMT 1
This does not alter the original point that 22/7 (rational number) does not directly translate using another number system because it turns out to be irrational in this case. This answers the question of the OP. It doesn't matter what number base you use, if a number is rational in base ten then it is rational in any base. ditto irrational numbers. That is not the issue here. We are dealing with a rational number (22/7) expressed in base 10 which turns out to be irrational (according to you).
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Post by eamonnshute on Jul 5, 2011 21:37:32 GMT 1
We are dealing with a rational number (22/7) expressed in base 10 which turns out to be irrational (according to you). According to me? Nonsense. A rational number is one that can be expressed as one integer divided by another integer. So 22/7 is rational (regardless of base).
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Post by abacus9900 on Jul 6, 2011 12:48:25 GMT 1
The decimal system is a number system that represents quantities in terms of tenths, which is why it is not always possible to perfectly and completely represent other number systems completely, no matter how close it gets. That is the answer to StuartG's question, sweet and simple. The fraction 22/7 does not know anything about tenths, only human beings do.
A way to look at it is that you are trying to combine two mathematical 'universes', if you like, which is always going to create difficulties.
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Post by speakertoanimals on Jul 6, 2011 14:06:26 GMT 1
Abacus, I'm afraid you have slipped back into your old ways, and are trying to castigate me for supposed sliips, when actually you are just totally incapable of grasping the relevant mathematical concepts, and talking absolute bollocks as a result.
Rational and irrational ARE absolutes. Saying 'nothing in maths is absolute' is just utter nonsense, and shows you have very little idea about maths.
So, rational means -- can be expressed as the ratio of two integers.
Irrational means the converse, CANNOT be expressed as the ratio of two integers.
So, unless you really think there is some alternative to can/cannot for numbers, then these are two distinct categories.
Wrong. Being rational or irrational has nothing to do with tenths, just that it is fairly simple to show that a rational number either has a finite or a recurring decimal expansion. And same goes for the expansion of that same number in any other number basis.
So, 1/7 is 0.1 in base 7 number system, but 0.142857 recurring in base 10. Finite or recurring, depending on the basis.
Frankly, abacus, as I've said before, I can't quiet believe that even you could actually be this stupid, when even the quickest google would have confirmed the meaning of rational and irrational numbers, and the existence of proofs that pi is irrational .
So, which is it abacus, as I've asked before, are you really that stupid, or just a total time-waster, because you're back to your old tricks of insulting people for stating what is just basic maths, as anyone could discover.
I'll be charitable, no one could actually be this stupid, he's just a wind-up merchant, back to his old tricks. I suggest we just ignore his posts.............................
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Post by abacus9900 on Jul 6, 2011 17:02:20 GMT 1
22/7 can be expressed as the ratio of two integers....I just did it! Pi cannot be expressed as a rational decimal and this is why it does not 'come out'. 5/10, for example, would be a rational decimal. What a peculiar idea. If being a rational or irrational number has nothing to do with tenths then why does Pi turn out to be irrational using the decimal system? But this is missing the whole point. 22/7 is a perfect and complete mathematical description of a ratio but when you try to convert 22/7 to a decimal it ceases being so. What you seem to be saying is that just because a particular fraction turns out being an irrational decimal that makes the fraction itself irrational yet: "In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero". en.wikipedia.org/wiki/Rational_numberYou keep missing the point. Using a simple fraction you can obtain a complete measure of Pi but using another system of counting you can't. Got it yet?
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