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Post by nickrr on Jul 6, 2011 17:59:30 GMT 1
Being rational or irrational has nothing to do with any base. If something is rational or irrational in one base, it is correspondingly rational / irrational in all bases.
Pi is irrational because it is irrational in all bases.
I presume you understand that a number with a repeating decimal (e.g. 22/7 in base 10) is rational, not irrational?
The fact that this is incorrect is the whole point - there is no fraction in any base that exactly denotes pi.
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Post by speakertoanimals on Jul 6, 2011 18:14:24 GMT 1
Stop pretending to be so stupid -- 22/7 is an APPROXIMATION to pi, as any fule knows...................................
Because pi would have an infinite, non-recurring expansion using ANY base, not just base 10..........................
See what I mean chaps! Either abacus is REALLY this stupid and confident, or (which is the option I prefer), he's just a wind-up merchant, who tries to derail any thread that he participates in. So, rather than actually discussing the maths of rational and irrational numbers, I have to descend to explaining that pi ISN'T 22/7. Okay, some american law-makers in Indiana apparently tried to pass legislation making pi rational, but that failed.
Anyway, let's look at a more interesting question. We know that pi is irrational, but does ANY possible finite sequence of digits occur somewhere in the decimal expansion of pi? So, say, is there a sequence of a billion fives somewhere in there? Seems most mathematicians believe that the answer is yes, but as far as I know, no one has managed to prove it one way or the other.
(In technical terms, this is related to pi being a normal number, but this hasn't been proven either!).
So, we swing between the two extremes -- rational numbers where the sequence of digits repeats ad infinitum at some point, and irrationals where it never repeats, then normal numbers where ANY sequence occurs at some point!
So, if pi is normal, at some point in the decimal expansion, the date of birth, date of death, age at death etc of every one of your ancestors for the past 100 generations occurs IN ORDER at some point. Or any other finite sequence of digits you care to define. Such as a list, in order, of the phone numbers of everyone you have ever slept with..................
Freaky isn't it! :-)
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Post by abacus9900 on Jul 7, 2011 9:32:41 GMT 1
Fractions and bases are not the same thing. This is why you cannot obtain an exact equivalence between the two. If you were able to obtain a perfect measuring instrument (which is impossible) then using a fraction would always produce an exact mathematical description of whatever you sought to measure. It is only when you seek to convert a fraction into a base that things become 'irrational'.
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Post by abacus9900 on Jul 7, 2011 9:39:51 GMT 1
Please see my latest response.
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Post by speakertoanimals on Jul 7, 2011 12:01:31 GMT 1
Wrong. There are numbers which cannot be expressed as fractions (using integers), which is the definition of an irrational number. The point about a decimal expansion is a consequence of that basic definition.
Simple example that even the greeks knew about (but some of them weren't that happy about it!). Suppose we have a square with unit sides, then we wish to measure the length of the diagonal. The length is square root two.
(The RATIO of the length of the diagonal to the length of the side, d/s if you like, defines root 2, but since d and s AREN'T both integers, possibly this is where yet another misunderstanding came in.....).
Which cannot be expressed as a fraction of the form a/b, although saying it is square root of two produces a complete mathematical description of the number you mean.
So, suppose you COULD write root 2 as a/b, where a and b are integers with no common factor.
Then you have that 2b^2 = a^2
Which means a^2 must be divisible by two (is an even number). The only way a^2 can be even is if a is even as well.
If a is even, it means that a^2 contains 2 as a factor at least twice. Hence b^2 must be even as well. The only way b^2 can be even is if b is also even. But then a and b have a common factor, whereas we said at the start that we had CANCELLED common factors out of a and b.
SO, it all falls apart, and we know then that our original assumption, that root 2 COULD be written as a/b must be false. Hence root 2 is NOT a rational number. QED and all that.
And bugger all to do with decimal expansions........................
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Post by abacus9900 on Jul 7, 2011 13:39:59 GMT 1
Wrong. There are numbers which cannot be expressed as fractions (using integers), which is the definition of an irrational number. The point about a decimal expansion is a consequence of that basic definition. Simple example that even the greeks knew about (but some of them weren't that happy about it!). Suppose we have a square with unit sides, then we wish to measure the length of the diagonal. The length is square root two. (The RATIO of the length of the diagonal to the length of the side, d/s if you like, defines root 2, but since d and s AREN'T both integers, possibly this is where yet another misunderstanding came in.....). Which cannot be expressed as a fraction of the form a/b, although saying it is square root of two produces a complete mathematical description of the number you mean. So, suppose you COULD write root 2 as a/b, where a and b are integers with no common factor. Then you have that 2b^2 = a^2 Which means a^2 must be divisible by two (is an even number). The only way a^2 can be even is if a is even as well. If a is even, it means that a^2 contains 2 as a factor at least twice. Hence b^2 must be even as well. The only way b^2 can be even is if b is also even. But then a and b have a common factor, whereas we said at the start that we had CANCELLED common factors out of a and b. SO, it all falls apart, and we know then that our original assumption, that root 2 COULD be written as a/b must be false. Hence root 2 is NOT a rational number. QED and all that. And bugger all to do with decimal expansions........................ I am unable to make a sensible response to this because it is so obtuse. What is the point you are trying to make? Whatever it is you are trying to say does not alter the fact 22/7 does not have an an exact equivalence in terms of decimals and, according to you and others, any other base, however, I will have to take your word for that since I am not a mathematician. The original question was: "Why doesn't Pi 'come out' exactly?" Obviously, if one stated 22/7 = 22/7 it would be true since you are working within the same number system (ordinary fractions) but once you step out of that system it will no longer always be true. This is the reason why certain fractions don't always come out exactly. Nothing to to with rational/irrational numbers, which refer to a given base, not an ordinary fraction of the form a/b.
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Post by louise on Jul 7, 2011 14:05:54 GMT 1
Abacus A quick look on Wikipedia will explain that Pi does not equal 22/7 e.g. from en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80Pi cannot be described as a ratio between two integers and so is classed as an irrational number. I think this is fully self explanatory without getting into the question of bases, etc.
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Post by speakertoanimals on Jul 7, 2011 14:43:48 GMT 1
Abacus, it's a (sketch) of a proof that root 2 is irrational. Because would be pointless talking about irrational numbers if I couldn't give you a simple example of a number that IS irrational. root 2 is one such, as is pi.
22/7 DOES have an exact equivalence in terms of decimals! Let's take a slightly easier case. 1/3 as a decimal is 0.3 recurring. That simple recurring tag then tells you ALL the digits of the decimal expansion. Okay, it doesn't terminate, but that isn't the point. In base 3, 1/3 would just be 0.1, which does terminate. Whether it terminates or recurs depends on your choice of basis.
Another example would be 0.12756 recurring -- you KNOW the value of any possible digit in the decimal expansion given that definition, so could reply simply if someone asked you to give the ninmillionth, or nine billionth digit of the number.
Whereas with pi, you couldn't give the value of a specific digit without actually doing the calculation. Knowing the digits BEFORE that wouldn't tell you the next one. Whereas with 0.12756 recurring, if someone told you that a digit somewhere was 7, you'd then KNOW that the next digit was a five, wherever that 7 occurred.
Except you're taking 'come out exactly' to mean something quite boring (like having a finite expansion), rather than the interesting case (for pi at least), which isn't that expansions terminate, but whether they are simply repeating or not. numbers that cab be written as a/b always repeat at some point (which includes the case where they terminate, we're just take that as 0 repeating for ever after the termination point), whereas numbers that can't beb written as a/b never repeat, and never terminate, whatever basis you use.
Abacus, I wish (as usual), that you'd go and look up the basics of what is being talked about, and the basic meanings of terms, before you're so keen to tell people they're talking nonsense! Or is it just your usual wind-em-up routine.........................
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Post by abacus9900 on Jul 7, 2011 17:35:11 GMT 1
Abacus Pi cannot be described as a ratio between two integers and so is classed as an irrational number. I think this is fully self explanatory without getting into the question of bases, etc. Louise, Pi is based on dividing something up into 7ths. (22/7), so while it is possible to use equivalent fractions by using multiples that are common to both denominator and divisor it's all really the same thing, i.e., expressions of 7ths. To make a conversion into another base (say 10) means you are no longer using 7ths, but 10ths. Well, 7ths. and 10ths. are nothing like one another and have nothing at all in common so all you can do is use as close as approximation as possible, even if this becomes vanishingly close to the the original. It's like trying to square a circle - can't ultimately be done. When people say Pi is an approximation what they really mean is that the decimal representation of Pi is an approximation, so this is a very misleading statement. Why people have to insist on measuring everything using the decimal system is an indictment of our historic, well tried and trusted imperial system of measurement. Presumably, it is pandering to the EEC!
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Post by abacus9900 on Jul 7, 2011 17:40:57 GMT 1
STA, to assert that trailing 0's are the same as recurring fractional integers is absurd and even comical! Either something comes out exactly or it does not and if it does not why should it matter whether you call it rational or irrational? It's all just mathematicians' jargon designed to create categories which do not in fact exist or matter.
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Post by speakertoanimals on Jul 7, 2011 18:40:25 GMT 1
Abacus, NO ONE can be that daft (except perhaps us for continuing to respond to you!).
22/7 is merely a USEFUL approximation to pi, a bit better than saying -- about 3, maybe a smidge more.
0 is a digit as good as any other, and 0 recurring is as valid as saying 1 recurring.
If you REALLY believe that rational/irrational doesn't matter, then you're as stupid as you're pretending to be!
God, you've only been back on here a few days, and already you have nearly surpassed your former depths of pretended stupidity!
Why don;t you just fuck-off again abacus, we managed to have some sensible discussions without you messing about all over these boards.................
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Post by abacus9900 on Jul 7, 2011 21:58:08 GMT 1
The problem here is that you are assuming 22/7 is any less accurate than umpteen decimal places thus far computed for Pi. From a purely mathematical perspective the latest value for Pi is as far from it as 22/7 so that we see transcendental numbers may never be attained, going on for infinity. This is why the value of 22/7 is as valid as any other value used for Pi. Pi is an ideal, an impossible goal that simply compels us to seek the unfathomable depths of the concept it represents. If you recall, I did try to point out that mathematical labels are just place holders for current ideas about numbers so that it is not the labels we should be really concerned about but the Godelian principle of incompleteness, which you seemed to have failed grasp.
Just to makes things a bit easier for you understand allow me to ask you this question:
Which is nearer to infinity: the number two or the number 1000000^1000000^1000000?
The answer, of course, is neither since it is a completely meaningless comparison as infinity cannot be measured, mathematically.
0 recurring has no other mathematical usefulness other than acting as a termination.
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Post by speakertoanimals on Jul 8, 2011 13:33:38 GMT 1
Now we see the final mad claims that Abacus would have us believe he is stupid enough to actually believe!
Abacus, in case you really ARE that stupid, take this case.
My first approximation to some number is 33.
What does that tell me? If I can agree my computations are correct, it says that the REAL number must lie somewhere between 33.00000......
and 33.9999999999
My next approximation to the same number is 33.5
This now tells me that the REAL number lies somewhere between 33.5000000000 and 33.59999999999999
First case, I've only narrowed it down to within some range of width 1, in the second case I've pinned it down to within some range of width 0.1
Since 0.1 is smaller than 1, the second approximation is closer.
pi is 3.14159................... The given digits are CORRECT
22/7 is 3.142857........... Hence 22/7 only agrees with pi to first 2 decimal places.
And you'd claim that that is AS GOOD as an approximation as a decimal expansion that agrees to a million (or a billion) decimal places.
Your counting argument is false, becase you have ignored the fact about decimals that each digit represents a distance that is 1/10 smaller than the same digit in a higher position.
Hence the line about which is closer to infinity, 1 or 6 million is utter bollocks.
The point being pi ISN'T just a random string of digits with no meaning, but a finite NUMBER. Hence the better comparison is that 1.25 is closer to 1.251 than 1.2 is.
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Post by abacus9900 on Jul 8, 2011 15:18:16 GMT 1
How is pi finite? Stop pulling my leg.
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Post by speakertoanimals on Jul 8, 2011 15:56:05 GMT 1
Pi lies somewhere between 3 and 4, hence it is FINITE.
quite clear that you're now pretending to not understand what finite and infinite mean...................
As regards infinity, we CAN say how far 1 and 1 million are from infinity We just need to know how to do arithmetic with infinite numbers -- of which there are MORE than one, and some infinities are demonstrably larger than others!
So, the number of integers 1,2,3,.......5 678 792,........... is aleph-0.
The number of even numbers, numbers of the form 2n, is also aleph-0
The number of odd numbers is also aleph-0.
Hence since every integer is either odd or even, we have that aleph-0 plus aleph=0 is equal to aleph-0.
similalrly, aleph-0 minus any finite number (like the distance between infinity and one million) is also aleph-0.
Hence 1 and 1 million are equally close to aleph-0.
Rational numbers are interesting, because you can show that numbers of the form p/q, there are also aleph-0 of those.
the continuum (which includes our new friends the irrational numbers as well), is a LARGER infinity than aleph-0.
thankyou Cantor for showing us that.
What you are messing about with now is trying to claim mistakenly that the fact that pi has an infnite, non-repeatuing decimal expansion means that pi is somehow infinite, or somehow 'hard to pin down', which is nonsense. We can specific exactly and precisely what the number pi IS, we could, if we had enough time and computational power, compute any specificed digit in the decimal expansion of pi.
Abacus, only a few days ago, you made some comment about I'd better be nicer to you this time -- except within just a few days, you've started coming out with worse nonsense than you did the first time round. What happened, did some other message board ban you that you had to come back here to try and get some sad little frisson of enjoyment from winding us up again?
Time to press the ignore button again I think, you lasted even less time than you did before..............................
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