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Post by speakertoanimals on May 24, 2011 15:12:53 GMT 1
I could get the number '1' by an infinite number of arithmetic processes, that isn't the point. The problem is, YOU are thinking of 1/3 as a PROCESS, a ratio via division. But it's just a number, and the SAME number as 0.3333333....... Don't let the infinite number of decimal digits fool you, it doesn't matter if you are trying to do it on paper, and having to do an infinite number of arithmetical processes to generate the decimal expansion. Actually, I could be smart, and SPOT as I did it that it was going to recur. Arithmetic isn't the point. The identity of the number IS. And 1/3 is the SAME number as 0.33333 recurring (which as I explained could be written as 0.1 in base 3, the recurring just happens because you have chosen an unfortunate base of 10 to write it down). but however you write it, it is the SAME number. Its the number as an abstract entity, not the particular base or alphabet you choose to write it down, just as 1/3 is the same as 'a third', 'or a thing equally divided into three parts and take one of those', or however else you want to communicate precisely the number that is a third. Different languages if you like, SAME number. To really warp your brain, lets consider the numbers 1, and 0.9 recurring. They are the SAME........................... lets say x = 0.999999..... then 10x = 9.999999......, just shift decimal point Now subtract. this gives us that 9x = 9, hence x = 0.999999.... = 1 Not arithmetic, maths! (hint, I used x hence MUST be algebra hence maths  ). SO, given that I have proven that 0.99999999.... is the same as 1, then you have to also accept that 0.3333333.... is 1/3, since I just divide both sides by 3. Job done! Damn, maths is fun, even the simple stuff! Makes STA a happy little bunny. P.S And of course, these infinite decimal expansions are actually infinite sums, and think of Achilles and the tortoise and all the fun you can have with infinite sums giving a finite answer. that one flummoxed the ole philsophers too! And then we have infinite decimals and the Cantor diagonal slash to prove that there are more reals than integers, hence different sizes of infinities! All that hidden in the simple 0.3333..... recurring expression. |
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Post by eamonnshute on May 24, 2011 16:27:17 GMT 1
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Post by speakertoanimals on May 24, 2011 16:33:46 GMT 1
Hmm, I think arctans don't count But the purely integer stuff, very pretty! What's even freakier are Ramanujan's infinite series for pi: planetmath.org/encyclopedia/RamanujansFormulaForPi.htmlNot so much the series themselves, as to HOW you get to them, or how Ramanujan did it......................... |
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Post by eamonnshute on May 25, 2011 11:55:01 GMT 1
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Post by StuartG on May 25, 2011 22:39:41 GMT 1
OK. I accept that my knuckles are rapped for PI. Another number that's always intrigued me is 127. For me 127 represents the number of teeth that has to be on one of the lathe changewheels to allow cutting of english threads with a metric leadscrew [or vicky verky]. That's one for the Wiki page.
Also the number of millimetres in 5 inches. So if constructing something with a ratio 3:4:5 it might well be easier to use inches rather mm's. Why? because it's easier to 'mark out' in inches [in this case] and less prone to making mistakes [bigger units]. The sums are easier, as smaller numbers used. Try it for yourselves, set a pair of dividers to 127 mm and then try 5 ins.
ASCII code 0-127.
Binary 1111111
Any more that appear in life?
StuartG
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Post by speakertoanimals on May 27, 2011 15:15:28 GMT 1
That is because they DEFINED the international inch to be 25.4mm. Hence multiply by 5 to get rid of the fraction, and there you go.
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Post by StuartG on May 27, 2011 17:20:13 GMT 1
The original conversion was 25.399 978 mm (in 1947) and was 'rounded up' in 1960 or thereabouts. Most modern operating systems for computers have an 'hosts' file. It is the first reference for the Domain Name Server to reconcile an address. eg. 'stuartg.org' if it finds that address it uses the ip address found associated with it, that is the address goes nowhere, because the first entry states 127.0.0.1 localhost The file is called 'hosts' whoever first used this title must have had some religious knowledge as '127' refers to... "There are many harmonics (multiples) of the prime Number 127 that relate to the Glory of God:" "Who is this King of glory? The LORD of hosts, he is the King of glory. Psalm 24.10 "The fundamental significance of the prime Number 127 is found in these words from Psalm 24:"..... ..hence the name and number. StuartG www.biblewheel.com/gr/gr_127.aspwww.biblewheel.com/gr/GR_TenC_C2.aspadded refs. |
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Post by principled on May 27, 2011 17:51:00 GMT 1
Which is exactly why 29 =30...  Three men go into a restaurant and have a meal. The bill comes and it is £30. They each put in £10. Minutes later the waiter comes back and says he made a mistake and the price is only £25 and gives them back £5. Impressed by his honesty, they each take back £1 and leave the remaining £2 as a tip. Where is the missing £1?  P With apologies to those who know this old conundrum! |
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Post by rjpageuk on Jun 30, 2011 13:28:57 GMT 1
One of the coolest properties of the set of irrational numbers is that despite the seeming rarety of them there are an uncountably large amount, such that there are "more" of them than standard rational numbers.
Between any two rational numbers (i.e numbers that can be expression as a fraction), no matter how close they are, there are an infinite number of irrational numbers.
Most of the issues humans have with irrational numbers is that they are hard to visualise because of their reliance on infinity. Assuming there is a finite minimum particle size there is no such thing as a perfect circle in the real world, so you can never really see pi. When you get a 1x1 piece of paper and cut down the diagonal it isnt really root2 in length because your piece of paper was not exactly 1x1 in the first place and you cant cut a perfect diagonal.
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Post by speakertoanimals on Jun 30, 2011 13:42:37 GMT 1
Yeah, the fact that there are an infinitude (and a bigger infinity that the number of intergers), of irrational numbers, squeezed into any section of the real line you care to take, is mind-boggling.
IF you can get your head round it, its when the cozy infinity of the integers gets blasted out of the water by LARGER infinities. The continuum is a strange place. But it means you have to go beyond the practicalities of measurement, in that any physical measurement of position is laways going to be to the nearest mm/inch/ etc, hence will always be a rational number. So, you can't put your finger on an irrational, but they're still there!
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Post by rjpageuk on Jun 30, 2011 13:52:28 GMT 1
They arent actually there in any real sense though are they?
If you ignore the practical issues with measurement where would you find an irrational number in the real world?
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Post by speakertoanimals on Jun 30, 2011 17:32:38 GMT 1
But if space (and time) are really continuous, then they ARE there, because that is what the continuum IS.
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Post by StuartG on Jun 30, 2011 19:18:05 GMT 1
When this question was originally posed it wasn't my intention to overturn a few thousand years of maths, but to try to see if my thoughts were valid. So here's what was said by me earlier at #7 "Really I was hoping that someone might say that it points to say, a physical reality, such that, when a circumference of some matter is examined at that miniscule level the matter would not be that precise in its boundary. I'll just have to settle for an 'irrational' number. " radio4scienceboards.proboards.com/index.cgi?action=gotopost&board=debate&thread=915&post=11295and took it that I was 'just not getting it', but then at #23, rjpageuk says... "Assuming there is a finite minimum particle size there is no such thing as a perfect circle in the real world" radio4scienceboards.proboards.com/index.cgi?action=gotopost&board=debate&thread=915&post=12551So perhaps maths is not just 'theory' but in fact Mathsdoes describe the 'real world', at least in this case, because the recurring digits indicate an 'indefinition' when working at that level? Does that fit all the concepts? Am I seeing it as it should be realised? Cheers, StuartG |
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Post by speakertoanimals on Jun 30, 2011 20:03:12 GMT 1
Perhaps the way we are looking at it here is too limited, in that we are just looking at numbers in terms of GEOMETRY (simple circles, or lines) and location in space or time. Hence we have the practical problem(s) that:
The resolution of actual physical measurements is always limited
Real objects are actually fractal in some sense (even the smoothest object has bumps on some level, depending how close you zoom in), until you reach a scale where the notion of THE surface becomes indistinct.
At the smallest scales, space and time themselves could become discrete, when quantum gravity comes into play.
If we think in terms of physical variables other than length, we tend to use the continuum to measure temperature, or pressure, or anything else you could name. Except the same issues come in when we think of physical measurements.
You could be given to think that things like energy are discrete as well, in that look at an atom, what do we all know, that it has discrete energy levels. Except we also have a continuum of non-bound states for the atom as well.
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.
After all, classical physics was based on the continuum and the maths of differential equations. And physicists weren't at all happy with early quantum theory formulated in terms of matrices (most had never comes across them way back then!), but a LOT happier when Schrodinger gave them a nice differential equation to describe the same physics, even if the thing that appeared in it was the slightly weird wavefunction.
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Post by principled on Jun 30, 2011 20:57:02 GMT 1
STA
That's pretty deep, even for you!
I have trouble with time. On an everyday level I see it as continuous, but as I break down the time period in my mind it changes to discrete "chunks" on the lines of De Morgan's verse...
But, then again, time as a fractal quantity certainly has appeal.
P
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