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An accurate representation of Irrational and rational numbers

  1. Jun 5, 2004 #1
    Is there an accurate way to write the value of an Irrational number?

    If there is no an accurate way to write the value of an Irrational number, then can we conclude that no irrational number has an exact place on the real line?

    And if there is an exact place to an irrational number on the real line, then how can we prove it?

    What base do we need to write the value of 2/3 in finitely many digits?
     
    Last edited: Jun 5, 2004
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  3. Jun 5, 2004 #2

    Integral

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    The current system provides for any precision you need.
    No, the point is fixed, there can be no doubt about that, it is the representation which requires an infinite number of digits, this is a result of the real line being dense.
    This can be done by defining a set of nested intervals each of which contain the number of interest.
    Base 3 works great

    [tex] \frac 2 3 = .2_3[/tex]
     
  4. Jun 5, 2004 #3

    Hurkyl

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    Is there something wrong with the representations 1/3 or √2?


    It's more or less a tautology; real numbers are the "places" on the real line.
     
  5. Jun 5, 2004 #4
    There are different ways to get numerical values for various irrational numbers to any arbitrary degree of precision. For example, pi can be found from the following infinite series:

    [tex]1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\dots=\frac{\pi}{4}[/tex]
     
    Last edited: Jun 6, 2004
  6. Jun 5, 2004 #5
    Dialog asks about something called "the value of an Irrational number". Is that supposed to be something different from just "an irrational number" itself (not furthur specified in particular)?
     
  7. Jun 6, 2004 #6
    Why there is no doubt about that?
    1/3 and √2 are arithmetical operations, and I am talking about an accurate representation of the right hand of the equations: 1/3 = ? or √2 = ?

    What is an irrational number itself, and how you define it if you have no way to represent it in an accurate way, which is not some arithmetical operation?

    Also I have another question:

    How can we be sure that different represetations of some irrational or rational number do not define new information which has an interesting Mathematical data, which is not cardinality or ordinality information?

    For example: The different information that we get when 1/3 is represented by base 3 or by base 10.
     
    Last edited: Jun 6, 2004
  8. Jun 6, 2004 #7

    Integral

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    Because this is the way the Real line in constructed. Read my post above, I told you how irrational numbers are "trapped".
    Why are you so infatuated with the representation of a point? Each point on the line can be represented in many different ways. The basic construction of the real line does not use digital representations. They have their uses, but are not required or needed to define real numbers.
     
  9. Jun 6, 2004 #8
    Can you explain in a non-formal way how you define pi for example?
    Please prove that pi is no more than a "point" on the real-line.

    An example:

    If circle's diameter = 1 then circle's perimeter = pi.

    Can we prove that there is an exact way to put pi on the real-line?

    Am I right if I say that we cannot represent pi because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any other number)?

    And if so then what kind of a number we get by the ratio between two irrational numbers like e/pi?
     
    Last edited: Jun 6, 2004
  10. Jun 6, 2004 #9

    Zurtex

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    How do you mean? Do you mean a polynomial expression with just integers, e.g:

    For x = 1/3:

    [tex]3x -1 = 0[/tex]

    For x = √2

    [tex]x^2 - 2 = 0[/tex]
     
  11. Jun 6, 2004 #10
    Hi Zurtex,

    I mean can you represent an Irrational number in an accurate way by not using arithmetical operations at all?
    Can you prove it?
     
    Last edited: Jun 6, 2004
  12. Jun 6, 2004 #11

    matt grime

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    Dialog, how do you propose to represent any number in a way "without using any arithmetic operations"? after all what does 35 mean but 3*10+5*1?

    This is mathematics, not aesthetics. Any representation of a real number is just a shorthand for stating *what the number does* which is its definition: sqrt(2) is the unique positive real number that squares to give 2, which is itself the result of adding 1 to 1, where 1 is the multiplicative identity of the unique totally ordered complete field of the real numbers. Got it? Real numbers are not decimals, that we can only write real numbers as finite length decimal expansions in certain cases is just a fact of the system of representation. It is not a deficiecny with the real numbers.

    You need to explain what you consider a 'reasonable' representation of a number, and then state which numbers satisfy the criteria. If you wish you may then make *subjective* statements about them, but the ones you've made so far just demonstrate your lack of mathematical sophistication.

    Do you even know what the real numbers are? I mean as a mathematical object?
     
  13. Jun 6, 2004 #12
    Hi Matt,

    I am talking about Irrational numbers (cannot be expressed by the ratio of two integers).

    If circle's diameter = 1 then circle's perimeter = pi.

    Am I right if I say that we cannot represent pi (by a ratio of two integers) because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any other number)?

    And if so then what kind of a number we get by the ratio between two irrational numbers like e/pi?
     
    Last edited: Jun 6, 2004
  14. Jun 6, 2004 #13

    matt grime

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    Why does it matter what kind of numbers you're talking about when you've not justified what you mean by having a 'proper' kind of representation? So, what's a proper representation? You've already discounted 1/3 as improper, and sqrt(2).

    THe stuff about pi is just plain wrong in any interpretation, not least because of the undefined subjective terms, and the observation that surely pi and 2pi must have a common subpart, whatever that might mean.
     
  15. Jun 6, 2004 #14

    Zurtex

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    Well as stated before you can represent irrational number by infinite sum etc..

    But the whole point of irrational numbers is you can not represent them with integers and your 4 basic mathematical operators.
     
  16. Jun 6, 2004 #15
    Can you represent pi in an accurate (detailed) way by using finitely many symbols?
     
  17. Jun 6, 2004 #16

    matt grime

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    yes, pi. now define what you mean by accurate, detail and symbol...
     
  18. Jun 6, 2004 #17
    My mistake, I mean:

    Am I right if I say that we cannot represent pi (by a ratio of two integers) because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any rational number)?
     
    Last edited: Jun 6, 2004
  19. Jun 6, 2004 #18

    matt grime

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    No, you are not because you have not stated what you mean by divisbility for numbes not in the ring of integers - there must be some ring in which you are asking for the divisibility to be examined, what is it? 2/5 is represented as the ratio of two integers, yet it is like a prime surely since it cannot be divided by any integer.

    We cannot represent pi as a ratio of two integers because it is irrational, that is the exact reason. It is moreover transcendental (and thus irrational), but so what? The real numbers aren't actually a line in any physical sense, this need to think of them as points, as dots on some infinite line, which you can't draw anyway, with some artificial and unjustified and unexplained meaning of 'accuracy' is slightly pointless if you don't even know what the real numbers are, and how mathematics works.
     
  20. Jun 6, 2004 #19
    And another qustion: Pi (when diameter = 1) is the value of a circle's perimeter .

    Can you prove that any value which is related to a circle also holds for a straight line?
     
    Last edited: Jun 6, 2004
  21. Jun 6, 2004 #20

    matt grime

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    Why, when asked to explain yourself more clearly, do you always introduce more unexplained subjective ideas? That question just does not make any sense.
     
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