About pi and repeating fractions

Discussion in 'Physics & Math' started by Cyperium, Mar 20, 2009.

  1. Cyperium I'm always me Valued Senior Member

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    3,058
    I took pi with 49980 decimals and added them together, then I divided the sum with 49980 to get the average, in order to get an average of 1 to 10 (5) I took each decimal + 1, so that the lowest would be 1 and the highest would be 10, I was only seeing if this was purely random (or rather evenly distributed) as this would produce a 5, and it did.

    This is the number it produced:

    5,5047619047619047619047619047619

    I noticed that after 5,5 the numbers was repeated over and over again.

    047619

    Why? Is there any theory why fractions behave this way?
     
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  3. Absane Rocket Surgeon Valued Senior Member

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    Consider 0.25.

    2+1 = 3
    5+1 = 6

    (3+6)/2 = 9/2 = 4.5.

    Not very interesting...
     
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  5. James R Just this guy, you know? Staff Member

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    All fractions (rational numbers) must repeat at some point.

    For example 3/4 = 0.75000000... (with the zero repeating indefinitely).
    1/7 = 0.142857 142857 142857 ....

    Non-repeating decimals that can't be written as fractions are called "irrational numbers". The most famous example is \(\pi\), but other examples are very easy to produce. For example, 0.101001000100001000001....
     
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  7. Cyperium I'm always me Valued Senior Member

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    I understand that zero repeats indefinitely since there is no end to the accuracy of 3/4, so we just omit the zeros, I also have no problem understanding 1.999999999999... since they show that it's very very very close to 2, in fact so close that it actually is 2 (as it goes to infinity).

    But why they repeat in some random manner baffles me, like 2.412342412342412342, so what is so special about 412342? Why is it repeated, what kind of accuracy does it show, and why?

    I hope you understand my question, I've thought about it alot.


    If we take a number; let's say 2.590839483
    My analysis is this:
    Very close to 2.6
    Though closer to 2.58
    Very close to 2.591
    Though not so close to 2.5909
    Though very close to 2.59084
    etc.

    So why would they repeat?

    My understanding is that each number shows how close that the former increases.
     
    Last edited: Mar 24, 2009
  8. Absane Rocket Surgeon Valued Senior Member

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    8,989
    Whenever the denominator does not divide any power of 10, the fraction will repeat.
     
  9. Cyperium I'm always me Valued Senior Member

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    But why does it repeat a specific sequence over and over again?
     
  10. Absane Rocket Surgeon Valued Senior Member

    Messages:
    8,989
    I can't give you a *why*.

    However, there is only one other alternative: the sequence does not repeat. Now, there are two ways this can happen: the sequence terminates or it goes on forever but not repeating.

    If it were the former case, we have a terminating decimal and that is not what concerns us. So if it goes on forever but never repeating, it must be an irrational number. But, irrational numbers cannot be represented by a fraction by definition.

    I'm sorry my explanations are crappy... I'm feeling "out of it" today :-/
     
  11. CheskiChips Banned Banned

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    3,538
    The average of the summed decimals should equal 5. 5 is the convergent point being both the median and the mean of a random sequence of numbers between and including 1 and 10. In general that's the normal distribution for an irrational, unless the irrational converges on another number.

    Repetitive decimal sequences will not converge necessarily to 5, sometimes they converge perfectly to a single number. Usually repetitious behavior signals a representative fraction. A truly non-repeating fraction would be void decimals in another base system, so it's a number system issue.

    For example take e (2.718281828) sum up the decimal places to equal 45 and divide by the count of 9 and you get 5.
     
  12. Absane Rocket Surgeon Valued Senior Member

    Messages:
    8,989
    But, as you notice... 1/7 = 0.142857 142857 142857 ....

    If we changed our base to 7, then 1/7 = 0.1

    It terminates.
     
  13. CheskiChips Banned Banned

    Messages:
    3,538
    Wrong scripting.

    \((\frac{1}{7})_{10} = (\frac{1}{10})_7 \)
     
  14. Absane Rocket Surgeon Valued Senior Member

    Messages:
    8,989
    ???

    Decimal representation.

    I must have used crappy notation.. again, I am feeling out of it today...

    0.142857 142857 142857 .... base 10 = 0.1 base 7
     
  15. CheskiChips Banned Banned

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    I re-edited my post to clarify my claim right before your response. It was a notation issue.
     
  16. Absane Rocket Surgeon Valued Senior Member

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    8,989
    Oh ok.. Sorry. Yea, I went for a long hike yesterday and I haven't had any carbohydrates since then... so my body is in ketosis... meaning I can't think.
     
  17. iceaura Valued Senior Member

    Messages:
    30,994
    Intuitive approach, ignoring the decimal point for a sec: The setup is that you are doing long division - check?

    If you divide a number into another, the biggest remainder you can get is one less than the divisor - check?

    There are only a finite number of possible remainders, then - counting down from the divisor, which is finite.

    As soon as you repeat a remainder, you are going to repeat the whole sequence that followed that remainder the first time - because you are starting with the same number.

    Does that help?
     
  18. Cyperium I'm always me Valued Senior Member

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    3,058
    Yes, it does. Thank you, your answer was what I was looking for.


    Thanks to everyone else too of course, alot of interesting information.
     
  19. Cyperium I'm always me Valued Senior Member

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    3,058
    This explains it!

    Found this article on wikipedia:

    http://en.wikipedia.org/wiki/Fraction_(mathematics)


    Converting repeating decimals to fractions
    Decimal numbers, while arguably more useful to work with when performing calculations, lack the same kind of precision that regular fractions (as they are explained in this article) have. Sometimes an infinite number of decimals is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions.

    For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. For example (the pattern is highlighted in bold):

    0.555555555555… = 5/9
    0.626262626262… = 62/99
    0.264264264264… = 264/999
    0.629162916291… = 6291/9999
    In case zeros precede the pattern, the nines are suffixed by the same number of zeros:

    0.0555… = 5/90
    0.000392392392… = 392/999000
    0.00121212… = 12/9900
    In case a non-repeating set of decimals precede the pattern (such as 0.1523987987987…), we must equate it as the sum of the non-repeating and repeating parts:

    0.1523 + 0.0000987987987…
    Then, convert both of these to fractions. Since the first part is not repeating, it is not converted according to the pattern given above:

    1523/10000 + 987/9990000
    We add these fractions by expressing both with a common divisor...

    1521477/9990000 + 987/9990000
    And add them.

    1522464/9990000
    Finally, we simplify it:

    31718/208125
     

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