Rational representation of arbitrary fractions.

Discussion in 'General Math' started by qWake, May 24, 2004.

  1. qWake

    qWake Guest

    Given an arbitrary fraction, is there a quick way to determine the most
    accurate denominator in a given range (say from 2 to 9999999) to represent
    the fraction in "numerator/denominator" format?

    By "quick" I mean faster than testing all possible denominators in that
    range for the smallest error in representation...
     
    qWake, May 24, 2004
    #1
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  2. Google for "FAREY sequence". A FAREY sequence is a sequence of optimal
    approximations to a given rational number with increasing
    denominators. Chose the element with highest denominator <=9999999. It
    will be the best approximation for all denominators up to 9999999.
     
    Horst Kraemer, May 25, 2004
    #2
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  3. qWake

    Don McDonald Guest


    it is called the euclidean algorithm
    continued fraction convergents.
    And you don't need anything else.

    as follows...
    elementary/ primary school math.

    don.lotto.
    1.6.04..

    From: (Don McDonald)
    Newsgroups: alt.math
    Subject: Re: Changing these fractions into decimals
    Date: 3 Feb 2004 18:24:40 -0800
    Organization: http://groups.google.com
    Lines: 259
    Message-ID: <>
    References: <>
    <>
    <401e45d1$0$18303$>
    ....

    From: Priscilla Periut ( )
    :Date: 2004-02-01 13:18:29 PST

    :Can you please help me change these fractions into decimals? The
    :fractions are 71/177 69/190 69/170 51/158 and 28/92. thanks a lot
    :for your effort and i am only in the 7th grade.

    03.02.04 22:06
    Thanks Priscilla.

    Well, I guess that is not the proper question.
    Not the question that may be in the book.

    A mathematician (well me) wouldn't put it that way.
    In my humble opnion.

    2x fractions are approx 2/5ths =.40
    which is greater, 71/177 or 69/170 ?? -- a very good problem.

    2 x fractions are approx .36. the problem may have been
    to arrange the fractions in increasing order.

    the problem could be the
    numerator is per cent mark in exam A and the
    denominator is sum of percent in exam papers A AND B.

    begin, enter start number (expression), q. end

    CALCULATE B / A =158 / 58 *****************

    = 2.72413793 recipr = 0.36708861
    CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
    fraction

    2 1/2= 0.5
    1 1/3= 0.33333333333333331
    2 3/8= 0.375
    1 4/11= 0.36363636363636365
    1 7/19= 0.36842105263157893
    1 11/30= 0.36666666666666664
    2 ** 29/79=*** 0.36708860759493672 ** REDUCES EXACT.
    (2) 58
    BEGIN, ENTER START NUMBER (Expression), Q. END

    CALCULATE B / A =92 / 28 ** *************
    = 3.28571429 recipr = 0.30434783
    CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
    fraction

    3 1/3= 0.33333333333333331
    3 3/10= 0.29999999999999999
    2 ** 7/23= 0.30434782608695654 ** reduces exact
    (4) 28
    BEGIN, ENTER START NUMBER (Expression), Q. END

    ?71
    (5) 71 ƒCTD FRACTION,
    CALCULATE B / A =177 / 71 ******************
    = 2.49295775 recipr = 0.40112994
    CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
    fraction
    2 1/2= 0.5
    2 2/5=********** 0.40000000000000002
    PREV. LINE IS(very) Economical APPROXN.
    ƒ

    35 71/177= 0.40112994350282488
    (6) 71 -?2


    (7) 69 ƒCTD FRACTION,
    RATIO TO DENOM 'A' (default 1)
    (ENTER NO. OR EXPRESSION) ?190
    190 = 190

    ( NEEDED TO SWAP SO THAT 0< A <= B )
    CALCULATE B / A =190 / 69 ****************
    = 2.75362319 recipr = 0.36315789
    CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
    fraction

    2 1/2= 0.5
    1 1/3= 0.33333333333333331
    3 *** 4/11***= 0.36363636363636365
    PREV. LINE IS(very) Economical APPROXN.
    ƒ

    17 69/190= 0.36315789473684212
    (8) 69 ƒCTD FRACTION,
    RATIO TO DENOM 'A' (default 1)
    (ENTER NO. OR EXPRESSION) ?170
    170 = 170

    ( NEEDED TO SWAP SO THAT 0< A <= B )
    CALCULATE B / A =170 / 69 ***
    = 2.46376812 recipr = 0.40588235
    CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
    fraction

    2 1/2= 0.5
    2 2/5=*** 0.40000000000000002
    PREV. LINE IS(very) Economical APPROXN.
    ƒ

    6 13/32= 0.40625
    2 28/69= 0.40579710144927539
    2 69/170= 0.40588235294117647

    (11) 69
    BEGIN, ENTER START NUMBER (Expression), Q. END

    ?Q. PROGRAM SERIcalc4S E N D.

    Message 2 in thread
    From: Henry ( )
    Subject: Re: Changing these fractions into decimals
    ....
    :
    :priscilla - are you any relation of Lillian?
    :
    :Try with your calculator: for example press "77" then the "divide
    key"
    :then "177" then "=" to get something like 0.4350282.

    What do I learn from that? / don.l

    From: Peter Webb ( )
    Subject: Re: Changing these fractions into decimals

    : Date: 2004-02-02 04:42:58 PST
    : If you want an approximate answer, use a calculator like Henry said.
    :
    : If you want an exact answer with
    repeating decimals then you should use long
    : division.
    :
    : Divide (say) 71.00000000... by 177 using long division. Keep
    carrying the
    : remainder until it starts to repeat. You might be doing this for a
    while; if
    : you divide by 177 there are 177 possible remainders each time (0 up
    to 176)
    : so you may have to do up to 177 carries until you get a repeat ....
    :
    : Maybe you should try on 2/7 to see how it works ... you should get a
    repeat
    : after 6 carries, giving 0.285714, so you know the full answer is
    0.285714
    : 285714 285714 285714 285714 ....

    Quite good.
    I think I just posted.. 03.02.04 today.

    Pi = 3.1 ... ... 713 999 9999 317 .... +(11 even digits.)
    Many palindromes. (Pi-search. angio net?)
    call it 5/7ths.

    but pi does not repeat with a fixed length.

    don.mcdonald.
    03.02.04 23:05
     
    Don McDonald, Jun 1, 2004
    #3
  4. qWake

    Don McDonald Guest

    very helpful. great. I learnt something liitle.

    I found this in groups.google.com
    (headers..)

    Search Result 6
    From: Dave Rusin ()
    Subject: Re: fractional representations of decimals
    View: Complete Thread (13 articles)
    Original Format
    Newsgroups: sci.math
    Date: 1999/01/14


    -...
    the article includes (Dr?) Rusin's websites?

    / don.lotto
    1.6.04
     
    Don McDonald, Jun 1, 2004
    #4
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