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B Heard of Another Kind of "Rational" (Fraction) Addition?

  1. Mar 22, 2016 #1

    PMH

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    I ran across this years ago.

    It’s defined by the addition rule: a/b + c/d = (a+c)/(b+d)

    (Sounds crazy – until you think of x/y as meaning There’s a pile of x things on one side, and one of y things on the other.)

    Applying it in almost a Pascal's Triangle sort of way generates ... er, a triangle (see below)..
    ..which I think has the properties of the sequence:

    1/2, 1/3, 2/3, 1/4, 3/4, ...

    (as you see, skipping non-reduced fractions)

    Properties: One person put it this way:

    "In 1873 Georg Cantor used an argument similar to this to show the existence, not of irrational numbers within every interval, but of transcendental numbers every interval."

    Start with 0/1 on the left, and 1/1 on the right of the top line.

    To form each item of the next line, rational-add the two fractions - anywhere above - that are closest on either side to its position in the line.

    (Hard to articulate: See the first four lines that of the result from that, below.)

    (I'm using dashes as separators since so many editors like this one collapse strings of blanks into one.)

    0/1---------------1/1
    ---------1/2
    ----1/3-------2/3
    -1/4-2/5--3/5-3/4

    I'm looking to references, but of course first-hand knowledge is great.
     
  2. jcsd
  3. Mar 22, 2016 #2

    pwsnafu

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    See Wikipedia. It's called the mediant. Problem with it is that it is not monotonic (does not preserve order).
     
  4. Mar 22, 2016 #3

    Mark44

    Staff: Mentor

    In addition to what pwsnafu said, this type of addition is sometimes called "baseball arithmetic." If a batter gets 2 hits from 6 times at bat in one game, his batting average for the game is 2/6, which is usually represented as .333. If he gets 1 hit for 4 at bats in the next game, his batting average for the two games is ##\frac{2 + 1}{6 + 4} = \frac 3 {10}## or .300.

    Note that this is different from adding ##\frac 2 6## and ##\frac 1 4##, which is ##\frac 7 {12}##.
     
  5. Mar 23, 2016 #4

    micromass

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    One of the issues with this addition (which I denote with ##\oplus##) is that it is not well-defined. For example, we have

    [tex]\frac{1}{3}\oplus \frac{1}{2} = \frac{2}{5}[/tex]

    While

    [tex]\frac{2}{6} \oplus \frac{1}{2} = \frac{3}{8}[/tex]

    This is not the same outcome although it should be.
     
  6. Mar 23, 2016 #5

    Ssnow

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    yes instead of the ratio it is much appropriate the vector notation because it is in fact the sum of two vectors ## \binom{a}{b}+\binom{c}{d}=\binom{a+c}{b+d}##...
     
  7. Mar 23, 2016 #6

    PMH

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    from Skipjack at myMathForum.com:

    "I've come across it before, this article can serve as a reference for you, and this interesting article makes use of it."

    Terms ("keywords"): Mediant, Farey sequence
     
  8. Mar 23, 2016 #7

    PMH

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    Thank you.

    It turns out that what you - and nearly everyone else, I wager - perceived here, seeing these things that look like fractions, as fractions (ie: that may be reduced or represented as decimals) are in fact not.
    (IOW, they are assumed to be ... not reducible)

    A better representation than "a/b", IMHO, would be "a:b".


    the answer I got that was what I had in mind:

    from Skipjack at myMathForum.com:
    "I've come across it before, this article can serve as a reference for you, and this interesting article makes use of it."
    Terms ("keywords"): Mediant, Farey sequence
     
  9. Mar 23, 2016 #8

    Mark44

    Staff: Mentor

    Not really.
    A ratio can be represented as a fraction, such as a/b, or using a colon, as a:b.
     
  10. Mar 23, 2016 #9

    PMH

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    Well, fine. What would you say it should be represented by? (not that that's all that important)

    I was just trying to get at the fact that its form should signify that it's not a fraction - ie: one that can be either reduced or multipied by k/k without having its meaning changed.
     
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