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

[PMH]

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.

 

[pwsnafu]

See Wikipedia. It's called the mediant. Problem with it is that it is not monotonic (does not preserve order).

 

[Mark44]

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

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 $\frac 7 {12}$.

 

[micromass]

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

\frac{1}{3}\oplus \frac{1}{2} = \frac{2}{5}

While

\frac{2}{6} \oplus \frac{1}{2} = \frac{3}{8}

This is not the same outcome although it should be.

 

[Ssnow]

yes instead of the ratio it is much appropriate the vector notation because it is in fact the sum of two vectors $\binom{a}{b}+\binom{c}{d}=\binom{a+c}{b+d}$...

 

[PMH]

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

 

[PMH]

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

 

[Mark44]

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

Not really.
A ratio can be represented as a fraction, such as a/b, or using a colon, as a:b.

 

[PMH]

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.