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

• B

## Main Question or Discussion Point

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
Mentor
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 $\frac 7 {12}$.

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

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

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".

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
Mentor
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.

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.