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## 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.

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.