It’s rather cliché that an alien message would use a value like π (actually I expect it to be 2π because Earthlings are weird), but in my story the message which contains the bootstrap information for reading the message has natural numbers as an atomic construct, and fractions, continued fractions, and other ways of expressing a non-whole positive value (let alone a transcendental value!) are built up out of more primitive elements.※
The lesson on the basic feature of natural numbers can illustrate counting, high enough to show every digit and how positional notation works.‡
But then it should give some large numbers, so the decoder/receiver can know he’s reading it properly. Now if you were to receive the value of e you’d recognize it as being something special, and out of all the possible numbers you’d know this was “right” because it is distinct and recognizable.
But, what numbers in ℕ are like that? What positive whole number, that’s not too short but needs a bunch of digits, will make the recipient recognize it as being a special somewhat unique value?
If a single value can’t do it, then perhaps a sequence. Primes are too dull and don’t grow fast enough. Something like a list of consecutive powers is too arbitrary.
Edit: To clarify, this number is not sent/presented in isolation with any need to show artificiality by itself. It appears in a huge message that’s already bootstrapped the main message low-level encoding as “pages” containing images and language encoded as binary files.
This number is to be a “cool” example to conclude the page explaining how natural numbers are encoded in the binary language file.
By “binary language” I mean similarly to how we would store a 32-bit number as 4 bytes (essentially base 256) and how you know to expect a number and how long it is. Not exactly. It’s like digits in some ways… but think of a text file where digits are bytes 0x30 – 0x39, not pictures of what our glyphs look like.
※ identifiers (“words” and “symbols”) other than numbers, and sentences such as needed to express relationships and algorithms are also build on top of this. Simple expressions like
2+3=5 start on the next page.
‡ details: (spoiler if you’d like to figure it out later when I post the completed image on puzzling, don’t look at the hidden parts.) The message is composed of 6-bit elements (hexets) and the way the channel coding was explained in the simpler/slower/cruder earlier part, it’s clear which code is
111111 etc. Well, maybe 4-way ambiguity as to the significance of the order and whether the solid mark is a 1 and hollow is 0 or vice-versa.
Anyway, one natural way of assigning numeric values to hexets will show o00 through o56 assigned to “digits” with the same value. A natural number is introduced with hexet o57 and followed by digits in little endian.
The page is ruled into boxes, with many small boxes on one row at the top, getting larger and larger until boxes fill a whole row and then continue to get taller.
Each box has spots in it. They are a few pixels across but vary in size and shape, being irregular in shape. They even have different “colors” (pixel value; reader chose palette arbitrarily). The spots are irregularly positioned as well, clumping here, rarified there.
The first cell has no spots; the next 1, then 2, etc. all the way up to 53 or so,
so it gets into numbers that need 2 hexets to represent. Also in each cell is the label, which is also positioned irregularly in each cell but never touching the rule lines. The labels show o57 o00 in the cell with no spots, o57 o01 in the cell with 1 spot, up through o57 o56 in the cell with 46 spots, then o57 o00 o01 in the cell with 47 spots, through o57 o05 o01 for 52.
You see directly each digit, then that it is positional and in what order.
But it does not carry on counting any higher; large enough to show anything useful would be impractical.
So I think it will go to some sequences after that. The big cool numbers come at the bottom of the page.