Alien Number Systems

Altairian Arithmetic

The method by which the Altairans write numbers is different from that adopted by any culture on Earth, but is nevertheless eminently practical. The symbols they use are those of their own alphabet but, for ease of understanding, I show the examples here in our own familiar alphabet, printed bold.

The values of the letters representing a number are combined additively (as in basic Roman numerals) and the letters can therefore be presented in any order without ambiguity, though for purposes of practical arithmetical calculation they are generally presented in alphabetical order.

As may be expected, the later letters in the alphabet represent higher numbers, but not in simple arithmetical progression (1, 2, 3, 4, ...). Beginning with a = 1 each letter represents twice the value of its predecessor. Thus b = 2, c = 4, d = 8, e = 16, f = 32, g = 64, and so on. (Thus, in our aphabet z would represent 2^25 = 33,554432.)

The important point to realise is that any number can be expressed uniquely as a sum of different powers of two. So any number has a unique name. Since their numbers are represented by the letters of their alphabet the Altairans are inveterate anagrammatists, numerologists and cabalists, associating names with numbers and vice versa.

The Altairans are not keen on high numbers, but their system allows for extension to any length by using a break symbol, which we can write ', to indicate that one complete alphabet has been exhausted and a new one has started. One of these breaks enables us to reach 'z = 2^51 = 2251,799813,685248 and two reaches ''z = 2^77 = 151115,727451,828646,838272.

Doubling

Since each letter represents twice its predecessor and letters placed together are added it follows that aa = b expresses the equation 1 + 1 = 2, and bb = c means 2 + 2 = 4. Multiplying by two is elementary: one simply replaces every letter in the number by its successor. For example bcdkm × b = cdeln.

The letters of two numbers to be added are arranged together in one alphabetical sequence, so that repeated letters are adjacent. Then, starting with the smaller numbers, pairs of repeated letters are replaced by their successor letters, until finally all repeated letters are eliminated. For example: bcdkm + acku = abccdkkmu = abddkkmu = abekkmu = abelmu. Of course with practice the number of steps can be reduced, by conflating steps or by carrying them out in a different order.

Reduplication

To multiply by a single letter-number one repeatedly doubles the number to be multiplied, and repeatedly reduces the multiplying letter to its preceding letter until a is reached. For example: e × akm = d × bln = c × cmo = b × dnp = a × eoq = eoq.

Multiplication

The rote method of multiplying by a compound number is to multiply by each of the letters of the multiplying number and then to add all the results.

Finding the Successor

To find the number that follows the number abcdk, we proceed by adding a unit (a) giving the improperly expressed number aabcdk. Then using aa = b we get bbcdk, and using bb = c we get ccdk and using cc = d we get ddk and using dd = e we get ek which is the final simplified form. In other words we have shown that abcdk + a = ek.

Finding the Predecessor

The example just given indicates that to subtract one from a power of two in this notation one replaces the letter by all its predecessors: ea = abcd. In our notation: 2^n – 1 = 1 + 2 + 4 + ... + 2^(n–1). More generally one replaces the highest letter in the name of the number by all its predecessors and then eliminates all the duplications by the addition process. Sometimes it is convenient to use a hyphen notation, writing a-k as an abbreviation for ‘all the letters a to k’.

Subtraction

To subtract a number from a larger number that uses the same letters one simply strikes out the duplicate letters. For example bcdkmudku = bcm. Where the larger number does not show all the letters of the smaller number we replace the largest digit by all its predecessors plus an extra a, then strike out the digits of the number to be subtracted, and tidy up the result. For example akmbfl = ak + a + abcdefghijklbfl = aaacdeghijkk = abcdeghijl. Using the hyphen notation this result can be expressed a-eg-jl.

Order of Magnitude

An important property of the notation is that we can tell from their names which of two numbers is the largest. If the last letter in one is different from that in the other then whichever number has the highest last letter is the highest. This is because the sum of all the lower letters is one less than the given letter so any number ending in x, say, is less than its successor y by at least one. Where both numbers have the same last letter we compare the preceding letters, and so on.

Conversion to Binary Notation

To express an Altairian number in our binary positional notation, as used in computers, write 1 for the last letter and, working backwards through the alphabet, write 1 for each letter present in the number and 0 for each letter absent. Thus bcdkm in binary is the number 1010000001110.

This study was first presented to the Outlanders (Leicester Science Fiction Group) in 2003. It has also appeared in my online magazine The Games and Puzzles Journal #27.