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.

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.

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**.

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.

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**.

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: **e** – **a** =
**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**’.

To subtract a number from a larger number that uses the same letters one simply
strikes out the duplicate letters. For example **bcdkmu** – **dku** =
**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 **akm** – **bfl** = **ak** + **a** + **abcdefghijkl**
– **bfl** = **aaacdeghijkk** = **abcdeghijl**. Using the hyphen
notation this result can be expressed **a-eg-jl**.

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.

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.*