Isn't base 27 more adapted to express numbers with letters?

Question

I thought about the importance of 0 in numeral systems, and base 26 doesn't have one, for example those numbers in base 10 are the same:

0342
342
000342

In base26, any word would need to be expressed as follow, for example the word

duck

would be equal to

aaaaaaduck

because a is considered to be zero.

Am I right ?

I know the question is not really interesting when thinking about maths, but I just wanted to know where I could be wrong with this.

Also the use of 0 allows to make actual sentences if you use it as a separator.

EDIT: of course when I ask about base 26, it implies letters.

Answer 1

1. In most place-value systems, including base 26, the convention is to run from 0 through 9 (as J.M. indicates in the comments), and then from A through Z, for as many digits as necessary. Thus:

   • for binary one uses the digits 0 and 1;
   • for duodecimal one uses 0–9 and A, B;
   • for base 26, one uses 0–9 and A–P;
   • for base 36, one uses 0–9 and A–Z.

   There is no solidly defined convention for bases larger than 36, although I have seen people run through greek letters (which obviously works better if you use lower-case letters, and don't manage to get far enough to encounter omicron or even iota).

2. In an absolute sense, you are not obligated to use any particular set of symbols at all. Indeed:

   • You can write binary numbers using white and black dots, such as $\bullet$ and $\circ$ , and decide for yourself which stands for zero or one.
   • You can write numbers in duodecimal using the symbols for the zodiacal constellations, setting ♈=0, ♉=1, ♊=2 (naturally), and so on up to ♓=11.
   • You can operate in base 256 using the extended ASCII table, although the non-printable characters may cause you some trouble (and you'll have difficulties expressing decimal points); if you restrict yourself to the printable characters (including the space character), you'll have to be content with base 225.

Number systems are just ways of representing numbers. They are in no sense absolute, but of course when communicating with others it is good to make clear what system you use, or (better still unless you have a good reason) simply adopt a system which is widespread. And if you are interested in the most widespread convention for base 26, it will be starting with 0–9 as J.M. indicated.

Answer 2

@gokoon above actually provides a link that has a thorough discussion of this question: http://en.wikipedia.org/wiki/Hexavigesimal

In short, it is true that in the most common convention, number bases tend to use 0-9A-Z for digits. And the answer above is certainly correct: the symbol choice is entirely arbitrary and has no effect on the calculations. However, in practical terms the question's a little more complex and fairly interesting. Basically, to humans the letters are a convenient and natural way of expressing a 26-based system. Convenience and naturalness are decent enough reasons for choosing a notation! In fact there are three everyday applications that use alphabetic base-26: serial numbers, where base-26 and base-36 are commonly found; slightly modified as the designation system for variable stars; and the 26-adic row and column numbering in Excel and similar spreadsheets.

Now I will admit that these are not deeply theoretical applications. And one must be careful not to carry the symbolism too far and end up in numerology. John Nash was somewhat obsessed by alphabetic base-26 and this was related to his interest in numerology. Nevertheless it's not absurd to look at A-Z for base-26 notation.

Answer 3

I'm impressed by how the answers and comments miss the point. Yes, using alphabetic base 27 with space as zero makes more sense for a few reasons.

• You most likely wouldn't ever want to represent anything that starts with many spaces, whereas you would very likely want to be able to start with one or more As without ambiguity
• We usually think of the numerical values of letters as being indexed from one, for instance we typically think of M as the 13th letter, so when converting base 27 to base 10 with a calculator it's more natural to convert M to 13 instead of 12
• It doesn't hurt for a text encoding this minimalistic to have an extra non-alphabetic character to separate words

For those who missed the point by saying that base 26 or 27 is whatever you want it to be, consider that there aren't many reasons to choose base 26 or 27 if not as a compact way to represent text, so for practical uses the options are limited and fairly obvious. It's a good way to choose a number of 3 to 5 digits based on a few letters. For instance let's say you make a program named "the ACME server" that needs its own arbitrary IP port number, take the string acme, in alphabetic base 26 you would get something ambiguous that would be interpreted back as cme, but in alphabetic base 27 you would get 22226, which is probably suitable as a port number. Such a way to decide on such numbers would help spread choices out evenly, which to me is the appeal.