# The Number System With Base 36

As we already know, the number of digits in any base (radix) N is exactly the same number N. For example, in the binary (N = 2) system there are two digits: 0 and 1; in the decimal (N = 10) there are ten of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. What if N exceeds 10?

If N > 10, the missing digits come from the alphabet (usually disregarding the case.) Thus A stands for the decimal 10 in any number system with base greater than 10. B stands for the decimal 11 in any number system with base greater than 11, and so on. The counter on the Content and Front pages cycles through representations of the same number in bases from 2 through 20. In base 20, we have the following 20 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J.

This is enough to occasionally get meaningful words. For example, FACE, BEEF, CAFE, DEED, BAD, CAD, FADE may appear even in the hexadecimal (N = 16) system. For a larger radix there are more examples: HIGH, FIG, JADE, JIBE, DIG.

The ultimate inclusion of the alphabet comes with the base 36 where we need 36 digits. In base 36, every single English word represents a number. So it becomes less of a fun to seek numbers represented by a meaningful combination of letters. We become more selective. We may look not for just any word but for words representing numbers with some special properties.

Chris Caldwell has put together an awesome page devoted to the prime numbers. Among other things he catalogued a collection of words that in base 36 represent prime numbers. For example,

(ADD)36 = (10·362 + 13·36 + 13)10 = (13,441)10

which is found in a list of the first 10,000 primes.

The collection is quite long and makes a delightful study. BROOKLYN is prime and so are the following professions: AGRONOMIST, BALLOONIST, CELLIST, CHRONOLOGIST, DRAMATIST, ECOLOGIST, HORTICULTURIST, HYPNOTIST, INSTRUMENTALIST, MYCOLOGIST, NEUROLOGIST, PALEONTOLOGIST, PARATROOP, PEDIATRIST, RADIOLOGIST, TAXIDERMIST, TRUCKMAN, VEXILLOLOGIST.

And there are a few more primes that end with "IST": ABORTIONIST, BUDDHIST, DARWINIST, DEIST, MISANTHROPIST, NUDIST, POLYGAMIST.

With the help of my son David I managed to compose whole sentences that consist entirely of prime words:

• TAUNTED TOTALITARIAN STALINIST TERRORIZED BEGUILED WORKMEN.
• CANDID TIGERISH OXEN OUTRAN PETULANT BABOONISH BITCH.
• MUTANT DEIST OUTSMARTED DARWINIST BREED.
• TWIN TOBOGGANIST TOURED TANZANIAN TOWN.
• YOU'D WANT EXPIRED FOOD, CONSUMED AT DAWN?