Definitions

# Base 36

Base 36 is a positional numeral system using 36 as the radix. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0-9 and the Latin letters A-Z. Base 36 is therefore the most compact case-insensitive alphanumeric numeral system using ASCII characters, although its radix economy is poor. (Compare with base 16 and base 64.)

From a mathematical viewpoint, 36 is a convenient choice for a base in that it is divisible by both 2 and 3, and by their multiples 4, 6, 9, 12 and 18. Each base 36 digit can be represented as two base 6 digits.

The most common latinate name for base 36 seems to be hexatridecimal, although sexatrigesimal would arguably be more correct. The intermediate form hexatrigesimal is also sometimes used. For more background on this naming confusion, see the entry for hexadecimal. Another name occasionally seen for base 36 is alphadecimal, a neologism coined based on the fact that the system uses the decimal digits and the letters of the Latin alphabet.

## Examples

Conversion table:

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Base 36 0 1 2 3 4 5 6 7 8 9 A B C D E F G H

Decimal 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Base 36 I J K L M N O P Q R S T U V W X Y Z

Some numbers in decimal and base 36:

Decimal Base 36
1 1
10 A
100 2S
1,000 RS
10,000 7PS
100,000 255S
1,000,000 LFLS
1,000,000,000 GJDGXS
1,000,000,000,000 CRE66I9S
Base 36 Decimal
1 1
10 36
100 1,296
1000 46,656
10000 1,679,616
100000 60,466,176
1000000 2,176,782,336
10000000 78,364,164,096
100000000 2,821,109,907,456
Fraction Decimal Base 36
1/2 0.5 0.I
1/3 0.333333333333… 0.C
1/4 0.25 0.9
1/5 0.2 0.777777777777…
1/6 0.166666666666… 0.6
1/7 0.142857142857… 0.555555555555…
1/8 0.125 0.4I
1/9 0.111111111111… 0.4
1/10 0.1 0.3LLLLLLLLLLL...

## Conversion

32- and 64-bit integers will only hold up to 6 or 12 base-36 digits, respectively. For numbers with more digits, one can use the functions mpz_set_str and mpz_get_str in the GMP arbitrary-precision math library. For floating-point numbers the corresponding functions are called mpf_set_str and mpf_get_str.

## Python Conversion Code

def base36decode(input):
`   CLIST="0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"`
`   rv = pos = 0`
`   charlist = list(input)`
`   charlist.reverse()`
`   for char in charlist:`
`       rv += CLIST.find(char) * 36**pos`
`       pos += 1`
`   return rv`

def base36encode(input):

`   CLIST="0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"`
`   rv = ""`
`   while input != 0:`
`       rv = CLIST[input % 36] + rv`
`       input /= 36`
`   return rv`

print base36decode("AQF8AA0006EH") print base36encode(1412823931503067241)

## Uses in practice

The Remote Imaging Protocol for bulletin board systems used base 36 notation for transmitting coordinates in a compact form. Many URL redirection systems like TinyURL also use base 36 integers as compact alphanumeric identifiers. Various systems such as RickDate use base 36 as a compact representation of Gregorian dates in file names, using one digit each for the day and the month.