Base 36

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic (Hindu numerals) Eastern Arabic Indian family Tamil	Burmese Khmer Lao Mongolian Thai
East Asian numerals
Chinese Japanese Suzhou	Korean Vietnamese Counting rods
Alphabetic numerals
Abjad Armenian Āryabhaṭa Cyrillic	Ge'ez Greek Georgian Hebrew
Other systems
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit	Kharosthi Mayan Quipu Roman Sumerian Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64
Balanced ternary
Non-positional system
Unary numeral system (Base 1)
List of numeral systems

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 (the ISO basic Latin alphabet). Base 36 is therefore the most compact case-insensitive alphanumeric numeral system using ASCII characters, although its radix economy is poor.

From a mathematical viewpoint, 36, as with all highly composite numbers, 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	0.3	0.C
1/4	0.25	0.9
1/5	0.2	0.7
1/6	0.16	0.6
1/7	0.142857	0.5
1/8	0.125	0.4I
1/9	0.1	0.4
1/10	0.1	0.3L

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.

Java implementation

public class Base36 {
  public long decode(final String value) {
    return Long.parseLong(value, 36);
  }
 
  public String encode(final long value) {
    return Long.toString(value, 36);
  }
}

PHP implementation

The decimal value of 12abcxyz is <?php print base_convert("12abcxyz",36,10); ?>

The base_convert function converts the value to a floating-point number, which loses accuracy for numbers above implementation-specific limits. For php 5.3.6 on 64-bit linux, the highest base-36 integer that can be represented accurately is 1y2p0ij32e8e7, equal to 2^63-1 or 9223372036854775807. Negative signs, decimal points and any characters outside the range 0-z are stripped prior to conversion, so -1.5 = 15 = f, rather than -1.i as might be expected.

Python implementation

def base36encode(number, alphabet='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'):
    """Convert positive integer to a base36 string."""
    if not isinstance(number, (int, long)):
        raise TypeError('number must be an integer')
 
    # Special case for zero
    if number == 0:
        return alphabet[0]
 
    base36 = ''
 
    sign = ''
    if number < 0:
        sign = '-'
        number = - number
 
    while number != 0:
        number, i = divmod(number, len(alphabet))
        base36 = alphabet[i] + base36
 
    return sign + base36
 
def base36decode(number):
    return int(number, 36)
 
print base36encode(1412823931503067241)
print base36decode('AQF8AA0006EH')

Ruby implementation

1412823931503067241.to_s(36)  #=> "aqf8aa0006eh"

Uses in practice

• The Remote Imaging Protocol for bulletin board systems used base 36 notation for transmitting coordinates in a compact form.
• The Siebel CRM application uses base 36 as the unique identifier generator for all the entities in its model (e.g., 1-8GH4DX).
• Many URL redirection systems like TinyURL or SnipURL/Snipr 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.
• Dell uses a 5 or 7 digit base 36 number (Service Tag) as a compact version of their Express Service Codes.
• The software package SalesLogix uses base 36 as part of its database identifiers.
• The TreasuryDirect website, which allows individuals to buy and redeem securities directly from the U.S. Department of the Treasury in paperless electronic form, serializes security purchases in an account using a 4 digit base 36 number. However, the Latin letters A-Z are used before the Arabic numerals 0-9, so that the purchases are listed as AAAA, AAAB... AAAZ, AAA0, AAA1... AAA9, AABA...
• The E-mail client program PMMail encodes the UNIX time of the email's arrival and uses this for the first six characters of the message's filename.
• MediaWiki stores uploaded files in directories with names derived from the base-36 representation of a uploaded file's checksum .
• Siteswap, a type of juggling notation, frequently employs 0-9 and a-z to signify the dwell time of a toss (which may roughly be thought of as the height of the throw). Throws higher than 'z' may be made but no notation has widespread acceptance for these throws.
• Minecraft stores level chunk files in directories with names derived from the base-36 representation of a chunk's position.
• In SEDOL securities identifiers, the check digit is computed from a weighted sum of the first six characters, each character interpreted in base-36.
• In the International Securities Identification Number (ISIN), the check digit is computed by first taking the value of each character in base-36, concatenating the numbers together, then doing a weighted sum.

External links

• A discussion about the proper name for base 36 at the Wordwizard Clubhouse
• The Prime Lexicon, a list of words that are prime numbers in base 36
• A Binary-Octal-Decimal-Hexadecimal-Base36 converter written in PHP
• A C# base36 encoder and decoder
• A short writeup on base36 numbering scheme with code in C#