Base (mathematics)

In mathematics, a base or radix is the amount of digits that a positional system of counting uses to represent numbers. For example, the most common number system used today is the decimal system. Decimal has 10 digits, 0 to 9, so it is a base 10 system.

A base is usually a whole number greater than 1, but non-integer bases are also mathematically possible. The base of a number may be written next to the number: for instance, $23_{8}$ means 23 in base 8, which is equal to 19 in base 10.

In computers

Binary (base 2) is used by computers because it is the simplest way to represent numbers. Hexadecimal or hex (base 16) is used by many software developers because it provides a shorthand of binary at a similar scale to decimal. Hex numbers can be converted to and from binary with no math, as every 1 hex digit is exactly equal to 4 binary digits.

Measurement

The oldest systems of counting used base one, unary. Making marks on a wall, using one mark for each item counted is an example of unary counting.

Some old systems of measurement used base twelve, or duodecimal. This is shown in English, as there are unique words for 12 (twelve or dozen), 12 * 12 (gross), and 12 * 12 * 12 (great gross). A foot is 12 inches. The 12-hour clock is also popular in English-speaking countries.

Angle measurement often uses a system adapted from the Babylonian numerals with base 60.

Writing bases

When typing a base, the small number indicating the base is usually in base ten. This is because if the radix were written in its own base, it would always be "10," so there would be no way of knowing what base it was supposed to be in.

Numbers in different bases

Here are some examples of how numbers are written in different bases, compared to decimal:

Decimal (Base 10)	Binary (Base 2)	Hex (Base 16)	Duodecimal (Base 12)	Senary (Base 6)	Unary (Base 1)
0	0	0	0	0	N/A
1	1	1	1	1	1
2	10	2	2	2	11
3	11	3	3	3	111
4	100	4	4	4	1,111
5	101	5	5	5	11,111
6	110	6	6	10	111,111
7	111	7	7	11	1,111,111
8	1,000	8	8	12	11,111,111
9	1,001	9	9	13	111,111,111
10	1,010	A	A	14	1,111,111,111
11	1,011	B	B	15	11,111,111,111
12	1,100	C	10	20	111,111,111,111
13	1,101	D	11	21	1,111,111,111,111
14	1,110	E	12	22	11,111,111,111,111
15	1,111	F	13	23	111,111,111,111,111
16	10,000	10	14	24	1,111,111,111,111,111
17	10,001	11	15	25	11,111,111,111,111,111