Base (mathematics)

In mathematics, a base or radix is the number of different digits or combination of digits and letters that a system of counting uses to represent numbers. For example, the most common base used today is the decimal system. Because "dec" means 10, it uses the 10 digits from 0 to 9. Most people think that we most often use base 10 because we have 10 fingers.

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

In computers

Different bases are often used in computers. Binary (base 2) is used because at the most simple level, computers can only deal with 0s and 1s. Hexadecimal (base 16) is used because of how computers group binary digits together. Every four binary digits turn into one hexadecimal digit when changing between them. Because there are more than 10 digits in hexadecimal, the six digits after 9 are shown as A, B, C, D, E, and F.

Measurement

The oldest systems of counting used base one. Making marks on a wall, using one mark for each item counted is an example of unary counting. Some old systems of measurement use the duodecimal radix (base twelve) since 12 is 2x6. This is shown in English, as there are words such as dozen (12) and gross (144 = 12×12), and lengths such as feet (12 inches). 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 some numbers are written in different bases, compared to decimals:

Decimal (Base 10)	Binary (Base 2)	Octal (Base 8)	Undecimal? (Base 11)	Sesary (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	10	8	12	11,111,111
9	1,001	11	9	13	111,111,111
10	1,010	12	A	14	1,111,111,111
11	1,011	13	10	15	11,111,111,111
12	1,100	14	11	20	111,111,111,111
13	1,101	15	12	21	1,111,111,111,111
14	1,110	16	13	22	11,111,111,111,111
15	1,111	17	14	23	111,111,111,111,111
16	10,000	20	15	24	1,111,111,111,111,111
17	10,001	21	16	25	11,111,111,111,111,111