Base

In Mathematics, a base or radix is the number of different digits that a system of counting uses to represent numbers. For example, the most commonly used base today is the decimal system. Because "dec" means 10, it uses the 10 digits from 0 through 9.

Bases must be a whole number bigger than 0. If it was 0, then there would be no digits.

Sometimes, to show that a number is in a different base to the decimal system, the base that it is in is typed after it, smaller than the number. For instance, [math]\displaystyle{ 23_8 \\ }[/math] means 23 in base 8 (which is equal to 19 in base 10).

Most people think that we most often use base 10 because we have 10 fingers.

Uses of Bases

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 every 4 digits can be used as short for binary. Every 4 binary digits turns into one hexadecimal digit when changing between them. Because there are more than 10 digits in hexadecimal, the 6 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). This is shown in English, as there are words such as dozen (12) and gross (144 = 12 x 12), and lengths such as feet (12 inches).

Writing bases

When typing a base, the small number which shows how many digits in that base is usually in base 10. This is because if the radix was written in the base it is, it would always be 10, so there would be no way of knowing what base it was.

Numbers in different bases

Here are some examples of how numbers are written in different bases.

Decimal base=10	Binary base=2	Hexadecimal base=16
1	1	1
2	10	2
3	11	3
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F
16	10000	10
17	10001	11
25	11001	19
50	110010	32
64	1000000	40
100	1100100	64
128	10000000	80

Adapted from Simple English Wikipedia: Base (mathematics)