# 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.

A base can be any whole number bigger than 0 (smaller than the number. For instance, ${\displaystyle 23_{8}\ }$ 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 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). 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).bases are used as a converter for various numbers.

## 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.

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