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. Decimal has 10 digits, 0 to 9, so it is a base 10 system.
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, means 23 in base 8 (which is equal to 19 in base 10).
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
changeThe 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
changeWhen 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
changeHere 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) | 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 | 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 |