Base (mathematics)

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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 can be any whole number bigger than 0 (if it was 0, then there would be no digits). 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).

Uses of bases[change | change source]

In computers[change | change source]

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[change | change source]

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 = 12x12), and lengths such as feet (12 inches).

Writing bases[change | change source]

When typing a base, the small number indicating the 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[change | change source]

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

Decimal Binary Undecimal Hexidecimal
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 8 8
9 001 9 9
10 1010 A A
11 1011 10 B
12 1100 11 C
13 1101 12 D
14 1110 13 E
15 1111 14 F
16 10000 15 10