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, 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) Roman Numerals Hex (Base 16) Sesary (Base 6) Unary (Base 1)
0 0 0 0 N/A 0 0 N/A
11 11 I1 1 1
210 22 II2 2 11
311 33 IIV3 3 111
4100 44 IV4 4 1111
5101 55 V5 5 11111
6110 66 VI6 10 111111
7111 77 VII7 11 1111111
81000 108 IIX8 12 11111111
91001 119 IX9 13 111111111
101010 12A XA 14 1111111111
111011 1310 XIB 15 11111111111
121100 1411 XIIC 20 111111111111
131101 1512 XIIID 21 1111111111111
141110 1613 XIVE 22 11111111111111
151111 1714 XVF 23 111111111111111
1610000 2015 XVI10 24 1111111111111111
1710001 2116 XVII11 25 11111111111111111
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