Base (mathematics)

In mathematics, a base is the number of single digits denoting different values in a positional numeral system, including zero. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever reach is 9, after this it is necessary to add another digit to achieve a higher number.
Sometimes, a subscript notation is used where the base number is written in subscript after the number. For example, a number, say 23, in base 8 would be written as 23_{8}. This notation will be used in this article.
Contents 
System
When one says "base b", the b refers to the decimal value of "10" in base b. For example, base 5 means that 10_{5} = 5_{10}. The largest digit in a base is therefore one less than the base itself, as after this largest digit, an extra digit must be added to make 10 in that base.
Bases work using exponentiation. A digit's value is the digit multiplied by the the value of is place. Place values are the number of the base raised to the nth power, where n is the number of digit to the left the units digit.
For example, the number 465 in its respective base (which is clearly at least base 7) is equal to:
 <math>4\times 10^2 + 6\times 10^1 + 5\times 10^0<math>
Numbers that are not integers use places beyond the decimal point. For every point behind the decimal point (and thus the units digit), the power n decreases by 1.
For example, the number 2.35 is equal to:
 <math>2\times 10^0 + 3\times 10^{1} + 5\times 10^{2}<math>
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5: 2 groups of 5² 4 groups of 5 1 group of 1 00000 00000 00000 00000 00000 00000 00000 00000 + + 0 00000 00000 00000 00000 00000 00000
241 in base 8: 2 groups of 8² 4 groups of 8 1 group of 1 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 + + 0 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
Conversion between bases
Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform the new base, for example:
241 in base 5: 2 groups of 5² 4 groups of 5 1 group of 1 00000 00000 00000 00000 00000 00000 00000 00000 + + 0 00000 00000 00000 00000 00000 00000
is equal to 107 in base 8: 1 groups of 8² 0 groups of 8 7 groups of 1 00000000 00000000 0 0 0 00000000 00000000 + + 0 0 00000000 00000000 0 0 0 00000000 00000000
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between nondecimal bases without using this intermediate step.
A number a_{n}a_{n1...a2a1a0 where a0, a1... an are all digits in a base B (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by: }
 <math>\sum_{i=0}^n \left( a_i\times B^i \right)<math>
Thus, in the example above:
 <math>241_5 = 2\times 5^2 + 4\times 5^1 + 1\times 5^0 = 25 + 20 + 2 = 71_{10}<math>
To convert a decimal number into a base is a slightly more complicated process. One must first find the largest power of the new base that will go into the number. Then, how many whole times the number will go into that power must be found, and the product of the two subtracted from the number. The process is then repeated until one reaches the end.
Thus, to convert 71_{10} into base 8:
8² goes into 71 once. 8² × 1 = 64; 71  64 = 7 units remaining. 8 does not go into 7, therefore still 7 units remaining. 7 goes into 1 seven times.
Therefore <math>47_{10} = 1\times 8^2 + 0\times 8^1 + 7\times 8^0 = 107_8<math>
Applications
The decimal system, base 10, is the base used in everyday life. It is believed that this came about because human beings have ten fingers. However, other civilisations and contexts used different bases.
Historical systems
The Babylonian civilisation used a base 60 system. There were not, however, 60 different symbols, as one would expect — each "digit" was represented by a somewhat decimal system, for example, "12 35 1" = 12×60^{2} + 35 ×60 + 1. The Babylonians had their own symbols.
Computing
In computing, the binary (base 2) and hexadecimal (base 16) bases are used. Computers, at the very simplest level, deal only with a series of conventional 1's and 0's, thus it is easier in this sense to deal with powers of two. The hexadecimal system came about as shorthand for binary  every 4 binary digits relates to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B... F.
References
O'Connor, J. J. and Robertson, E. F. Babylonian numerals (http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Babylonian_numerals.html). Retrieved 26 April 2005.