2.1 Numbers in Different Bases

In most cultures the concept of quantity and numbers is usually introduced with counting on our fingers. To represent each of these unique quantities we use a symbol or combination of symbols. For example five is represented with the symbol 5 using the Arabic numerals and the symbol V using the Roman numerals. To represent larger quantities, a combination of symbols is used; for example, thirteen is represented using a combination of two symbols 1 and 3 as 13 in Arabic numerals and a combination of X and I as XIII in Roman numerals. The use of combinations is very helpful because it eliminates the need for an infinite symbol set. Thus, making the concept of using combinations a very convenient tool for counting. The number of symbols available for counting defines the base of a number. The numbers that are in common use today use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Since there are ten symbols, this system of representation is called base ten or denary. The base ten representation is also commonly referred to as the decimal system. The prefix deci- is a reference to ten in Latin.  When the unique symbols are exhausted, we begin to use a combination of numbers, as shown below:

0,1,2,3....8,9,10,11,12...19,20,21.....98,99,100

In the above set of numbers, the combinations involving two symbols begin after the unique symbols are used up or not available, and after all the two symbol combinations are used up the three symbol combinations begin and so on. This method of using combinations allows for the representation of any quantity or of infinitely many numbers using a finite set of symbols. Each symbol within a combination is commonly referred to as a digit, and as the number of digits increases, the value of the number also increases.

The system of counting using ten symbols, described earlier, is the most common form of counting, and is probably the only form of counting that many of us are familiar with. The question then is: is it possible to count using a symbol set with a different number of symbols? For example, can four symbols be used instead of ten symbols to perform counting? The answer to the question is yes. If the concept of counting using symbols and combination of symbols is extended to symbols sets of any size, then counting can be done with any number of symbols. Imagine a system where the symbols available are (0, 1, 2, 3), and using the same concept of combination we achieve the following result

Since there are four symbols in the symbol set, the above example would be counting in base four. The symbolic representation of the number five in base four should not be confused with the number eleven in base ten. To avoid this confusion, numbers in bases other than ten are read in terms of the individual digits. For example, “one one base four” instead of eleven. To avoid confusion in the written form, a subscript is used to indicate the base of the number. For example, five in base four is written as 114 so that it will not be confused with eleven in base ten, which also has the same two symbols. A number without a base subscript is assumed to be base ten.

The idea of bases could be extended beyond base 10 as well, but this leads to a problem with the symbols available. The question is how can the symbol set be extended above the ten symbols (0-9) that are commonly used for representing numbers? This problem is solved by using letters along with numeric symbols. For example, base 16 uses 16 symbols, 0-9 and A-F. This base is commonly referred to as the hexadecimal base. When a base subscript is not present on a number with alphabetical symbols then it is assumed to be hexadecimal number.

Here again, like the previous case of base 4, when the available symbols are used up combination of symbols are used to represent numbers larger than the base. From the two examples of counting involving different symbols sets, it can be seen that counting can be accomplished using any number of symbols. In the world of computers the binary system or base 2 is the basis on which the computer operates, and by performing calculations in base 2 the computer is able to perform a variety of tasks. The reason why base 2 is used in computers is because the two symbols (1 and 0) can be represented using electrical signals (on and off).