Intruduction to Number Systems
Number systems provide the basis for conveying and quantifying information. Weather data, stocks, pagination of books, weights and measures - these are just a few examples of the use of numbers that affect our daily lives. For this purpose we find the decimal (or Arabic) number system to be reliable and easy to use. This system evolved presumably because early humans were equipped with a crude type of calculator, their 10 fingers. But a number system that is appropriate for humans may be intractable for use by a machine such as a computer. Likewise, a number system appropriate for a machine may not be suitable for human use.
Before concentrating on those number systems that are useful in computers, it will be helpful to review those characteristics that are desirable in any number system. There are four important characteristics in all:
- Distinguishability of symbols
- Arithmetic operations capability
- Error control capability
- Tractability and speed
To one degree or another the decimal system of numbers satisfies these characteristics for hard-copy transfer of information between humans. Roman numerals and binary are examples of number systems that do not satisfy all four characteristics for human use. On the other hand, the binary number system is preferable for use in digital computers. The reason is simply put: current digital electronic machines recognize only two identifiable
states, physically represented by a high voltage level and a low voltage level. These two physical states are logically interpreted as binary symbols 1 and 0.
A fifth desirable characteristic of a number system to be used in a computer should be that it have a minimum number of easily identifiable states. The binary number system satisfies this condition. However, the digital computer must still interface with humankind. This is done by converting the binary data to a decimal and character-based form that can be readily understood by humans. A minimum number of identifiable characters (say 1 and 0, or true and false) is not practical or desirable for direct human use. If this is difficult to understand, imagine trying to complete a tax form in binary or in any number system other than decimal. On the other hand, use of a computer for this purpose would not only be practical but, in many cases, highly desirable.
Positional and polynominal representations
The positional form of a number is a set of side-by-side (juxtaposed) digits given generally in fixed-point form as
where the radix (or base), r, is the total number of digits in the number system, and a is a digit in the set defined for radix r. Here, the radix point separates n integer digits on the left from m fraction digits on the right. Notice that an-1 is the most significant (highest order) digit called MSD, and that a-m is the least significant (lowest order) digit denoted by LSD.
The value of the number in Eq. (2.1) is given in polynomial form by
where ai is the digit in the i-th position with a weight ri.
Applications of Eqs. (2.1) and (2.2) follow directly. For the decimal system r = 10, indicating that there are 10 distinguishable characters recognized as decimal numerals 0, 1, 2, . . . , r-1(= 9).
Examples of the positional and polynomial representations for the decimal system are
where di is the decimal digit in the i-th position. Exclusive od possible leading and trailing zeros, the MSD and LSD for this number are 3 and 8, respectively, This number could have been written in a form such as N10 = 03017.52800 without altering its value but implying greater accuracy of the fraction portion.