Number Representation Systems

At a digital level, for the processing and transmission of information, we need first to convert information into numbers. A digital photograph observed on a computer screen or in a color print give us high-level information that our brain interprets at a holistic level: for example, as a red poppy in a corn field. For the computer, instead, this image is nothing but a long sequence of numbers encoding the intensity and the color of any image point called a pixel. Moreover, the numbers representing pixels are in a particular number representation called the binary system (Ore 1988). Binary digits are the only numbers that our computers are able to manage at the hardware level. This technological choice is due to the fact that binary numbers are represented using only two digits, i.e. 0 and 1. The binary representation system pertains to the class of positional power representation systems. In positional representation systems, any ordered digit in the representation represents the number of times that a given power of an integer number called the base (b) is included in the additive decomposition of the number. The position of the digit determines the power of the base by which the digit is multiplied: the nth digit is multiplied by bn. The digit position is computed starting with the rightmost one, because the rightmost digit is the least significant, corresponding to a positional weight of 1 = b0. In a particular power representation, the digits should take values from 0 to b - 1 in order to assure that the representation is univocal: given a number, its representation is unique (there is only one sequence of signs), and given a sequence of signs, these specify through additive decomposition only this same number. Of course it is necessary also that all numbers may be represented, and this property is assured by the fact that the sum up to order n of all the powers of the base times the greatest possible digit, i.e. (b - 1), equals the following power of the base, (n + 1), minus one, i n=o (b -1)-bn=bk-i

Thus, the only thing that is needed in order to completely specify a positional power representation system is the value of the base b. Our usual number representation system, the decimal one, pertains also to this class of representation systems with the value of the base equal to 10. The same is true for other common number representation systems, for example, b = 2 for the binary system, b = 8 for the octal one, and so on.

As an example we show the representation of the number 421 in the usual decimal system (base 10). As stated above, the digits can take only the values from 0 to b - 1 = 9. From right to left (see Table 1), any digit is multiplied by the corresponding power of 10 and the result sums to 421.

For the case of the binary system, as remarked above, the digits can take only the values 0 and 1. For this reason it is the system preferred in computing applications: the two digits can be represented by two complementary states of an electronic system, for example, an electronic switch with the "on"-"off" states representing the 1 and 0 values of the binary digits. Really, there is no limitation to the number of stable states that an electronic system can have, but historical reasons have determined the use of the binary system as a privileged one for computing applications, for example, thanks to the development of a very consistent theoretical corpus known as Boolean Algebra (Kohavi, 1970) that allows a great simplification in the design of logical functions.

As an example we can see in Table 2 the representation of the decimal number 21 in the binary system (1,0,1,0,1)

Table 1 Representation in base 10 of the number 421. The decimal representation system is one example of positional power representation systems, which include also the binary system

Base powers 102 = 100 1 01 = 10 100 = 1 Total x x x

Multiplicative 4 2 1

digits

Table 2 Representation of the decimal number 21 in the binary system. As in the decimal case the digits (0 or 1), are multiplied by the powers of the base (2) from right to left. The sum of these products equals 21

Base powers

24=16

23=8

22=4

21=2

20=1 Total

x

x

x

x

X

Multiplicative

1

0

1

0

1

digits

Partial result

16

0

4

+ 0 +

1 = 21

The univocal property of positional power representations precludes the existence of redundancy in the represented numbers. As we will see later, redundancy is a fundamental ingredient for detecting and correcting eventual errors in a communication process. For this reason, if we want to implement error correction codes, redundancy has to be artificially introduced. It may be remarked that, although redundancy is absent in power representation systems, other number representation systems exist that introduce redundancy in a natural fashion. This is the case for the non-power positional representation systems. An example of the utility of this particular kind of number representation for the description of biological information is given in this same volume (Chapter 6, this volume).

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