Number systems

The study of number systems is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a digital computer. It is one of the most basic topics in digital electronics. In this chapter we will discuss different number systems commonly used to represent data. We will begin the discussion with the decimal number system. Although it is not important from the viewpoint of digital electronics, a brief outline of this will be given to explain some of the underlying concepts used in other number systems. This will then be followed by the more commonly used number systems such as the binary, octal and hexadecimal number systems.

1.2 Decimal number system
The decimal number system is a radix-10 number system and therefore has 10 different digits or symbols. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All higher numbers after ‘9’ are represented in terms of these 10 digits only. The process of writing higher-order numbers after ‘9’ consists in writing the second digit (i.e. ‘1’) first, followed by the other digits, one by one, to obtain the next 10 numbers from ‘10’ to ‘19’. The next 10 numbers from ‘20’ to ‘29’ are obtained by writing the third digit (i.e.‘2’) first, followed by digits ‘0’ to ‘9’, one by one. The process continues until we have exhausted all possible two-digit combinations and reached ‘99’. Then we begin with three-digit combinations. The first three-digit number consists of the lowest two-digit number followed by ‘0’ (i.e. 100), and the process goes on endlessly.
The place values of different digits in a mixed decimal number, starting from the decimal point, are 100 , 101 , 102 and so on (for the integer part) and 10?1 , 10?2 , 10?3 and so on (for the fractional part). The value or magnitude of a given decimal number can be expressed as the sum of the various digits multiplied by their place values or weights.
As an illustration, in the case of the decimal number 3586.265, the integer part (i.e. 3586) can be expressed as

3586 = 6 × 100 + 8 × 101 + 5 × 102 + 3 × 103 = 6 + 80 + 500 + 3000 = 3586 and the fractional part can be expressed as
265 = 2 × 10?1 + 6 × 10?2 + 5 × 10?3 = 0 2 + 0 06 + 0 005 = 0 265

We have seen that the place values are a function of the radix of the concerned number system and the position of the digits. We will also discover in subsequent sections that the concept of each digit having a place value depending upon the position of the digit and the radix of the number system is equally valid for the other more relevant number systems.