4
Digital Electronics
Yet another significant advantage of this number system was that all kinds of data could be
conveniently represented in terms of 0s and 1s. Also, basic electronic devices used for hardware
implementation could be conveniently and efficiently operated in two distinctly different modes. For
example, a bipolar transistor could be operated either in cut-off or in saturation very efficiently.
Lastly, the circuits required for performing arithmetic operations such as addition, subtraction,
multiplication, division, etc., become a simple affair when the data involved are represented in the
form of 0s and 1s.
1.5 Octal Number System
The octal number system has a radix of 8 and therefore has eight distinct digits. All higher-order
numbers are expressed as a combination of these on the same pattern as the one followed in the case
of the binary and decimal number systems described in Sections 1.3 and 1.4. The independent digits
are 0, 1, 2, 3, 4, 5, 6 and 7. The next 10 numbers that follow ‘7’, for example, would be 10, 11, 12,
13, 14, 15, 16, 17, 20 and 21. In fact, if we omit all the numbers containing the digits 8 or 9, or both,
from the decimal number system, we end up with an octal number system. The place values for the
different digits in the octal number system are 8
0 , 8
1 , 8
2 and so on (for the integer part) and 8
−1 , 8
−2 ,
8
−3 and so on (for the fractional part).
1.6 Hexadecimal Number System
The hexadecimal number system is a radix-16 number system and its 16 basic digits are 0, 1, 2, 3,
4, 5, 6, 7, 8, 9, A, B, C, D, E and F. The place values or weights of different digits in a mixed
hexadecimal number are 16
0 , 16
1 , 16
2 and so on (for the integer part) and 16
−1 , 16
−2 , 16
−3 and so on
(for the fractional part). The decimal equivalent of A, B, C, D, E and F are 10, 11, 12, 13, 14 and 15
respectively, for obvious reasons.
The hexadecimal number system provides a condensed way of representing large binary numbers
stored and processed inside the computer. One such example is in representing addresses of different
memory locations. Let us assume that a machine has 64K of memory. Such a memory has 64K (= 2
16
= 65 536) memory locations and needs 65 536 different addresses. These addresses can be designated
as 0 to 65 535 in the decimal number system and 00000000 00000000 to 11111111 11111111 in the
binary number system. The decimal number system is not used in computers and the binary notation
here appears too cumbersome and inconvenient to handle. In the hexadecimal number system, 65 536
different addresses can be expressed with four digits from 0000 to FFFF. Similarly, the contents of the
memory when represented in hexadecimal form are very convenient to handle.
1.7 Number Systems – Some Common Terms
In this section we will describe some commonly used terms with reference to different number systems.
1.7.1 Binary Number System
Bit is an abbreviation of the term ‘binary digit’ and is the smallest unit of information. It is either ‘0’
or ‘1’. A byte is a string of eight bits. The byte is the basic unit of data operated upon as a single unit
in computers. A computer word is again a string of bits whose size, called the ‘word length’ or ‘word
size’, is fixed for a specified computer, although it may vary from computer to computer. The word
length may equal one byte, two bytes, four bytes or be even larger.
Digital Electronics
Yet another significant advantage of this number system was that all kinds of data could be
conveniently represented in terms of 0s and 1s. Also, basic electronic devices used for hardware
implementation could be conveniently and efficiently operated in two distinctly different modes. For
example, a bipolar transistor could be operated either in cut-off or in saturation very efficiently.
Lastly, the circuits required for performing arithmetic operations such as addition, subtraction,
multiplication, division, etc., become a simple affair when the data involved are represented in the
form of 0s and 1s.
1.5 Octal Number System
The octal number system has a radix of 8 and therefore has eight distinct digits. All higher-order
numbers are expressed as a combination of these on the same pattern as the one followed in the case
of the binary and decimal number systems described in Sections 1.3 and 1.4. The independent digits
are 0, 1, 2, 3, 4, 5, 6 and 7. The next 10 numbers that follow ‘7’, for example, would be 10, 11, 12,
13, 14, 15, 16, 17, 20 and 21. In fact, if we omit all the numbers containing the digits 8 or 9, or both,
from the decimal number system, we end up with an octal number system. The place values for the
different digits in the octal number system are 8
0 , 8
1 , 8
2 and so on (for the integer part) and 8
−1 , 8
−2 ,
8
−3 and so on (for the fractional part).
1.6 Hexadecimal Number System
The hexadecimal number system is a radix-16 number system and its 16 basic digits are 0, 1, 2, 3,
4, 5, 6, 7, 8, 9, A, B, C, D, E and F. The place values or weights of different digits in a mixed
hexadecimal number are 16
0 , 16
1 , 16
2 and so on (for the integer part) and 16
−1 , 16
−2 , 16
−3 and so on
(for the fractional part). The decimal equivalent of A, B, C, D, E and F are 10, 11, 12, 13, 14 and 15
respectively, for obvious reasons.
The hexadecimal number system provides a condensed way of representing large binary numbers
stored and processed inside the computer. One such example is in representing addresses of different
memory locations. Let us assume that a machine has 64K of memory. Such a memory has 64K (= 2
16
= 65 536) memory locations and needs 65 536 different addresses. These addresses can be designated
as 0 to 65 535 in the decimal number system and 00000000 00000000 to 11111111 11111111 in the
binary number system. The decimal number system is not used in computers and the binary notation
here appears too cumbersome and inconvenient to handle. In the hexadecimal number system, 65 536
different addresses can be expressed with four digits from 0000 to FFFF. Similarly, the contents of the
memory when represented in hexadecimal form are very convenient to handle.
1.7 Number Systems – Some Common Terms
In this section we will describe some commonly used terms with reference to different number systems.
1.7.1 Binary Number System
Bit is an abbreviation of the term ‘binary digit’ and is the smallest unit of information. It is either ‘0’
or ‘1’. A byte is a string of eight bits. The byte is the basic unit of data operated upon as a single unit
in computers. A computer word is again a string of bits whose size, called the ‘word length’ or ‘word
size’, is fixed for a specified computer, although it may vary from computer to computer. The word
length may equal one byte, two bytes, four bytes or be even larger.