CSC 1103: Digital Logic. Lecture Six: Data Representation

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1 CSC 1103: Digital Logic Lecture Six: Data Representation Martin Ngobye Mbarara University of Science and Technology MAN (MUST) CSC / 32

2 Outline 1 Digital Computers 2 Number Systems Conversion Between Binary and Decimal Octal and Hexadecimal Numbers 3 Summary MAN (MUST) CSC / 32

3 Digital Computers Humans have many symbolic forms to represent information. Alphabet, numbers, pictograms Several formats are used to store data. Deep down inside, computers work with just 1 s and 0 s. Because a computer uses binary numbers all these formats are patterns of 1 s and 0 s. MAN (MUST) CSC / 32

4 Data Types Binary information in digital computers is stored in memory or processor registers. Registers contain either data or control information. Control information is a bit or a group of bits used to specify the sequence of command signals needed for manipulation of the data in other registers. Data are numbers and other binary-coded information that are operated on to achieve required computational results. Present common types of data found in digital computers Show how the various data types are presented in binary-coded form in computer registers. MAN (MUST) CSC / 32

5 Data Types Data types found in registers of digital computers may be classified as: 1 Numbers used in arithmetic computations 2 Letters of the alphabet used in data processing 3 Other discrete symbols used for specific purposes. All types of data except binary numbers are presented in computer registers in the binary-coded form. MAN (MUST) CSC / 32

6 Data Representation A bit is the most basic unit of information in a computer. A byte is a group of eight bits. It is the smallest possible addressable unit of computer storage. A word is a contiguous group of bytes. Words can be any number of bits or bytes. Word sizes of 16, 32, or 64 bits are most common. MAN (MUST) CSC / 32

7 Number Systems A number system of base or radix r is a system that uses distinct symbols for digit representation. Numbers are represented by a string of digit symbols. To determine the quantity that a number represents, it is necessary to multiply each digit by an integer power of r and then form the sum of all weighted digits. For example, the decimal number system in everyday use employs the radix 10 system. The 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. MAN (MUST) CSC / 32

8 Number Systems The strings of digits is interpreted to represent the quantity: 7x x x x10 1 That is 7 hundreds, plus 2 tens, plus 4 units, plus 5 tenths. MAN (MUST) CSC / 32

9 Number Systems The binary number system used the radix 2. The two digit symbols used are 0 and 1. The string of digits is interpreted to represent the quantity 1x x x x x x2 0 = 45 MAN (MUST) CSC / 32

10 Number Systems Besides the decimal and binary number systems, the octal (radix 8) and hexadecimal (radix 16) are important in digital computer work. The eight symbols of the octal system are 0, 1, 2, 3, 4, 5, 6, and 7. The 16 symbols of the hexadecimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. The symbols A, B, C, D, E, F correspond to the decimal numbers 10, 11, 12, 13, 14, 15 respectively. MAN (MUST) CSC / 32

11 Number Systems Conversion A number in the radix r can be converted to a decimal system by forming the sum of weighted digits. For example, octal is converted to decimal as follows: (736.4) 8 = 7x x x x8 1 = 7x64 + 3x8 + 6x = (478.5) 10 The equivalent decimal number of hexadecimal F 3 is obtained from the following calculation: (F 3) 16 = Fx x16 0 = 15x = (243) 10 MAN (MUST) CSC / 32

12 Conversion Between Binary and Decimal There are two methods for base conversion: the subtraction method and the division remainder method. The subtraction method The division remainder method MAN (MUST) CSC / 32

13 Conversion Decimal to Binary We can convert from base 10 to base 2 by repeated divisions by 2. The remainders and the final quotient, 1, give us, the order of increasing significance and the binary digits of a number N. To convert a decimal integer into binary, keep dividing by 2 until the quotient is 0. Collect the remainders in reverse order. Example One: Convert to base 2. 11/2 = 5 rem 1 5/2 = 2 rem 1 2/2 = 1 rem 0 1/2 = 0 rem 1 Therefore, = MAN (MUST) CSC / 32

14 Conversion Decimal to Binary Example Two: Convert to base 2. 21/2 = 10 rem 1 10/2 = 5 rem 0 5/2 = 2 rem 1 2/2 = 1 rem 0 1/2 = 0 rem 1 Therefore, = MAN (MUST) CSC / 32

15 Conversion Decimal to Binary To convert a decimal number, for example into binary is done by first separating the number into its integer part 162 and the fraction part.375. The integer part is converted by dividing 162 by 2 to get reminders. The fraction part is converted by multiplying it by 2 until it becomes 0. Collect the integer parts in forward order to get the decimal part of the number. MAN (MUST) CSC / 32

16 Conversion Decimal to Binary Example One: Convert into binary. 162/2 = 81 rem 0 81/2 = 40 rem 1 40/2 = 20 rem 0 20/2 = 10 rem 0 10/2 = 5 rem 0 5/2 = 2 rem 1 2/2 = 1 rem 0 1/2 = 0 rem 1 MAN (MUST) CSC / 32

17 Conversion Decimal to Binary Example One: Convert into binary. The decimal part is converted as: 0.375x2 = x2 = x2 = Therefore, = MAN (MUST) CSC / 32

18 Octal and Hexadecimal Number The conversion from and to binary, octal and hexadecimal representation plays an important part in digital computers. Since 2 3 = 8 and 2 4 = 16, each octal digit corresponds to three binary digits and each hexadecimal digit corresponds to four binary digits. The conversion from binary to octal is easily accomplished by partitioning the binary number into groups of three bits each. The corresponding octal digit is then assigned to each group of bits and the strings of digits obtained gives the octal equivalent of the binary number. MAN (MUST) CSC / 32

19 Octal and Hexadecimal Numbers Consider a 16-bit register shown as: Starting from the low-order bit, we partition the register into groups of three bits. Each group of three bits is assigned its octal equivalent and placed on top of the register. The string of octal digits obtained represents the octal equivalent of the binary number. The octal representation of the number is shown as: 1 {}}{ 1 2 {}}{ {}}{ {}}{ {}}{ {}}{ MAN (MUST) CSC / 32

20 Octal and Hexadecimal Numbers Conversion from binary to hexadecimal is similar except that the bits are divided into groups of four. The corresponding hexadecimal digit for each group of four bits is written as shown: A {}}{ F {}}{ {}}{ {}}{ MAN (MUST) CSC / 32

21 Octal and Hexadecimal Numbers The corresponding octal digit for each group of three bits is easily remembered after studying the first eight entries in the table. Octal Binary-coded Decimal number octal equivalent MAN (MUST) CSC / 32

22 Octal and Hexadecimal Numbers The correspondence between a hexadecimal digit and its equivalent 4-bit code can be found in the table. Hexadecimal Binary-coded Decimal number hexadecimal equivalent A B C D E F MAN (MUST) CSC / 32

23 Octal and Hexadecimal Numbers The registers in a digital computer contain many bits. Specifying the content of registers by their binary values will require a long string of binary digits. It is more convenient to specify content of registers by their octal or hexadecimal equivalent. The number of digits is reduced by one-third in the octal designation and by one-fourth in the hexadecimal equivalent. For example, the binary number has 12 digits. It can be expressed in octals of 7777 (four digits) or in hexadecimal as FFF (three digits). MAN (MUST) CSC / 32

24 Decimal Representation The binary number system is the most natural system for a computer. However, people are accustomed to the decimal system. One way of solving this conflict is to convert all input decimal numbers into binary numbers and let the computer perform all arithmetic operations in binary and then convert the binary results back to decimal for the human to understand. It is also possible for the computer to perform arithmetic operations directly with decimal numbers provided they are placed in their registers in a coded form. The bit assignment most commonly used for the decimal digits is the referred to as Binary-coded Decimal (BCD). MAN (MUST) CSC / 32

25 Binary-coded Decimal Decimal Binary-coded decimal number (BCD) number MAN (MUST) CSC / 32

26 Binary-coded Decimal The advantage of BCD format is that it is closer to the alphanumeric codes used for I/Os. Numbers in text data formats must be converted from text form to binary form. This conversion is usually done by converting text input data to BCD, converting BCD to binary, do calculations then convert the result to BCD, then BCD to text output. MAN (MUST) CSC / 32

27 Binary-coded Decimal For example, the number 99 when converted to binary is represented by a string of bits However, when 99 is represented as a binary-coded decimal, it is shown as: The difference between a decimal number represented by familiar digit symbols 0, 1, 2,..., 9 and the BCD symbols 0001, 0010,..., 1001 is in the symbols used to represent the digits. The number itself is exactly the same. MAN (MUST) CSC / 32

28 Alphanumeric Representation Many applications of digital computers require the handling of data that consists not only of numbers, but also of letters of the alphabet and certain special symbols. An alphanumeric character set is a set of elements that includes the 10 decimal digits, the 26 letters of the alphabet and a number of special characters, such as +, and =. The two most prominent alphanumeric codes are: - EBCDIC (Extended Binary Coded Decimal Interchange Code); this is mostly used by IBM. - ASCII: (American Standard Code for Information Interchange); used by other manufacturers MAN (MUST) CSC / 32

29 Alphanumeric Representation ASCII represents each character with a 7 bit string. The total number of characters that can be represented is 2 7 = 128. For example, J O H N = Since most computers manipulate an 8 bit quantity, the extra bit when 7 bit ASCII is used depends on the designer. It can be set to a particular value or ignored. Binary codes can be formulated for any set of discrete elements such as the musical notes and chess pieces and their positions on a chess board. Binary codes are also used to formulate instructions that specify control information for a computer. MAN (MUST) CSC / 32

30 Exercise One Convert the following decimal numbers into binary Convert the following binary numbers into decimal MAN (MUST) CSC / 32

31 Exercise Two Convert the following hexadecimal numbers into decimal. 1 F 4 2 6E 3 B7 4 ABC 5 6E 6 FEC Convert the following octal numbers into decimal MAN (MUST) CSC / 32

32 Summary 1 Data Representation 2 Bases: Binary, Decimal, Octal and Hexadecimal 3 Conversion between bases - Conversion from Binary to Decimal - Conversion from Decimal to Decimal - Conversion from octal to Decimal - Conversion from hexadecimal and Decimal MAN (MUST) CSC / 32

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