Data Representation
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This document discusses number systems and data representation in computers. It covers topics like binary, decimal, hexadecimal, and ASCII number systems. Some key points covered include: - Computers use the binary number system and positional notation to represent data precisely. - Different number systems have different bases (like binary base-2, decimal base-10, hexadecimal base-16). - Methods for converting between number systems like binary to decimal and hexadecimal to decimal. - Signed and unsigned integers, ones' complement, twos' complement representation of negative numbers. - ASCII encoding of characters and how to convert between character and numeric representations.
![Character Representation
All data, characters must be coded in binary to be
processed by the computer.
ASCII:
American Standard Code for Information Interchange
Most popular character encoding scheme.
Uses 7 bit to code each character.
27 = 128 ASCII codes.
Single character Code = One Byte [7 bits: char code, 8th
bit set to zero]
32 to 126 ASCII codes: printable
0 to 31 and 127 ASCII codes: Control characters
27](https://image.slidesharecdn.com/3-datarepresentation-140905055307-phpapp01/85/Data-Representation-27-320.jpg)
Data Representation
- 1. CSC 222: Computer Organization & Assembly Language 3 – Data Representation Instructor: Ms. Nausheen Majeed
- 2. Number System Any number system using a range of digits that represents a specific number. The most common numbering systems are decimal, binary, octal, and hexadecimal. Numbers are important to computers represent information precisely can be processed For example: to represent yes or no: use 0 for no and 1 for yes to represent 4 seasons: 0 (autumn), 1 (winter), 2(spring) and 3 (summer) 2
- 3. Positional Number System A computer can understand positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. A value of each digit in a number can be determined using The digit The position of the digit in the number The base of the number system (where base is defined as the total number of digits available in the number system). 3
- 5. Binary Number System 5
- 7. Conversion Between Number Systems Converting Hexadecimal to Decimal Multiply each digit of the hexadecimal number from right to left with its corresponding power of 16 or weighted value. Convert the Hexadecimal number 82ADh to decimal number. 7
- 8. Conversion Between Number Systems Converting Binary to Decimal Multiply each digit of the binary number from right to left with its corresponding power of 2 or weighted value. Convert the Binary number 11101 to decimal number. 8
- 9. Conversion Between Number Systems Converting Decimal to Binary Divide the decimal number by 2. Take the remainder and record it on the side. REPEAT UNTIL the decimal number cannot be divided into anymore. 9
- 10. Conversion Between Number Systems Converting Decimal to Hexadecimal Divide the decimal number by 16. Take the remainder and record it on the side. REPEAT UNTIL the decimal number cannot be divided into anymore. 10
- 11. Conversion Between Number Systems Converting Hexadecimal to Binary Given a hexadecimal number, simply convert each digit to it’s binary equivalent. Then, combine each 4 bit binary number and that is the resulting answer. Converting Binary to Hexadecimal Begin at the rightmost 4 bits. If there are not 4 bits, pad 0s to the left until you hit 4. Repeat the steps until all groups have been converted. 11
- 12. Binary Arithmetic Operations Addition Like decimal numbers, two numbers can be added by adding each pair of digits together with carry propagation. 12 11001 + 10011 101100 647 + 537 1184 Binary Addition Decimal Addition
- 13. Binary Arithmetic Operations Subtraction Two numbers can be subtracted by subtracting each pair of digits together with borrowing, where needed. 13 11001 - 10011 00110 627 - 537 090 Binary Subtraction Decimal Subtraction
- 14. Hexadecimal Arithmetic Operations Addition Like decimal numbers, two numbers can be added by adding each pair of digits together with carry propagation. 14 5B39 + 7AF4 D62D Hexadecimal Addition
- 15. HexaDecimal Arithmetic Operations Subtraction Two numbers can be subtracted by subtracting each pair of digits together with borrowing, where needed. 15 D26F - BA94 17DB Hexadecimal Subtraction
- 16. MSB and LSB In computing, the most significant bit (msb) is the bit position in a binary number having the greatest value. The msb is sometimes referred to as the left-most bit. In computing, the least significant bit (lsb) is the bit position in a binary integer giving the units value, that is, determining whether the number is even or odd. The lsb is sometimes referred to as the right-most bit. 16
- 17. Unsigned Integers An unsigned integer is an integer at represent a magnitude, so it is never negative. Unsigned integers are appropriate for representing quantities that can be never negative. 17
- 18. Signed Integers A signed integer can be positive or negative. The most significant bit is reserved for the sign: 1 means negative and 0 means positive. Example: 00001010 = decimal 10 10001010 = decimal -10 18
- 19. One’s Complement The one’s complement of an integer is obtained by complementing each bit, that is, replace each 0 by a 1 and each 1 by a 0. 19
- 20. 2’s Complement Negative integers are stored in computer using 2’s complement. To get a two’s complement by first finding the one’s complement, and then by adding 1 to it. Example 11110011 (one's complement of 12) + 00000001 (decimal 1) 11110100 (two's complement of 12) 20
- 21. Subtract as 2’s Complement Addition Find the difference of 12 – 5 using complementation and addition. 00000101 (decimal 5) 11111011 (2’s Complement of 5) 00001100 (decimal 12) + 11111011 (decimal -5) 00000111 (decimal 7) No Carry 21
- 22. Example Find the difference of 5ABCh – 21FCh using complementation and addition. 5ABCh = 0101 1010 1011 1100 21FCh = 0010 0001 1111 1100 1101 1110 0000 0100 (2’s Complement of 21FCh) 0101 1010 1011 1100 (Binary 5ABCh) + 1101 1110 0000 0100 (1’s Complement of 21FCh) 10011 1000 1100 0000 Discard Carry 22
- 23. Decimal Interpretation How to interpret the contents of a byte or word as a signed and unsigned decimal integer? Unsigned decimal interpretation Simply just do a binary to decimal conversion or first convert binary to hexadecimal and then convert hexadecimal to decimal. Signed decimal interpretation If msb is zero then number is positive and signed decimal is same as unsigned decimal. If msb is one then number is negative, so call it -N. To find N, just take the 2’s complement and then convert to decimal. 23
- 24. Example Give unsigned and signed decimal interpretation FE0Ch. Unsigned decimal interpretation 163 ∗ 15 + 162 ∗ 14 + 161 ∗ 0 + 160 ∗ 12 = 61440 + 3584 + 0 + 12 = 65036 Signed decimal interpretation FE0Ch = 1111 1110 0000 1100 (msb is 1, so number is negative). To find N, get its 2’s complement 0000 0001 1111 0011 (1’s complement of FE0Ch) + 1 N = 0000 0001 1111 0100 = 01F4h = 500 24So, -N = 500
- 25. Decimal Interpretation For 16 – bit word, following relationships holds between signed and unsigned decimal interpretation From 0000h – 7FFFh, signed decimal = unsigned decimal From 8000h – FFFFh, signed decimal = unsigned decimal – 65536. Example: Unsigned interpretation of FE0Ch is 65036. Signed interpretation of FE0Ch = 65036 – 65536 = - 500. 25
- 26. Binary, Decimal, and Hexadecimal Equivalents. Binary Decimal Hexadecimal Binary Decimal Hexadecimal 0000 0 0 1000 8 8 0001 1 1 1001 9 9 0010 2 2 1010 10 A 0011 3 3 1011 11 B 0100 4 4 1100 12 C 0101 5 5 1101 13 D 0110 6 6 1110 14 E 0111 7 7 1111 15 F
- 27. Character Representation All data, characters must be coded in binary to be processed by the computer. ASCII: American Standard Code for Information Interchange Most popular character encoding scheme. Uses 7 bit to code each character. 27 = 128 ASCII codes. Single character Code = One Byte [7 bits: char code, 8th bit set to zero] 32 to 126 ASCII codes: printable 0 to 31 and 127 ASCII codes: Control characters 27
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- 29. How to Convert? If a byte contains the ASCII code of an uppercase letter, what hex should be added to it to convert to lower case? Solution: 20 h Example: A (41h) a (61 h) If a byte contains the ASCII code of a decimal digit, What hex should be subtracted from the byte to convert it to the numerical form of the characters? Solution: 30 h Example: 2 (32 h) 29
- 30. Character Storage ASCII Representation of “123” and 123 30 ' 1 ' ' 2 ' ' 3 ' 00110001 00110010 00110011 123 01111011 = = "1 2 3" 1 2 3