The representation of data within the computer

Transcription

1 The representation of data within the computer Digital and Analog System of Coding Digital codes. Codes that represent data or physical quantities in discrete values (numbers) Analog codes. Codes that represent data in a continuous form. They are more difficult to interpret than digital codes but they are more gradual. Example of digital device Vs analog device: Traditional watch with an hour hand and a minute hand revolving Vs digital watch. Number systems Under different number systems, quantities may be represented in different ways. Different number systems have different numbers of digit symbols. Each digit in a number has its values. Decimal (Denary) It is the most familiar one. It originated from counting numbers by fingers. 10 digit symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and each digit has a place value in a power of 10. e.g : digit 8 has a place value 10 0 digit 2 has a place value 10 1 digit 4 has a place value 10 2 Binary The number of meaningful electronic states is only two An electronic pulse is on or off. The most natural coding for computer is binary. The binary number system has only 2 digit symbols, i.e. 0 and 1, with place values in power of 2. Conversions Octal From binary to decimal, = 1* * * * * * 2 0 = From decimal to binary, Therefore, the result is Programming in binary is however very tedious and prone to error. The octal system is used as it makes programming simpler (closer to 10) and not very difficult for the computer (easy conversion between binary and octal). DATA REPRESENTATION Page 1

2 This system has 8 digit symbols (0 to 7) and each digit has a place value in powers of 8. Knowing the place values, we can convert octal to decimal and vice versa = 2* * *8 0 = CH/NSSICT/Feb., i.e Conversions between octal and binary numbers are much simpler, just grouping the binary codes 3 in a group and then change each group into an octal digit. e.g = Hexadecimal The octal number system has a disadvantage in coding computer instructions. Computers work in twos, or power of 2, thus it is not efficient for them to deal with numbers in 3-bit groups. A new number system, the hexadecimal system, is developed with 16 digit symbols (0-9, A-F) and each hexadecimal digit is equivalent to a 4-digit binary number. Table represent numbers of different number system Decimal Binary Hexadecimal Decimal Binary Hexadecimal A B C D E F Conversions between hexadecimal and binary, and hexadecimal and decimal are similar to those of octal. Conversions between octal and hexadecimal are done by first converting the number to binary. Simple Arithmetic on Binary Number Addition Subtraction Multiplication Division * ) DATA REPRESENTATION Page 2

3 Number Representation inside the computer Sign-and-magnitude representation If we use 8 bits (one bit means one 1 or one 0 ) to store an integer, the 1st bit is reserved as a sign bit ( 1 for -ve, 0 for +ve) and the rest are used to represent the magnitude. and = ( ) sign = = -( ) sign = 's Complement representation Under 1's complement representation, all the negative numbers are represented by replacing all the digits to their complements. (i.e. 0 to 1, 1 to 0) The first digit can still be regarded as sign bit. 2 s Complement representation Two steps to represent a negative number: Step 1 : change from its positive value to its 1 s complement. Step 2 : add 1 to the last digit and take care of the carry. e.g = (binary value) = (1 s complement) = (2 s complement) is represented as in 2 s complement. Advantages i. complement operation is easy ii. no need to place sign bit explicitly iii. easy to perform addition and subtraction iv. no redundancy representation to 0 (in 8-bit sign-and-magnitude representation, both and represent zero) Subtraction between 2 s complement integer. for A - B, (same as A + (-B)) step 1 : change B to its 2 s complement step 2 : adding the result to A step 3 : ignore the extra digit in front of the resulting number (if necessary) e.g (i.e ) step 1 : 2 s complement of = step 2 : = step 3 : ignore the extra bit, the result is (i.e ) DATA REPRESENTATION Page 3

4 Exercise Convert the following into 8-bit with 2 s complement representation as negative number. i = (binary value) ii = (binary value) = (1 s comp.) = (1 s comp.) = (2 s comp.) = (2 s comp.) iii = (binary value) iv = (binary value) = (1 s comp.) = (2 s comp.) Using the representation in part i to calculate the following binary arithmetic expressions i = = (binary value) = (1 s comp.) = (2 s comp.) = ( ) 2 + ( ) 2 = (discarding leading zero) = ii = = (binary value) = (1 s comp.) = (2 s comp.) = = (2 s comp.) = (1 s comp.) = = Convert the following bit pattern using the representation in part i into decimal integers: i = (1 s comp.) ii = (1 s comp.) = (binary value) = (binary value) = = -99 iii = iv = (1 s comp.) = (binary value) What is the maximum value and minimum value of such an integer? = Maximum representation : = = Minimum representation : = (1 s comp.) = = -2 7 = DATA REPRESENTATION Page 4

5 Data representation of binary fraction Interconversion between binary and decimal fraction Sample conversion from binary to decimal fraction: = = Sample conversion from decimal fraction, e.g , using repeated multiplication: = and = = Range and Accuracy Range The range is the set of all numbers that can be represented by a particular system. e.g. for the 8-bit integer, the range using 2 s complement is -128 to 127. Accuracy. This is a measure of the closeness of an approximation to the exact or true value. Precision. This is the term associated with the word length, i.e. available to represent a given number. e.g. Suppose that we have to i. represent the number 7 in pure binary using four digits, i.e. 0111, ii. represent in 8 bits fixed point, i.e Then the first one is more accurate and the second representation is more precise. Resolution. This is simply the magnitude of the difference between the last tow two adjacent digits or numbers. DATA REPRESENTATION Page 5

6 Limitation of number system Overflow and Underflow error Every number representation has its own range so that every number in the system fall inside the range. A value of the result from an arithmetic operation which is larger than the upper limit of the range of the system will cause an overflow error. On the other hand, if the value is too small to be represented by the number of bit, it will cause an underflow error. Rounding errors After certain calculation (e.g. multiplication and division), the number of significant figures of the resulting value may be too large to be hold by the number of bits. Round becomes necessary. Conversion errors Error may be evolved from conversions. REVISION e.g = i. Convert the following into 11-bit with 2 s complement representation as negative number. a b c d ii. Using the representation in part i to calculate the following binary arithmetic expressions a b = = = = = = iii. Convert the following bit pattern using the representation in part i into decimal integers: a b c d DATA REPRESENTATION Page 6

7 Character Set Characters also stored as binary code in computer. Usually, one character is stored in 1 byte, so there can be 256 different characters. The most common character set used in computer is the American Standard Coding for Information Interchange (ASCII). Decimal Binary Hexa. Code Character Decimal Binary Hex Character Code Code Code Code a. Code (space) P ! Q " R # S $ T % U & V ' W ( X ) Y A * A Z B B [ C C \ D D ] E E ^ F / F _ ` a b c d e f g h i A : A j B ; B k C < C l D = D m E > E n F? F o p A q B r C s D t E u F v G w H x I y A J A z B K B { C L C D M D } E N E ~ F O F (del) DATA REPRESENTATION Page 7

8 Chinese Text Processing Under a Chinese text system, we can operate the computer with Chinese characters. That means we can use the computer to i. store the character which represented by certain codes (internal code). ii. use the English keyboard to input Chinese characters (input method). iii. display the Chinese characters in different shapes (graphical composition of Chinese characters of different fonts). All these features are provided by the Chinese operating system. (Windows XP) Character set In an English processing system, the internal code of a character may be an ASCII code. E.g. A s internal code is In a Chinese processing system, the internal code may be the Chinese National Standard Codes for Information Interchange, Big 5 or other codes. (The internal code used in Chinese Windows is Big 5.) In an English processing system, the character set includes the alphabets ( a to z, and A to Z ), numerals ( 0 to 9 ) and symbols, such as $, &,?, #, *, etc, having over 200 characters in total. In a Chinese processing system, the character set includes all characters in an English processing system and all the commonly used Chinese characters, over in total. The character set is much larger and thus two bytes are used to store a single Chinese character. (Double Byte Character Set, DBCS). Big-5. i. For Traditional Chinese characters ii. iii. Regions: Taiwan, Hong Kong and Malaysia HK government has developed an extension to the set. (Hong Kong Supplementation Character Set). GuoBiao (GB). i. For Simplified Chinese characters ii. Regions: China, Singapore and Malaysia. Unicode consists of CJK (Chinese, Japanese and Korean) characters. Thus Simplified Chinese characters, Traditional Chinese characters and Japanese characters can be identified in the same document. Unicode. It contains the characters from the world s alphabets, ideographs and symbols. Its first 256 codes are the same as that of ASCII. The latest version, Unicode 6.2, contains graphic and format characters. One character may be stored in 1 byte, 2 bytes or 4 bytes. ۍ DATA REPRESENTATION Page 8