ࡱ> z|yG bjbjَ 1]****8bT*,~h"$^ 3   ^**  inj\`0-F**y NUMBER SYSTEMS AND CODES Topics to be covered: Number systems Number notations Arithmetic Base conversions Signed number representation Codes Decimal codes Error detection code Gray code ASCII code Number Systems The decimal (real), binary, octal hexadecimal number systems are used to represent information in digital systems. Any number system consists of a set of digits and a set of operators (+, (, (, (). The radix or base of the number system denotes the number of digits used in the system. Decimal (base 10)0 1 2 3 4 5 6 7 8 9Binary (base 2)0 1 Octal (base 8)0 1 2 3 4 5 6 7Hexadecimal (base 16)0 1 2 3 4 5 6 7 8 9 A B C D E F DecimalBinaryOctalHexadecimal00000000001000101102001002203001103304010004405010105506011006607011107708100010809100111910101012A11101113B12110014C13110115D14111016E15111117F Number notations A number can be represented in either positional notation or polynomial notation. Positional notation It is convenient to represent a number using positional notation. A positional notation is written as a sequence of digits with a radix point separating the integer and fractional part.  EMBED Equation.3  where r is the radix, n is the number of digits of the integer part, and m is the number digits of the fractional part. Polynomial notation A number can be explicitly represented in polynomial notation.  EMBED Equation.3  where rp is a weighted position and p is the position of a digit. In decimal number system  EMBED Equation.3  In binary number system  EMBED Equation.3  In octal number system  EMBED Equation.3  In hexadecimal number system  EMBED Equation.3  Arithmetic Addition In binary number system, (101101)2 +(11101)2 :1111 1+ 101101 111011001010 In octal system, (6254)8 +(5173)8 :1 1+ 6254 517313447 In hexadecimal system, (9F1B)16 +(4A36)16 : 1 1+ 9F1B 4A36 D951 Subtraction In binary number system, (101101)2 -(11011)2 : 10 10- 101101 11011 10010 In octal system, (6254)8 -(5173)8 : 8- 6254 5173 1061 In hexadecimal system, (9F1B)16 -(4A36)16 : 16- 9F1B 4A36 54E5 Multiplication In binary number system, (1101)2 ( (1001)2 : ( 1101 1001 1101 0000 000011011110101 Division In binary number system, (1110111)2 ((1001)2 : 1101100111101111001 1011 1001 1011 1001 10 Base conversions Convert (100111010)2 to base 8  EMBED Equation.3  or  EMBED Equation.3  Convert (100111010)2 to base 10  EMBED Equation.3  Convert (100111010)2 to base 16  EMBED Equation.3  or  EMBED Equation.3  Convert (372)8 to base 2  EMBED Equation.3  Convert (372)8 to base 10  EMBED Equation.3  Convert (372)8 to base 16  EMBED Equation.3  Convert (9F2)16 to base 2  EMBED Equation.3  Convert (9F2)16 to base 8  EMBED Equation.3  Convert (9F2)16 to base 10  EMBED Equation.3  Binomial expansion (series substitution) To convert a number in base r to base p. Represent the number in base p in binomial series. Change the radix or base of each term to base p. Simplify. Convert base 10 to base r Convert (174)10 to base 8 81746LSB8215822MSB0 Therefore (174)10 = (256)8 Convert (0.275)10 to base 8 8(0.275(2.200MSD8(0.200(1.6008(0.600(4.8008(0.800(6.4008(0.400(3.200LSD Therefore (0.275)10 = (0.21463()8 Convert (0.68475)10 to base 2 2(0.68475(1. 3695MSD2(0.3695(0.73902(0.7390(1.47802(0.4780(0.95602(0.9560(1.9120LSD Therefore (0.68475)10 = (0.10101()2 Signed Number Representation There are 3 systems to represent signed numbers: Signed-magnitude 1's complement 2's complement In binary number system Signed-magnitude system In signed-magnitude systems, the most significant bit represents the number's sign, while the remaining bits represent its absolute value as an unsigned binary magnitude. If the sign bit is a 0, the number is positive. If the sign bit is a 1, the number is negative.  1's Complement system A 1's complement system represents the positive numbers the same way as in the signed-magnitude system. The only difference is negative number representations. Let be N any positive integer number and  EMBED Equation.3  be a negative 1's complement integer of N. If the number legnth is n bits, then  EMBED Equation.3  For example in a 4-bit system, 0101 represents +5 and  EMBED Equation.3  1010 represents (5 2's Complement System A 2's complement system is similar to 1's complement system, except that there is only one representation for zero. Let be N any positive integer number and  EMBED Equation.3  be a negative 2's complement integer of N. If the length of the number is n bits, then  EMBED Equation.3  For example in a 4-bit system, 0101 represents +5 and  EMBED Equation.3  1011 represents (5  Adding and subtracting signed numbers Signed-magnitude system (a)5 +2 0101 +00107 0111 (b)-5 -2 1101 +1010-7 1111 (c)5 -2 0101 +10103 0011 (d)-5 +2 1101 +0010-3 1011 1's complement system (a)5 +2 0101 +00107 0111 (b)-5 -2 1010 +1101-7 1 0111 1 1000 (c)5 -2 0101 +11013 1 0010 1 0011 (d)-5 +2 1010 +0010-3 1100 2's complement system (a)5 +2 0101 +00107 0111 (b)-5 -2 1011 +1110-7 1 1001 (c)5 -2 0101 +11103 1 0011 (d)-5 +2 1011 +0010-3 1101 Overflow conditions Carry-in ( carry-out 5 +3 0111 0101 +0011 -8 1000 -5 -4 1000 1011 +1100 7 1 0111 Carry-in = carry-out 5 +2 0000 0101 +0010 7 0111 -6 -2 1110 1010 +1110 -8 1 1000 Codes Decimal codes Decimal DigitBCD 8421Excess-324210000000110000100010100000120010010100103001101100011401000111010050101100010116011010011100701111010110181000101111109100111001111 Error detection code Parity bit Odd ParityEven ParityPMessagePMessage1000000000000011000100010100101001100011001001010010101001011011000110001111011101000110001100101001110100101001011110111110001100011011110101110111101111101111 Gray code Decimal EquivalentBinary CodeGray Code000000000100010001200100011300110010401000110501010111601100101701110100810001100910011101101010111111101111101211001010131101101114111010011511111000 ASCII code PAGE  PAGE 8 Number Systems and Codes 0 !"278GWXn;ES], - \ o jCJEHUj@: UVmHjCJEHUj@: UVmH jCJU CJOJQJ6CJ jCJ jCJ j-CJCJ5CJD01@Q\n !$$$h$ & F$ & F $ & F $$01@Q\n !"278GWXnĻ                           C!"278GWXnX$$$$$$Tl0 X$$$Tl0 X$$$$$Tl0 X888:$$Tlp\T $$:$$Tlp\T  88888$$:$$Tlp\T  !"%*-/038;=>AFIKLOTWYZ]beghkpsuvy~[ \ p % & ? @ X Y q r  % , -     `!"%*-/038;=>AFIKLOTW8888:$$Tlp\T $$WYZ]beghkpsuvy~8888:$$Tlp\T $$[ \ p $$ $ & F h$:$$Tlp\T  & ' : ; < = @ Y Z m n o p r   ! 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