CSC 320 Chapter Conversion, Spring 2015

Number Systems and Codes Dr. Sanaa Sharafeddine CSC 320 Computer Organization Positional Number Systems � Number represented by a string of digits H Each digit has an associated weight H Number = weighted sum of digits H Each weight is a power (-ve or +ve) of the radix H Radix r (also called base) maybe any integer ≥ 2 � General form of a number is: H dp-1dp-2 … d1d0.d-1d-2 … d-n H dp-1 = Most significant digit H d-n = Least significant digit � The value of the number is: ∑ − =− = 1 p n i diri D Positional Number Systems � Radix 10: Decimal numbers H Used in everyday calculations � Radix 2: Binary numbers H Used to represent numbers in a digital system � Octal (Radix 8) and Hexadecimal (Radix 16) numbers H Used to represent multibit numbers H Octal: Digits from {0,1,2,3,4,5,6,7} H Hexadecimal: Digits from {0,…,9,A,B,C,D,E,F} Positional Number System Conversions � Binary to Octal: Bits are grouped in “Threes” from right to left, and the binary value of every group represents the corresponding Octal Digit � Example: Convert 1001111 from Binary to Octal? Example: 1 0 0 1 1 1 0 0 1 0 1 2 3 4 5 Result: 100111001012 = 23458 Positional Number System Conversions Example: 1 1 0 1 1 1 0 0 1 0 1 1 1 0 3 7 2 E Result: 110111001011102 = 372E16 � Binary to Hexadecimal: Bits are grouped in “Fours” from right to left, and the binary value of every group represents the corresponding Hexadecimal Digit � Example: Convert 1001111 from Binary to Hexadecimal? � Example: Convert 10.1011 to Octal and Hexadecimal? � Example: Convert 9F.4 from Hexadecimal to Binary? Positional Number System Conversions � Conversion to Decimal (radix-10) H To convert a number D to decimal, we convert each digit to radix-10 equivalent and expand the weighted- sum formula using radix-10 arithmetic H EX1: 1CE8.116 = 1x163 + 12x162 + 14x16 + 8x1 + 1x16-1 = 7400.062510 H EX2: F1AC16 = ? Positional Number System Conversions � Decimal to Binary: It is done by successive divisions by r until a zero quotient is obtained. The remainders taken in reverse order constitute the binary representation Example: Convert 46 from Decimal to Binary 46 / 2 = 23 and Remainder = 0 23 / 2 = 11 and Remainder = 1 11 / 2 = 5 and Remainder = 1 5 / 2 = 2 and Remainder = 1 2 / 2 = 1 and Remainder = 0 1 / 2 = 0 and Remainder = 1 Result: 4610 = 1011102 Positional Number System Conversions � Decimal fraction to Binary: It is done by successive multiplications taking out each time the integer part of the result Example: Convert 0.3125 to binary 0.3125 x 2 = 0.625 � 0 0.625 x 2 = 1.25 � 1 0.25 x 2 = 0.5 � 0 0.5 x 2 = 1 � 1 Result: 0.312510 = 0.01012 Addition and Subtraction: Binary Numbers 190 10111110 +141 10001101 331 101001011 229 11100101 - 46 00101110 183 10110111 � Subtraction can be used to compare numbers Quiz Question � Determine radix x: (12)x = 1810 Codes � Binary numbers are used in internal computations of digital systems, but H Some digital devices process/display decimal numbers H Some digital devices process events, actions, conditions, etc. � Solution: Use codes (mappings) H Map decimal numbers to bits (binary sequences) H Map events, actions, and conditions to bits H Codes are composed of codewords � E.g., how many bits are required to represent the ten decimal digits? Binary Codes for Decimal Numbers � There are many possibilities to build binary codes for decimal digits. How many? Table 2-9 Decimal codes. Decimal digit BCD (8421) 2421 Excess-3 Biquinary 1-out-of-10 0 0000 0000 0011 0100001 1000000000 1 0001 0001 0100 0100010 0100000000 2 0010 0010 0101 0100100 0010000000 3 0011 0011 0110 0101000 0001000000 4 0100 0100 0111 0110000 0000100000 5 0101 1011 1000 1000001 0000010000� 6 0110 1100 1001 1000010 0000001000 7 0111 1101 1010 1000100 0000000100 8 1000 1110 1011 1001000 0000000010 9 1001 1111 1100 1010000 0000000001 Unused code words 1010 0101 0000 0000000 0000000000 1011 0110 0001 0000001 0000000011 1100 0111 0010 0000010 0000000101 1101 1000 1101 0000011 0000000110 1110 1001 1110 0000101 0000000111 1111 1010 1111 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ �