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Chapter 3 Number System and Codes Decimal and Binary Numbers Decimal and Binary Numbers Converting Decimal to Binary 1. Sum of powers of 2 Converting Decimal to Binary 1. Repeated Division Binary Numbers and Computers Hexadecimal Numbers Converting decimal to hexadecimal Converting binary to hexadecimal Converting hexadecimal to binary? Hexadecimal numbers Binary arithmetic Binary addition Representing Integers with binary Some of challenges:Integers can be positive or negative Each integer should have a unique representation The addition and subtraction should be efficient. Representing a positive numbers Representing a negative numbers using Sign-Magnitude notation -5 = 1101 4-bits sign-manitude -55= 10110111 8-bits sign-magnitude 1’s Complement The 1’s complement representation of the positive number is the same as sign-magnitude. +84 = 01010100 1’s Complement The 1’s complement representation of the negative number uses the following rule: Subtract the magnitude from 2n-1 For example: -36 = ??? +36 = 0010 0100 1’s Complement Example : - 57 +57 = 0011 1001 -57 = 1100 0110 Converting to decimal format 2’s Complement For negative numbers: Subtract the magnitude from 2n. Or Add 1 to the 1’s complement Example Convert to decimal value Positive values: 0101 1001 = +89 Negative values Two's Complement Arithmetic Adding Positive Integers in 2's Complement Form Overflow in Binary Addition Overflow in Binary Addition Overflow in Binary Addition Overflow in Binary Addition Adding Positive and Negative Integers in 2's Complement Form Adding Positive and Negative Integers in 2's Complement Form Subtraction of Positive and Negative Integers Digital Codes Binary Coded Decimal (BCD) BCD BCD 4221 Code Gray Code In pure binary coding or 8421 BCD then counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously. Gray coding avoids this since only one bit changes between subsequent numbers Binary –to-Gray Code Conversion Gray –to-Binary Conversion Gray –to-Binary Conversion The Excess-3- Code Parity The method of parity is widely used as a method of error detection. Extar bit known as parity is added to data word The new data word is then transmitted. Two systems are used: Even parity: the number of 1’s must be even. Odd parity: the number of 1’s must be odd. Parity Example: 11001 11110 Even Parity 110011 111100 11000 110000 Odd parity 110010 111101 110001