Chapter 3 Number System and Codes Decimal and

Chapter 3 Number System and Codes

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 n Binary addition

Representing Integers with binary n Some of challenges: Integers can be positive or negative n Each integer should have a unique representation n The addition and subtraction should be efficient. n

Representing a positive numbers

Representing a negative numbers using Sign-Magnitude notation -5 = 1101 4 -bits sign-manitude -55= 10110111 8 -bits sign-magnitude

s Complement’ 1 n The 1’s complement representation of the positive number is the same as sign-magnitude. n +84 = 01010100

s Complement’ 1 n The 1’s complement representation of the negative number uses the following rule: n n Subtract the magnitude from 2 n-1 For example: n -36 = ? ? ? n +36 = 0010 0100

s Complement’ 1 n Example : n - 57 n n +57 = 0011 1001 -57 = 1100 0110

Converting to decimal format

s Complement’ 2 q For negative numbers: v Subtract the magnitude from 2 n. Or v Add 1 to the 1’s complement

Example

Convert to decimal value n Positive values: n n 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

Adding Positive and Negative Integers in 2's Complement Form

Subtraction of Positive and Negative Integers

Digital Codes n Binary Coded Decimal (BCD)

BCD

BCD

4221 Code

Gray Code n n 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

The Excess-3 - Code

Parity n The method of parity is widely used as a method of error detection. Extar bit known as parity is added to data word n The new data word is then transmitted. n n Two systems are used: Even parity: the number of 1’s must be even. n Odd parity: the number of 1’s must be odd. n

Parity n Example: 11001 11110 Even Parity 110011 11110000 Odd parity 110010 111101 110001